Gérard Favier
email: favier@i3s.unice.fr

An overview of system modeling and identification

Introduction

System identification consists in building mathematical models of dynamical systems from experimental data. Such a methodology was mainly developed for designing modelbased control systems. More generally, parameter estimation is at the heart of many signal processing applications aiming to extract information from signals, like radar, sonar, seismic, speech, communication, or biomedical (EEG, ECG, EMG) signals. Nowadays, dynamical models and identification methods play an important role in most of disciplines such as automatic control, signal processing, physics, economics, medicine, biology, ecology, seismology, etc. In this plenary talk, an overview of the principal models and identification methods will be presented.

Since the pioneering works of Gauss and Legendre who introduced the least squares (LS) method, at the end of the eighteenth century, in the field of astronomy to predict the motion of planets and comets from telescopic measurements, quite a lot of papers and books on the parameter estimation problem have been published, so that it is not easy to get a general view of available model identification methods. Due to time limitation, some restrictive choices have been made as summarized below, with the attempt to have the best coverage as possible.

First, we will answer to the following basic questions:

-Why to use a model? -How to classify the models? -How to build a model? Then, several model-based applications will be briefly discussed, with particular emphasis on simulation/synthesis, filtering, prediction/forecasting, interpolation/smoothing, source separation, adaptive equalization and adaptive control. It is important to note that depending on the considered application, a model is also called filter, predictor, interpolator, mixture, channel, equalizer or controller.

In a second part of the talk, we will present different classes of models.

Discrete-time deterministic linear models will be first considered, both in state-space (SS) and input-output (I/O) forms. Then, several discrete-time stochastic linear models will be presented in a unified way. These models can be viewed as -either linear dynamical filters allowing the generation/synthesis, analysis and classification of random signals, as it is the case of the autoregressive (AR), moving average (MA) and ARMA models, -or linear models with a random additive noise that represents measurement noise, external disturbances, and modeling errors, like ARX, ARMAX, and ARARX models.

Real-life systems being often time-varying or/and nonlinear in nature, two other classes of models are currently used: linear parameter-varying (LPV) and nonlinear (NL) models. These two classes will be described, with a focus on Volterra and block-structured (Wiener, Hammerstein, Wiener-Hammerstein...) models.

LPV models are suited to modeling linear time-varying (LTV) systems, the dynamics of which are functions of a measurable, time-varying parameter vector p(t), called the scheduling variable. They can also be used for representing nonlinear systems linearized along the trajectory p(t). LPV models can be viewed as intermediate descriptions between linear time invariant (LTI) models and nonlinear time-varying models.

Nonlinear models are very useful for various application areas including chemical and biochemical processes (distillation columns, polymerization reactors, biochemical reactors, see [START_REF] Doyle | Identification and control using Volterra models[END_REF]), hydraulic plants, pneumatic valves, radio-over-fiber (RoF) wireless communication systems (due to the optical/electrical (O/E) conversion), high power amplifiers (HPA) in satellite communications, loudspeakers, active noise cancellation systems, physiological systems [START_REF] Marmarelis | Nonlinear dynamic modeling of physiological systems[END_REF], vibrating structures and more generally mechatronic systems like robots [START_REF] Kerschen | Past, present and future of nonlinear system identification in structural dynamics[END_REF].

Block oriented NL models are composed of a concatenation of LTI dynamic subsystems and static NL subsystems. The linear subsystems are generally parametric (transfer functions, SS representations, I/O models), whereas the NL subsystems may be with memory or memoryless. The different subsystems are interconnected in series or parallel.

In our talk, finite-dimensional discrete-time Volterra models, also called truncated Volterra series expansions, that allow to approximate any fading memory nonlinear system with an arbitrary precision, as shown in [START_REF] Boyd | Fading memory and the problem of approximating nonlinear operators with Volterra series[END_REF], will be considered in more detail; see [START_REF] Favier | Signaux aléatoires : Modélisation, estimation, détection, chapter 7: Estimation paramétrique de modèles entrée-sortie[END_REF], and [START_REF] Guglielmi | Signaux aléatoires : Modélisation, estimation, détection, chapter 3: Modèles monovariables linéaires et non linéaires[END_REF]. The importance of such models for applications is due to the fact that they represent a direct nonlinear extension of the very popular finite impulse response (FIR) linear model, with guaranteed stability in the bounded-input bounded-output (BIBO) sense, and they have the advantage to be linear in their parameters, the kernel coefficients [START_REF] Schetzen | The Volterra and Wiener theories of nonlinear systems[END_REF]. Moreover, they are interpretable in terms of multidimensional convolutions which makes easy the derivation of their z-transform and Fourier transform representations [START_REF] Mathews | Polynomial signal processing[END_REF]. The main drawback of these models concerns their parametric complexity implying the need to estimate a huge number of parameters. So, several complexity reduction approaches for Volterra models have been developed using symmetrization or triangularization of Volterra kernels, or also their expansion on orthogonal bases (like Laguerre, Kautz and GOB bases), or their Parafac decomposition. These two last ap-proaches lead to the so-called Volterra-Laguerre and Volterra-Parafac models; [START_REF] Campello | A note on the optimal expansion of Volterra models using Laguerre functions[END_REF], [START_REF] Campello | Optimal expansions of discrete-time Volterra models using Laguerre functions[END_REF], [START_REF] Favier | Identification de modèles de Volterra basée sur la décomposition PARAFAC de leurs noyaux et le filtre de Kalman étendu[END_REF], [START_REF] Kibangou | Laguerre-Volterra filters optimization based on Laguerre spectra[END_REF], [START_REF] Kibangou | Selection of generalized orthonormal bases for second order Volterra filters[END_REF] The third part of the talk will be concerned with the model identification problem which is strongly related to statistics, stochastic processes (also called random signals), time series analysis, signal processing, information/detection/estimation theories, optimization, linear algebra, machine learning, and more recently with compressive sensing, also known as compressive sampling and sparse sampling, that consists in finding sparse solutions to an underdetermined system of linear equations, i.e. characterized by more unknowns than equations [START_REF] Donoho | Compressed sensing[END_REF]. The increasing interest in this last topic is due to the fact that most signals are sparse, i.e. with many coefficients close to or equal to zero, as it is the case of images for instance.

A complete identification procedure is composed of six main steps [START_REF] Ljung | System identification: Theory for the user[END_REF]:

-Experiment design, -Input-output measurements, -Model structure choice, -Structure parameter determination, -Model parameter estimation, -Model validation.

This procedure is generally iterative in the sense that, starting from a priori information about the system to be identified, the different above-listed sub-problems are successively addressed with the need to revise some choices until the model is validated.

The experiment design consists in making several choices (optimal design of input sequence, i.e. excitation signal, sampling period, etc.).

The determination of the structure parameters is an important problem, for which several tests are proposed in the literature, the most popular ones being the Akaike information criterion [START_REF] Akaike | A new look at the statistical model identification[END_REF], abbreviated to AIC, and the minimum description length (MDL) criterion of Rissanen [START_REF] Rissanen | Modeling by shortest data description[END_REF].

The model validation consists in testing whether the model is valid, e.g. "good enough". Such tests are based on a priori information about the system, the intended utilization of the model, its fitness to real I/O data, etc.

The parameter estimation algorithms are depending on two choices:

-The cost function (criterion) to be optimized.

-The optimization method or the algorithm utilized to compute the optimal solution.

Among the most popular parameter estimation methods, we can cite:

-The LS variants like the weighted least squares (WLS) method, with the Gauss-Markov estimator as particular case that is a best linear unbiased estimator (BLUE) under certain assumptions, the generalized least squares (GLS) method, introduced by Aitken (1935), the extended least squares (ELS) and the total least squares (TLS) methods [START_REF] Van Huffel | The total least squares problem: Computational aspects and analysis[END_REF]. -The maximum-likelihood (ML) method, introduced by Gauss and Laplace, and then popularized by Fisher at the beginning of the 19th century, which consists in maximizing the probability density of the observations conditioned upon the parameters. -The maximum a posteriori (MAP) method which consists in maximizing the probability density function of the unknown random parameters conditioned upon the knowledge of measured signals.

-The minimum mean squared error (MMSE) estimation method.

-The M-estimation methods, introduced by Huber (1964) in the context of robust statistics [START_REF] Huber | Robust statistics[END_REF], and more generally robust identification methods in presence of outliers in the data or small errors in the hypotheses [START_REF] Box | A bayesian approach to some outlier problems[END_REF], [START_REF] Goodwin | Robust identification of process models from plant data[END_REF]. -The instrumental variable (IV) methods [START_REF] Söderström | Instrumental variable methods for system identification[END_REF], [START_REF] Young | Some observations on instrumental variable methods of time-series analysis[END_REF].

-The bounding approach that consists in determining a feasible set (also called membership set) for the parameters or states when the additive output error is assumed to be bounded; see [START_REF] Milanese | Bounding approaches to system identification[END_REF] for a review of bounding techniques both for linear and nonlinear models; in [START_REF] Favier | Bounding approaches to system identification, chapter 4: Review and comparison of ellipsoidal bounding algorithms[END_REF], a review and a comparison of ellipsoidal outer bounding algorithms are made. -The subspace methods whose the concept was introduced with the MUSIC (MUltiple SIgnal Classification) algorithm, which is a super resolution technique for array processing [START_REF] Schmidt | Multiple emitter location and signal parameter estimation[END_REF]. They are also strongly connected with deterministic and stochastic realization theories [START_REF] Faurre | Opérateurs rationnels positifs[END_REF], [START_REF] Favier | Filtrage, modélisation et identification de systèmes linéaires stochastiques à temps discret[END_REF], [START_REF] Ho | Effective construction of linear state-variable models from input/output functions[END_REF], [START_REF] Katayama | Subspace methods for system identification[END_REF].

Many books discuss estimation theory and system identification. In the field of engineering sciences, we can cite the fundamental contributions of [START_REF] Aström | Numerical identification of linear dynamic systems for normal operating records[END_REF], [START_REF] Box | Time series analysis: Forecasting, and control[END_REF], [START_REF] Eykhoff | Some fundamental aspects of process-parameter estimation[END_REF], [START_REF] Eykhoff | System identification. Parameter and state estimation[END_REF], [START_REF] Sage | System identification[END_REF].

See also [START_REF] Gelb | Applied optimal estimation[END_REF], [START_REF] Goodwin | Dynamic system identification: Experiment design and data analysis[END_REF], [START_REF] Heuberger | Modelling and identification with rational orthogonal basis functions[END_REF], [START_REF] Kay | Fundamentals of statistical signal processing: Estimation theory[END_REF], [START_REF] Kumar | Stochastic systems: Estimation, identification and adaptive control[END_REF], [START_REF] Ljung | System identification: Theory for the user[END_REF], [START_REF] Ljung | Theory and practice of recursive identification[END_REF], [START_REF] Mendel | Lessons in digital estimation theory[END_REF], [START_REF] Norton | An introduction to identification[END_REF], [START_REF] Söderström | System identification[END_REF], [START_REF] Young | Recursive estimation and time-series analysis[END_REF] for linear systems, and [START_REF] Doyle | Identification and control using Volterra models[END_REF], [START_REF] Favier | Signaux aléatoires : Modélisation, estimation, détection, chapter 7: Estimation paramétrique de modèles entrée-sortie[END_REF], [START_REF] Giannakis | A bibliography on nonlinear system identification[END_REF], [START_REF] Keviczky | Nonlinear system identification. Input-ouput modeling approach[END_REF], [START_REF] Janczak | Identification of nonlinear systems using neural networks and polynomial models: A block-oriented approach[END_REF], [START_REF] Mathews | Polynomial signal processing[END_REF], [START_REF] Nelles | Nonlinear system identification: From classical approaches to neural networks and fuzzy models[END_REF] for nonlinear systems.

Identification methods can be classified in different ways depending on:

-The class of systems to be identified (linear/nonlinear, SISO/MIMO, ...).

-The assumptions made about model uncertainties (probabilistic description/unknown but bounded (UBB) error description). -The domain of the used information (time-domain/frequency domain).

-The experiment configuration (open-loop/closed-loop).

-The I/O data processing (non iterative/iterative, non recursive/recursive or non adaptive/adaptive). -The knowledge or not of the input signals (supervised/unsupervised or blind approaches).

Concerning this last point, we have to note that, unlike control applications for which the input signals are optimized, and therefore measured, in such a way that the system to be controlled has a desired behaviour, most of signal processing applications are characterized by the fact that input signals can not be measured, as it is the case of seismic, astronomical, audio, digital communication, or biomedical applications. That leads to unsupervised or blind identification methods. Such blind approaches are used for blind seismic deconvolution, blind channel equalization in communications, or blind source separation in a more general context of MIMO (multi input -multi output) systems like antenna arrays in underwater acoustics, electrode arrays in electroencephalography or electrocardiography, audio mixtures etc. See [START_REF] Comon | Handbook of blind source separation. Independent component analysis and applications[END_REF].

Adaptive estimation is closely linked with adaptive filtering [START_REF] Haykin | Adaptive filter theory[END_REF], [START_REF] Sayed | Fundamentals of adaptive filtering[END_REF], [START_REF] Widrow | Adaptive signal processing[END_REF]. Adaptive modeling and processing are very useful for a wide variety of applications like adaptive control, adaptive noise cancelling, adaptive channel equalization, adaptive receiving arrays for various types of signals (seismic, acoustic, electromagnetic), among many others.

In our talk, we will begin by presenting the cost functions generally considered for parameter estimation.

Then, two standard optimization methods will be introduced: The Newton-Raphson method, with its variant the Levenberg-Marquardt algorithm, and the Gradient descent method.

Five parameter estimation methods will be presented for discrete-time Volterra models:

-The MMSE method, that leads to the Wiener solution, -The non recursive deterministic LS method, -The recursive least squares (RLS) method, -The least mean squares (LMS) algorithm, -The normalized least mean squares (NLMS) algorithm.

Other identification methods for Volterra systems can be found in [START_REF] Doyle | Identification and control using Volterra models[END_REF], [START_REF] Kibangou | Identification of fifth-order Volterra systems using i.i.d. inputs[END_REF].

In the case of a Pth-order Volterra system and an i.i.d. (independently and identically distributed) input signal, we will give the necessary and sufficient condition for system identifiability using the MMSE solution. Moreover, in the case of a second-order Volterra system, we will give the decoupling conditions for the parameter estimation of the linear and quadratic kernels. The necessary condition to be satisfied by the step size for ensuring the LMS algorithm convergence will be also given.

As already mentioned, we have to note that a tensor-based approach has been recently proposed for reducing the parametric complexity of Volterra models, leading to the socalled Volterra-Parafac models [START_REF] Favier | Identification de modèles de Volterra basée sur la décomposition PARAFAC de leurs noyaux et le filtre de Kalman étendu[END_REF]. Tensor-based methods have also been developed for blind identification of SISO and MIMO linear convolutive systems, and for block structured nonlinear system identification [START_REF] Favier | Nonlinear system modeling and identification using tensor approaches[END_REF], [START_REF] Fernandes | Blind channel identification algorithms based on the PARAFAC decomposition of cumulant tensors: The single and multiuser cases[END_REF], [START_REF] Fernandes | PARAFAC -based blind identification of convolutive MIMO linear systems[END_REF], [START_REF] Kibangou | Wiener-Hammerstein systems modeling using diagonal Volterra kernels coefficients[END_REF], [START_REF] Kibangou | Identification of parallel-cascade Wiener systems using joint diagonalization of third-order Volterra kernel slices[END_REF], [START_REF] Kibangou | Identification of fifth-order Volterra systems using i.i.d. inputs[END_REF], [START_REF] Kibangou | Tensor analysis-based model structure determination and parameter estimation for block-oriented nonlinear systems[END_REF].

Some simulation results will be presented to illustrate the behaviour and to compare the performance of the (RLS, LMS, NLMS) algorithms and the (EKF, LMS, NLMS) algorithms for the parameter estimation of linear FIR models and Volterra-Parafac models, respectively.

Finally, our plenary talk will be concluded by giving several topics to be addressed in future research works.


Olivier Rodriguez
email: olivier.rodriguez@univ-montp2.fr

ON THE TEICHM ÜLLER GEODESIC GENERATED BY THE L-SHAPED TRANSLATION SURFACE TILED BY THREE SQUARES

We study the one parameter family of genus 2 Riemann surfaces defined by the orbit of the L-shaped translation surface tiled by three squares under the Teichmüller geodesic flow. These surfaces are real algebraic curves with three real components. We are interested in describing these surfaces by their period matrices. We show that the only Riemann surface in that family admitting a non-hyperelliptic automorphism comes from the 3-square-tiled translation surface itself. This makes the computation of an exact expression for period matrices of other Riemann surfaces in that family by the classical method impossible. We nevertheless give the solution to the Schottky problem for that family: we exhibit explicit necessary and sufficient conditions for a Riemann matrix to be a period matrix of a Riemann surface in the family, involving the vanishing of a genus 3 theta characteristic on a family of double covers.

INTRODUCTION

A translation surface can be defined as an assembling of Euclidean polygons with appropriate identifications of sides or, in an equivalent manner, as a pair (X, ω) where X is a compact Riemann surface and ω a holomorphic 1-form on X. Such a pair can be considered as an element of a rank g vector bundle ΩT g → T g over the Teichmüller space T g Date: October 2011. 1 of genus g Riemann surfaces. The moduli space of holomorphic 1-forms with a unique (double) zero on a genus 2 Riemann surface is denoted by H (2).

There exists a natural action of the group GL + 2 (R) on translation surfaces. The projections of the SL 2 (R)-orbits into the Riemann moduli space M g are called Teichmüller disks. It may happen that the stabilizer of a translation surface under the SL 2 (R)-action is a lattice. Passing to the quotient, it gives rise to a Teichmüller curve, that is, an algebraic curve in the Riemann moduli space, isometrically immersed for the Teichmüller metric.

To date very few is known about how one passes explicitly from one description of a complex structure to another under the GL + 2 (R) action. For example, how does the period matrix of a Riemann surface vary under this action? What about the equations defining the corresponding algebraic curve?

Note that after this paper was written, M. Möller wrote an important result on a closely related family of examples, see [START_REF]Prym covers, theta functions and Kobayashi curves in Hilbert modular surfaces[END_REF]Theorem 0.1].

In this paper we study the family of Riemann surfaces defined by the SL 2 (R)-orbit of the L-shaped translation surface tiled by three squares (see Figure 1). this family is the Teichmüller curve of discriminant 9 and, following McMullen's notation, will be denoted by W 9 (comprehensive overviews on Teichmüller curves can be found e.g. in [START_REF] Curtis | Billiards and Teichmüller curves on Hilbert modular surfaces[END_REF] and [START_REF] Lochak | On arithmetic curves in the moduli spaces of curves[END_REF]). We will sometimes denote by W M 9 the Teichmüller geodesic generated by the 3-square-tiled surface, that is, its orbit under the diagonal subgroup 1 {( t 0 0 t -1 )} t>0 . The 3-square-tiled surface admits an order 4 automorphism; period matrices of Riemann surfaces defined by such translation surfaces in H (2) were computed by R. Silhol in [START_REF]Genus 2 translation surfaces with an order 4 automorphism, The geometry of Riemann surfaces and abelian varieties[END_REF]§3]. We will first show the following.

Proposition 1. The only Riemann surface in the family W M 9 admitting a non-hyperelliptic automorphism is the one defined by the L-shaped translation surface tiled by three squares.

As automorphisms provide precious informations in order to compute the period matrix of a Riemann surface, that prevents to compute an exact expression for period matrices of these surfaces by the classical method described, for instance, in [BL04, §11.7] (see Remark 4 below). Nevertheless, we consider a certain family of double covers of those surfaces (whose construction will be described in detail) admitting many automorphisms, for which we obtain the following characterization of their period matrices.

Theorem A. Let (X, ω) be a translation surface in H (2), then X is in the family W 9 if and only if a certain explicit double cover X admits a period matrix Ẑ of the form

Ẑ =   z 1 1 2 z 1 z 13 1 2 z 1
for which ϑ 1 1 1 1 0 1 ( Ẑ) = 0. Recall that for all m, n ∈ ( Z / 2Z ) g , the theta characteristic ϑ m n is defined by

ϑ m n (Z) = ∑ k∈Z g exp πi t k + 1 2 m Z k + 1 2 m + t k + 1 2 m n
for all Z ∈ M g (C) such that t Z = Z and ℑm(Z) > 0.

Riemann surfaces in the Teichmüller geodesic generated by the 3-square-tiled surface correspond to real algebraic curves defined by polynomials with real roots. The situation being more rigid in this case, this allows us to establish a form for their period matrices and deduce necessary and sufficient conditions for a Riemann surface to be in the family W M 9 . Theorem B. Let (X 1 , ω 1 ) be the L-shaped translation surface tiled by three squares. For any real number t ≥ 1, let (X t , ω t ) = 1 0 0 t • (X 1 , ω 1 ). Then Z t is a period matrix of X t associated to a certain explicit homology basis if, and only if there exists a unique real number y t > 2t/3 such that Z t = 1 + i(2y tt) iy t iy t i(y t /2 + t) and satisfying

(1)

∑ (k 1 ,k 2 ,k 3 )∈Z 3 expπ t 2 -y t + 1 2 i 3 ∑ l=1 k 2 ℓ + (y t -t + i) ∑ 1≤ℓ<m≤3 k ℓ k m + 3 2 i - t 2 3 ∑ ℓ=1 k ℓ = 0.
The construction of the homology basis will be described in detail.

Acknowledgements. I would like to express my gratitude toward my Ph.D. advisor Robert Silhol for helpful conjectures as well as fulfilling discussions. I also wish to thank Emmanuel Royer, Pascal Hubert and Guillaume Bulteau for various advices and useful discussions and comments.

PRELIMINARIES

Translation surfaces.

A translation surface is a finite collection of Euclidean polygons in the complex plane such that • the boundary of each polygon is oriented counterclockwise;

• for every side of a polygon, there exists another side (possibly of the same polygon) parallel and of the same length: both sides are then identified by translation. Such a collection of polygons defines a topological surface admitting a translation structure, that is, away from a finite set of points, a maximal atlas whose transition functions are translations.

As a non-trivial example consider a compact Riemann surface equipped with a holomorphic 1-form. Integrating the form yields, away from its zeros, an atlas with polygonal charts and transition functions that are translations. Conversely, a translation structure defines a complex structure since translations are biholomorphic. Pulling back the 1-form dz on the complex plane by the charts gives a holomorphic 1-form on the surface.

As a consequence, we can define in an equivalent manner a translation surface as a pair (X, ω) where X is a compact Riemann surface and ω a holomorphic 1-form on X. At a regular point, in the local coordinate defined by integrating the form, we have ω = dz. At an order k zero of ω, we have

ω = z k dz = d z k+1
k + 1 so that the Riemann surface X is locally a (k + 1)-fold cover over the complex plane. This means that an order k zero corresponds to a cone-type singularity of angle 2π(k + 1) for the locally Euclidean metric |ω|. More details can be found in [START_REF] Masur | Ergodic theory of translation surfaces[END_REF]§1] concerning the equivalence of these definitions. See also [START_REF] Zorich | Flat surfaces[END_REF] for a general survey on translation surfaces.

We will use the notation (X, ω) = ( P / ∼ , dz) where the quotient P / ∼ designates an assembling of Euclidean polygons with appropriate identifications of sides by translation and dz is the holomorphic 1-form on C.

Example 1 (Translation surface tiled by three squares). Consider the L-shaped polygon obtained by assembling three copies of the Euclidean unit square equipped with the glueings specified in Figure 1. The black dots are identified to the cone-type singularity of angle 6π, hence defining a translation surface (X, ω) in H (2). Rotation by angle π/2 around the center of the bottom left square induces an order 4 automorphism on the Riemann surface X: the corresponding algebraic curve then admits an equation of the form

y 2 = x(x 2 -1)(x -a)(x -1/a)
and the order 4 automorphism is

(x, y) → 1 x , iy x 3 .
Following [START_REF]Genus 2 translation surfaces with an order 4 automorphism, The geometry of Riemann surfaces and abelian varieties[END_REF]§3&4], in this example we have a = 7+4 √ 3 and the holomorphic 1-form ω is

ω = µ dx y - xdx y with µ ∈ C * .
Remark 1. From a more general point of view, translation surfaces are a specific case of half-translation surfaces, for which the transition functions are of the form z → ±z + c for c ∈ C. Such a surface can be defined as a pair (X, q) where q is a holomorphic quadratic differential on X. If (X, ω) is a translation surface, then the quadratic differential defining the complex structure is ω 2 .

GL +

2 (R)-action. There exists a natural action of the linear group on translation structures.

An element M = ( a b c d ) ∈ GL + 2 (R) acts on z ∈ C by M • z = ax + by + i(cx + dy).
This is just the affine action on the complex plane identified to R 2 . When a translation surface is defined by a collection of polygons ( P / ∼ , dz), then the group GL + 2 (R) operates naturally on the polygons, giving a new translation surface. The action is well defined since linear applications transform parallelograms into parallelograms.

On a form (X, ω), the action of M is defined as follows: let

η := aℜe(ω) + bℑm(ω) + i cℜe(ω) + dℑm(ω) ,
then η is a harmonic form on X. There exists a unique complex structure on the underlying topological surface for which ω is holomorphic, so that we obtain a new Riemann surface Y and we set M • (X, ω) = (Y, η).

Remark 2. For every translation surface (X, ω), the complex structure defined by (X, ω) is stable under the action of the subgroup R * + • SO 2 (R), since this action corresponds to multiplying ω by a non zero scalar complex.

Teichmüller geodesics. Let t be a real number such that t > 1 and consider the R-linear application M t : C → C defined by

z → 1 2 (1 + t)z + 1 2 (1 -t)z.
Let (X, ω) be a translation surface and (X t , ω t ) = M t • (X, ω). The change of complex structure yields a natural application f t : X → X t verifying d f t = M t and having constant complex dilatation:

∀P ∈ X, µ f t (P) := ∂ f t (P) ∂ f t (P) = 1 -t 1 + t .
The maximal dilatation is then

K( f t ) := sup P∈X 1 + |µ f t (P)| 1 -|µ f t (P)| = t.
The homeomorphism f t : X → X t is then a Teichmüller extremal map. We call the Riemann surface X t the Teichmüller deformation of X of dilatation t with respect to ω and the family {X t } t≥1 a Teichmüller geodesic (see e.g. [Abi80, chap. 1] for more details).

Remark 3. Let (X, ω) be a L-shaped translation surface of the form indicated in Figure 2. Then X admits an order 4 automorphism induced by rotation of angle π/2 around 0. Let t

✻ ❄ ✲ ✛ λ λ ❄ s s s s s s s s ✲ ✛ ✻ β 2 β 1 α 1 α 2 0 (X, ω) ✻ ❄ ✲ ✛ 1 1 FIGURE 2.
An L-shaped translation surface with an order 4 automorphism be a real number such that t > 0, we define the translation surfaces

(X g t , ω g t ) = ( t 0 0 t -1 ) • (X, ω), (X h t , ω h t ) = ( t 0 0 1 ) • (X, ω), and (X v t , ω v t ) = ( 1 0 0 t ) • (X, ω).
Then by Remark 2 it is readily checked that (1) the Riemann surfaces X h t and X v t are isomorphic;

(2) X g t and X h t 2 (resp. X g t -1 and X v t 2 ) are isomorphic;

(3) X h t and X h t -1 (resp. X v t and X v t -1 ) are isomorphic.

2.3. Period matrix. Let X be a compact Riemann surface of genus g and let B = (α 1 , . . . , α g , β 1 , . . . , β g ) be a symplectic basis for H 1 (X, Z), that is, such that the matrix of the intersection product is 0 -I g I g 0

Let (ω 1 , . . . , ω g ) be a basis of holomorphic 1-forms on X, we define

A = α j ω k j,k and B = β j ω k j,k
.

Then Z = AB -1 is the period matrix associated to B. Note that Z does not depend on the choice of the basis (ω 1 , . . . , ω g ).

It is well known that Z verifies the Riemann bilinear relations, that is, Z is symmetric and ℑm(Z) is positive definite. The Siegel upper half-space is defined by

S g = {Z ∈ M g : t Z = Z and ℑm(Z) > 0}. Let α, β , γ, δ ∈ M g (R), an element M = ( α β γ δ ) of the symplectic group Sp 2g (R) = M ∈ M 2g (R) : t M 0 -I g I g 0 M = 0 -I g I g 0
acts on an element Z ∈ S g as follows:

M(Z) = (αZ + β )(γZ + δ ) -1 .
If B ′ is another symplectic basis for H 1 (X, Z) and Z ′ the period matrix associated to B ′ , let M be the base change matrix from B ′ to B; then M is a symplectic matrix with integer entries and Z and Z ′ are related by

Z ′ = t M(Z).
Recall that if M is symplectic, then t M is also symplectic.

Moreover, by the Torelli theorem, the period matrix characterizes the complex structure: if X and X ′ are compact Riemann surfaces with period matrices Z and Z ′ respectively, then X and X ′ are isomorphic if, and only if there exists M ∈ Sp 2g (Z) such that M(Z) = Z ′ (see [START_REF] Birkenhake | Complex abelian varieties[END_REF]Theorem 11.1.7]).

Example 2 (Translation surface with an order 4 automorphism). Let (X, ω) be the L-shaped translation surface defined in Figure 2. The rotation by angle π/2 around 0 induces an order 4 automorphism on the Riemann surface X. Then following [Sil06, §3],

Z = i 2λ 2 -2λ +1 2λ -1 -2λ (λ -1) 2λ -1 -2λ (λ -1) 2λ -1 2λ 2 -2λ +1 2λ -1
is the period matrix of X associated to {α 1 , α 2 , β 1 , β 2 }.

Remark 4. The main tool in Silhol's calculation is the existence of a non-hyperelliptic involution whose action permutes ω with another linearly independent holomorphic 1-form that can be explicitly described (enabling to compute the whole period matrix, following [BL04, §11.7] for example). However, on the one hand, generic algebraic curves have no non-trivial automorphisms, and on the other hand it is not known in general how to provide a description of another holomorphic 1-form.

Hyperelliptic Riemann surfaces.

Here we describe a construction of a period matrix of a hyperelliptic Riemann surface. If X is hyperelliptic, then the associated algebraic curve is defined by an equation of the form

y 2 = m ∏ j=1 (x -x j )
where the x j are distinct complex numbers and m = 2g

+ 1 or 2g + 2. If m = 2g + 1, then let x 2g+2 = ∞
Let ε 1 be a simple arc in P 1 (C) joining x 1 to x 2 and not passing through any of the other x j for j ≥ 3. Let ε 2 a second simple arc joining x 2 to x 3 and not passing through x 1 nor any of the other x j 's for j ≥ 4, and such that ε 2 only intersects ε 1 in x 2 . In the same way, we construct simple arcs ε 3 , . . . , ε 2g+1 , ε 2g+2 joining respectively x 3 to x 4 , . . . , x 2g+1 to x 2g+2 and x 2g+2 to x 1 , so that ε j only intersects ε j+1 in one point ( j mod 2g + 2). Let π : X → P 1 (C) denote the projection (x, y) → x and, for j = 1, . . . , 2g + 2, let δ j := π -1 (ε j ): this is a simple closed curve in X.

We choose on each ε j a holomorphic determination of √ P so that the induced orientation on δ j is such that (δ j • δ j+1 ) = 1 ( j mod 2g + 2), all other intersection numbers being zero. With this convention, up to homology, we thus have ∑ g j=1 δ 2 j = -δ 2g+2 and ∑ g j=1 δ 2 j-1 = -δ 2g+1 (this is a readily checked by a topological sketch of the situation, the reader may also refer to [FK92, §VII.1.1] where this construction is described).

For j = 1, . . . , g, we define

α j = - j-1 ∑ ℓ=1 δ 2ℓ -δ 2g+2 , β j = δ 2 j-1 .
It is readily checked that B := (α 1 , . . . , α g , β 1 , . . . , β g ) is a symplectic basis for H 1 (X, Z). Moreover, as X is a hyperelliptic Riemann surface,

dx y , xdx y , . . . , x g-1 dx y is a basis of holomorphic 1-form on X (see [FK92, §III.7.5, Corollary 1]). Let A = - j-1 ∑ ℓ=1 ε 2ℓ x k-1 dx P(x) - ε 2g+2 x k-1 dx P(x) j,k B = ε 2 j-1 x k-1 dx P(x) j,k
then Z = AB -1 is the period matrix associated to B.

Real algebraic curves.

Definitions. F. Klein observed that a complex algebraic curve X is defined by real polynomial equations if, and only if X admits an anti-holomorphic involution σ (see [START_REF] Klein | On Riemann's theory of algebraic functions and their integrals. A supplement to the usual treatises[END_REF]§21]), called a real structure. Moreover, these polynomials can always be chosen so that σ is induced by complex conjugation. A real algebraic curve is a couple (X, σ ). When it is clear from the context, we omit σ and simply say that X is a real curve, in particular when X is defined by a real polynomial equation and σ is the complex conjugation.

If X is of genus g, then the connected components of the fixed point set of σ are said to be real. We say that X is an M-curve if it admits the maximum number of real components, which is g + 1 (see [GH81, Proposition 3.1]).

Real hyperelliptic M-curves. Suppose that X is a hyperelliptic curve defined by a polynomial equation y 2 = P(x) with deg P = 2g + 1 or 2g + 2, and such that σ is induced by complex conjugation. Then (X, σ ) is a real M-curve if, and only if all roots of P are real. Composing σ with the hyperelliptic involution h X yields a second real structure denoted by -σ . The connected components of the fixed point set of -σ are said to be pure imaginary and the Weierstrass points of X are exactly the intersection points of the real and pure imaginary components.

Example 3 (M-curve defined by a translation surface). Let (X, ω) = ( P / ∼ , dz) be a translation surface in H (2) obtained from four mirror images of a L-shaped polygon (see Figure 3). We can always assume that the polygon P admits 0 as a center of symmetry and is stable by complex conjugation. The latter defines an anti-holomorphic involution σ on X that fixes pointwise the three simple closed curves coming from the horizontal axis of symmetry of P and its horizontal sides: we thus obtain a real M-curve (X, σ ). The Weierstrass points are represented by black dots (identified to the double zero of ω) and small circles in Figure 3. Cutting and reassembling the cross-shaped polygon in the left part of Figure 3 gives the L-shaped polygon in the right part of Figure 3, where the segments corresponding to the simple closed curves fixed by σ are the horizontal and vertical segments that join the black dots and those passing through the circles. In what follows, if (X, σ ) is a real M-curve defined by a translation surface (X, ω), then the real structure σ will always be the one defined by complex conjugation as in Example 3. If (X 1 , ω 1 ) denotes the L-shaped translation surface tiled by three squares, then it corresponds to the unique, so-called splitting prototype in [START_REF]Teichmüller curves in genus two: discriminant and spin[END_REF]§3] and by [START_REF]Teichmüller curves in genus two: discriminant and spin[END_REF]§6], this translation surface generates the whole Teichmüller curve W 9 . The above convention on the definition of the real structure fixes the orientation of the L-shaped polygon, then by Remark 3 the set of M-curves in the family W 9 thus coincides with the Teichmüller deformations of X 1 with respect to ω 1 .

Automorphisms of genus 2 real M-curves. If (X, σ ) is a real algebraic curve, we denote by Aut(X, σ ) the group of its real automorphisms, that is, biholomorphic applications φ : X → X such that φ • σ = σ • φ . The group Aut X is sometimes called the group of complex automorphisms of (X, σ ).

F.-J. Cirre computed the groups of real and complex automorphisms of a real M-curve as stated below, where D n denote the dihedral group of order 2n.

Theorem 2 (Cirre). Let a, b, c be three real numbers such that 0 < a < b < c < 1 and let (X, σ ) be the real M-curve defined by the equation

y 2 = P(x) = x(x -a)(x -b)(x -c)(x -1)
Then Aut(X, σ ) = AutX if, and only if a = b(c -1)/(b -1). Moreover, we have the following two cases:

(1) Suppose that Aut(X, σ ) = Aut X: (a) if we have a = bc or a = (b -c)/(c -1) or a = 1 + c -c/b, then Aut X ≃ D 2 ; (b) if we have bc = a = (b -c)/(c -1) or bc = a = 1 + c -c/b or (b -c)/(c -1) = a = 1 + c -c/b, then Aut X ≃ D 6 ; (c) else, Aut X ≃ Z/2Z. (2) Suppose that Aut(X, σ ) = Aut X: (a) if we have a = bc or a = (b -c)/(c -1) or a = 1 + c -c/b, then Aut(X, σ ) ≃ D 2 and AutX ≃ D 4 ; (b) if we have bc = a = (b -c)/(c -1) or bc = a = 1 + c -c/b or (b -c)/(c -1) = a = 1 + c -c/b, then (a, b, c) = 1 3 , 1 2 , 2 3 ,
Aut(X, σ ) ≃ D 6 and Aut X is isomorphic to the group G 24 of order 24 admitting the following presentation:

(2) G 24 = r, s|r 4 , s 6 , (rs) 2 , (r -1 s) 2 ;

(c) else, Aut(X, σ ) ≃ Z/2Z and Aut X ≃ D 2 .
The details of the proof can be found in [Cir01, Theorem 4.2] and [Cir03, Proposition 3.5].

This classification can also be translated in terms of the period matrices and of the hyperbolic structure of the algebraic curve. See [Rod10, chap. 3] for more details, see also [Nat78].

REAL M-CURVES WITH AUTOMORPHISMS IN W 9

3.1. Description of the family W 9 . The SL 2 (R)-orbit of the L-shaped translation surface tiled by three squares is denoted by ΩW 9 . It projects into the Riemann moduli space M 2 as an irreducible algebraic curve, which is denoted by W 9 . According to [HL05, Remark, p. 3], the latter is the modular curve defined by the quotient Γ \ H where Γ is the level 2 congruence subgroup generated by 1 2 0 1 and 0 -1 1 0 .

In [Möl05, §4], M. Möller provides equations for the family W 9 . R. Silhol gives in [Sil07, Theorem A] a description of the algebraic curves in this family and shows that a translation surface (X, ω) is an element of ΩW 9 if, and only if the curve X admits an equation of the form

y 2 = P u (x) = x(x -1) x 3 + ux 2 - 8 3 ux + 16 9 u (3) with ω = λ xdx y
for u ∈ C \ {-9, 0} and for some constant λ ∈ C * . The degree 3 polynomial

x 3 + ux 2 - 8 3 ux + 16 9 u
admits exactly three real distinct roots if, and only if its discriminant is positive, that is, if, and only if u is a real number such that u < -9. Thus, a genus 2 curve X = X u defined by an equation of the form (3) is a real M-curve if, and only if u < -9.

The following fact was observed in [START_REF]On some one parameter families of genus 2 algebraic curves and half twists[END_REF] but not stated.

Proposition 3. There exists a bijection between the set of real automorphism classes of real M-curves in W 9 and the interval [-18; -9[.

Proof. In [Sil07, Proposition 4.1], the author considers a certain family whose image in moduli space is shown to be the quotient of P 1 (C) \ {-9, 0, ∞} under u → -9u/(u + 9) (see also [START_REF]On some one parameter families of genus 2 algebraic curves and half twists[END_REF]Remarks 4.11]). In particular, there is a bijection between this image and P 1 (C) minus two points and a cone point of order 2, namely the one corresponding to u = -18. Is is then shown in [Sil07, Lemma 5.4] that the aforementioned family is W 9 .

If we now consider the case of real M-curves, then by the above discussions there is a bijection between W M 9 and [-18; -9[ on the one hand, and between W M 9 and ] -∞; -18] on the other hand. Furthermore, it results from [Sil07, Proposition 4.1 and Lemma 5.4] that if X u is a real M-curve defined by an equation y 2 = P u (x) as in Equation (3), then the transformation x → x/(x -1) induces a (complex, but non-real) isomorphism between X u and X u ′ with u ′ = -9u/(u + 9). Defining

f 3 : x → x + √ 3 - √ 3x + 1 ,
we note that the Möbius transformation f 3 is of order 3 and fixes the points i and -i. The orbits of 0 and ∞ under f 3 are respectively

{ √ 3, - √ 3, 0} and {- √ 3/3, √ 3/3, ∞}.
We consider a genus 2 algebraic curve defined by an equation of the form

y 2 = Q s (x) = x(x + 1)(x -s 2 ) x -f 3 (s) 2 x -f 3 f 3 (s) 2 .
If we apply the transformation x → x + 1 to the roots of Q s (x), then a direct calculation shows that this curve also admits an equation of the form (3) with

u = -81(s 2 + 1) 3 (3s + √ 3) 2 (3s - √ 3) 2 .
We note that the function

s → g(s) = -81(s 2 + 1) 3 (3s + √ 3) 2 (3s - √ 3) 2
is even and invariant under f 3 . This leads to the following: Proposition 4. Let (X, ω) be a translation surface in H (2). Then (X, ω) is an element of ΩW 9 if, and only if the algebraic curve X admits an equation of the form

y 2 = Q s (x) = x(x + 1)(x -s 2 ) x -f 3 f 3 (s) 2 x -f 3 (s) 2 (4) with ω = λ dx y + xdx y with s ∈ C \ {- √ 3, - √ 3/3, -1, 0, √ 3/3, √ 3
, -i, i} and for some constant λ ∈ C * .

Proof. According to [Sil07, Theorem A], a translation surface (X, ω) in H (2) is an element of ΩW 9 if, and only if X admits an equation of the form y 2 = P u (x) as defined by (3) with u ∈ C \ {-9, 0} and ω admitting a double zero at 0. The condition in the statement is already proven to be sufficient by the discussion above.

Conversely, let (X, ω) be an element of ΩW 9 , then X is defined by an equation of the form y 2 = P u (x) with u ∈ C \ {-9, 0}. From the study of the function g we can take u = g(s) for some

s ∈ C \ {- √ 3, - √ 3/3, -1, 0, √ 3/3, √ 3, -i, i}.
Applying the transformation x → x -1 to the roots of the equation y 2 = P u (x) then leads to the announced form (4) of the equation y 2 = Q s (x). Moreover, up to a non-zero, complex scalar multiple, dx/y + xdx/y is the only holomorphic 1-form on X admitting a double zero at (-1, 0).

Noting that we have g(s) ≤ -9 for every s ∈ R and that

∀s ∈]0; √ 3/3[, s 2 < f 3 f 3 (s) 2 < f 3 (s) 2 ,
we obtain another description of this set.

Lemma 5. Let s be a real number such that 0

< s < √ 3/3, implying √ 3 3 < f 3 f 3 (s) < √ 3 < | f 3 (s)|.
Let X be the real M-curve defined by the equation

y 2 = Q s (x) = x(x + 1) x -a(s) x -b(s) x -c(s) , with a(s) = s 2 , b(s) = f 3 f 3 (s)
2 and c(s) = f 3 (s) 2 . Then X is in the family W 9 . Conversely, every real M-curve in the family W 9 admits such a description.

As a consequence, we have the following: Proposition 6. There exists a bijection between the set of real automorphism classes of real M-curves in W 9 and the interval ]0; √ 3/3[.

3.2.

Automorphisms. The preceding discussion encourages to consider the curve defined by an equation of the form (3) with u = -18, that is

y 2 = P u (x) = x(x -1)(x -8 + 4 √ 3)(x -2)(x -8 -4 √ 3).
This curve admits an order 4 automorphism induced by the Möbius transformation x → x/(x -1). Applying the transformation x → x -1 to the set of the roots of the polynomial P u , we then get the equation

y 2 = x(x 2 -1)(x -a)(x -1/a) with a = 7 -4 √ 3.
By Example 1 this curve is defined by the 3-square-tiled translation surface. We note that 7 -4

√ 3 = (2 - √ 3) 2 with 0 < 2 - √ 3 < √ 3/3, and that f 3 (2 - √ 3) 2 = 7 + 4 √ 3 = 1 a and f 3 f 3 (2 - √ 3) 2 = 1,
hence this curve is defined by an equation of the form (4) with s = 2 -√ 3. By Example 2, the Riemann surface defined by the 3-square-tiled translation surface admits an order 4 automorphism induced by an affine transformation. In order to weaken this condition, we could consider lower order automorphisms of the Riemann surface, for example non-hyperelliptic holomorphic involutions. However, we observe that in genus 2 such automorphisms can not be induced by affine transformations: Lemma 7. Let (X, ω) be a translation surface in H (2) and ϕ be an order 2 automorphism defined by an order 2 automorphism affine with respect to ω. Then ϕ is the hyperelliptic involution h X .

Proof. Let ϕ be such an automorphism: since X is hyperelliptic, then following [FK92, Theorem V.2.13], ϕ must satisfy one of the following: (a) ϕ has no fixed point; (b) ϕ only fixes non-Weierstrass points; (c) ϕ is the hyperelliptic involution. Now ϕ must fix the double zero of ω, which is necessarily a Weierstrass point, hence the conclusion.

We now prove Proposition 1 restated as follows.

Proposition 8. Let s be a real number such that 0 < s < √ 3/3 and let X be the real M-curve defined by the equation

y 2 = Q s (x) = x(x + 1)(x -s 2 ) x -f 3 f 3 (s) 2 x -f 3 (s) 2
Then Aut X = h X if, and only if s = 2 -√ 3.

Proof. Considering Theorem 2, it is sufficient to consider Möbius transformations that could give an order 2 automorphism. By applying a suitable Möbius transformation to the roots of Q s , we can easily adapt the conditions ensuring the existence of non-trivial automorphisms stated in the theorem to equations of the form (4) with 0 < a(s) < b(s) < c(s) for all s ∈]0; √ 3/3[. The possibilities are the following: (a) the Möbius transformation

x → c(s) -b(s) b(s) -a(s) x -a(s) x -c(s) , maps (a, b, c) to (0, -1, ∞)
and induces a holomorphic involution on X if, and only if we have

a(s) = b(s) -1 + b(s) c(s) . Calculations lead to b(s) c(s) = - 16 √ 3x(x 2 + 1) (x + √ 3) 2 ( √ 3x + 1) 2 then to b(s) -1 + b(s) c(s) -a(s) = - (x 2 + 1)(3x 4 + 8 √ 3x 3 + 18x 2 + 16 √ 3x -9) (x + √ 3) 2 ( √ 3x + 1) 2 .
The degree 4 polynomial in the numerator admits two real roots, namely -2 -√ 3 and 2 -√ 3, the latter being the only one in the interval ]0; √ 3/3[. According to the above discussion at the beginning of this section, this curve has an order 4 automorphism;

(b) the Möbius transformation x → a(s)

x + 1 xa(s) induces an automorphism on X if, and only if

a(s) = b(s)c(s) 1 + b(s) + c(s)
.

The only real solutions lead to the singular curve of equation

y 2 = x(x + 1)(x - √ 3/3)(x + √ 3/3) corresponding to s = √ 3/3; (c) the Möbius transformation x → c(s) -x
x + 1 induces an automorphism if, and only if

a(s) = c(s) -b(s) 1 + b(s) ,
which gives the singular curve defined by the equation

y 2 = x 2 (x + 1)(x - √ 3)(x + √ 3)
corresponding to s = 0. Note that the Möbius transformation x → 1/x induces an automorphism between the two curves defined by s = 0 and s = √ 3/3; (d) lastly, the curve X has a holomorphic involution induced by

x → a(s) -b(s) b(s) -c(s) x -c(s)
xa(s) if, and only if

a(s) = b(s) 1 -b(s) + c(s) ,
which also gives the solution s = 2 -√ 3. Calculations are tedious, but routine.

PERIODS OF REAL M-CURVES IN W 9

4.1. Theta characteristics.

Definitions and elementary properties. Let z ∈ C g and Z ∈ S g , the Riemann ϑ function is defined by

(5) ϑ (z, Z) = ∑ k∈Z g exp(πi t kZk + 2πi t kz).
One shows that the above series defines a holomorphic function on

C g × S g (see [Mum83, chap. II, Proposition 1.1]).
Let m, n ∈ 1 2 Z g , we define order 2 theta characteristics by

ϑ 2m 2n (z, Z) = ∑ k∈Z g exp πi t (k + m)Z(k + m) + 2πi t (k + m)(z + n) = exp πi t mZm + 2πi t m(z + n) ϑ (z + Zm + n, Z),
which, by the above identity, are also holomorphic functions on C g × S g . The following proposition points out well known properties of the function z → ϑ 2m 2n (z, Z).

Proposition 9. Let Z ∈ S g , m, n ∈ 1 2 Z g and p, q ∈ Z g , then for every z ∈ C g , the functions ϑ 2m 2n satisfy:

(6) ϑ 2m 2n (z, Z) = exp πi t pZ p + 2πi t p(z + n) -2πi t mq ϑ 2m 2n (z + Z p + q, Z)
and

(7) ϑ 2m + 2p 2n + 2q (z, Z) = exp(2πi t mq)ϑ 2m 2n (z, Z).
One also has:

(8) ϑ 2m 2n (-z, Z) = exp(4πi t mn)ϑ 2m 2n (z, Z).
See for example [START_REF] Mumford | Tata lectures on theta[END_REF]p. 123] for ( 6) and ( 7), and [Mum83, §II.3, Proposition 3.14] for (8).

Remark 6. According to identity (8), for all m, n ∈ 1 2 Z g , the function z → ϑ 2m 2n (z, Z) is even (resp. odd) if and only if 4 t mn ≡ 0 mod 2 (resp. 4 t mn ≡ 1 mod 2). There are exactly 2 2g order 2 theta characteristics, among which 2 g-1 (2 g + 1) are even and 2 g-1 (2 g -1) are odd functions (see [FK92, Corollary VI.1.5]).

Modular transformation formula.

The following describes how Z → ϑ 2m 2n (z, Z) transforms under the action of the symplectic group.

Theorem 10. Let m, n ∈ 1 2 Z g and M = ( α β γ δ ) ∈ Sp(2g, Z). For every (z, Z) ∈ C g × S g we define M(z, Z) = t (γZ + δ ) -1 z, (αZ + β )(γZ + δ ) -1 .
Then the following transformation formula holds:

ϑ 2m ′ 2n ′ M(z, Z) = ζ M exp πi t z(γZ + δ ) -1 γz det(γZ + δ ) 1 2 ϑ 2m 2n (z, Z) with m ′ n ′ = δ m -γn -β m + αn + 1 2 diag(γ t δ ) diag(α t β ) ,
where diag(N) = (N 11 , . . . , N gg ) for N ∈ M g (C), and where ζ M ∈ C * is a eighth root of the unity only depending on M.

See [Mum83, §II.5, pp. 189-197] for a proof.

Theta characteristics and hyperelliptic curves.

Let X be a genus g curve and Z ∈ S g a period matrix of X. Let m, n ∈ ( Z / 2Z ) g , from now on we will denote

ϑ m n (Z) = ϑ m n (0, Z).
The theta characteristics of Z are the values ϑ m n (Z). A theta characteristic is said to be even (resp. odd) if t mn is even (resp. odd). In particular, every odd theta characteristic is zero.

Remark 7. If ϑ m n (Z) is zero, then by ( 7), for all p, q ∈ (Z/2Z) g such that p, q ≡ 0 mod 2, the theta characteristic ϑ m+p n+q (Z) is also zero. The important role played by theta characteristics in the theory of hyperelliptic curves is illustrated by the following result, stated here in the specific case of genus 3.

Theorem 11. Let Z be a period matrix of a genus 3 curve X. If X is not hyperelliptic, then no even theta characteristic of Z is zero, and X is hyperelliptic if, and only if exactly one even theta characteristic of Z is zero.

The reader may refer to [Mum84, §IIIa.9, Theorem 9.1]. 4.2. Double cover associated to a curve in W 9 . In this section we describe the construction of a non ramified double cover of a translation surface in H (2) tiled by three parallelograms.

Construction. Let (X, ω) be a translation surface in the SL 2 (R)-orbit of the L-shaped surface tiled by three squares. Such a surface is geometrically defined by the quotient ( P / ∼ , dz) where P is the Euclidean hexagon obtained by assembling three copies of a parallelogram of unit area, as decribed in Figure 4.

( X, ω) ✲ ✛ ✲ ✛ t (X,
✎ ✗ ✎ ✗ ✎ ❞ ❞ ❞ ❞ ❞ ❞ ❞ ❞ ❞ ❞ ❞ ❞ FIGURE 4.

Double cover: identifications and Weierstrass points

In this representation, the six Weierstrass points of X correspond to:

• the cone-type singularity of angle 6π, represented by black dots in Figure 4;

• the center of the parallelograms and the middle points of the two horizontal and vertical sides, pairwise identified, of the two non-adjacent parallelograms, all represented by small circles in Figure 4. We construct the non-ramified double cover p : ( X, ω) → (X, ω) by assembling (X, ω) with its image by the symmetry whose center is one of the two Weierstrass points that is neither the singularity, nor the center of a parallelogram. More precisely, we can suppose that, up to a translation, the point denoted by P 1 in Figure 4 is 0 and we denote by -P the image of the polygon P by z → -z. Note that the choice of one or another of the two Weierstrass points P 1 or P 2 has no incidence on the construction, as is readily checked by reassembling the parallelograms. Then P ∪ (-P) is an Euclidean dodecagon and we consider the identifications by translation specified in Figure 4. The quotient P ∪ (-P) / ∼ , dz defines a staircase-shaped translation surface tiled by six squares, whose vertices are identified to two cone-type singularities of angle 6π, marked by black disks and squares in Figure 4. We thus obtain a genus 3 translation surface ( X, ω) such that ω has two double zeros on X.

Automorphisms. By a classical result of algebraic geometry, the algebraic curve X defined above is hyperelliptic, as a non-ramified cover of a genus 2 curve; its eight Weierstrass points are

• the two cone-type singularities;

• the centers of the parallelograms, represented by small circles in Figure 4. The central symmetry defines an involution that fixes exactly four points. These points are represented by small crosses in Figure 4 and correspond to:

• the center of symmetry of the assembling of parallelograms;

• the middle points of the two horizontal sides of the two central parallelograms;

• the middle points of the four vertical sides of the two parallelograms located at the extremities. The central symmetry then induces a non-hyperelliptic order 2 automorphism on X, which will be denoted by ψ 2 . The curve X then admits an equation of the form

w 2 = P(z) = (z 2 + 1)(z 2 -a 2 )(z 2 -b 2 )(z 2 -c 2 ) with ω = dz w + z 2 dz w ,
for which ψ 2 is defined by (z, w) → (-z, w).

Composing ψ 2 with the hyperelliptic involution h X : (z, w) → (z, -w) yields an extra involution τ : (z, w) → (-z, -w), that is fixed point free and such that X = X/ τ ; the covering map p : X → X is then given by (z, w) → (z 2 , zw). The algebraic curve X then admits the equation

y 2 = P(x) = x(x + 1)(x -a 2 )(x -b 2 )(x -c 2 ) with ω = 1 2 dx y + xdx y .
Furthermore, the curve X admits an order 3 automorphism defined as follows. Consider the affine transformation induced by rotation of angle π around each of the two cone-type singularities, operating by circular permutation of the three parallelograms belonging to a same diagonal row. Letting this transformation act three times is equivalent to replace ω byω. Now this operation leaves invariant the quadratic differential ω2 that induces the complex structure on X (see Remark 1). The automorphism thus defined on X is of order 3 and will be denoted by ψ 3 .

Homology basis and periods. Let X be a genus 2 algebraic curve defined by

y 2 = P(x) = x(x + 1)(x -a 2 )(x -b 2 )(x -c 2 ).
Let π : X → P 1 C be the projection (x, y) → x. Using the construction presented in p. 7 applied to

x 1 = -1, x 2 = 0, x 3 = a 2 , x 4 = b 2 and x 5 = c 2 , we obtain simple closed curves δ 1 , . . . , δ 6 in X, oriented such that the intersection numbers are (δ j • δ j+1 ) = 1 ( j mod 6), all others being zero, and such that ∑ 3 j=1 δ 2 j = 0 and ∑ 3 j=1 δ 2 j-1 = 0. Let X be the genus 3 hyperelliptic curve defined by ( 9)

w 2 = P(z) = (z 2 + 1)(z 2 -a 2 )(z 2 -b 2 )(z 2 -c 2 )
and let π : X → P 1 C be the projection (z, w) → z. We proceed the same way by applying the construction in p. 7 to

x 1 = -c, x 2 = -b, x 3 = -a, x 4 = i, x 5 = -i, x 6 = a, x 7 = b and x 8 = c.
We denote by γ 1 , . . . , γ 8 the obtained cycles, satisfying (γ j • γ j+1 ) = 1 ( j mod 8), all others being zero, such that ∑ 4 j=1 γ 2 j = 0 and ∑ 4 j=1 γ 2 j-1 = 0.

✻ ✻ ✲ α 1 β 1 β 2 α1 β1 β2 β3 α2 ❲ X X ✻ ✲ α3 ✲ α 2 ✲ ❄ ❄ FIGURE 5
. Homology bases for X and X Now let

(10) α1 = γ 1 , β1 = -γ 2 , α2 = γ 1 + γ 3 + γ 4 , β2 = -γ 4 , α3 = γ 7 , β3 = γ 6 and (11) α 1 = δ 1 -δ 6 , β 1 = δ 1 , α 2 = δ 4 , β 2 = δ 3 .
Then B = ( α1 , α2 , α3 , β1 , β2 , β3 ) and B = (α 1 , α 2 , β 1 , β 2 ) are symplectic bases for H 1 ( X, Z) and H 1 (X, Z) respectively, represented in Figure 5.

Remark 8. If a 2 , b 2 and c 2 are positive real numbers, then X is a real genus 2 M-curve and X is a real genus 3 curve with three real components: this case corresponds to the situation where the translation surfaces (X, ω) and ( X, ω) are tiled by rectangles. We can then specialize the above construction: we call δ 1 the pullback of [-1; 0] to X by π, δ 2 the pullback of [0; a 2 ] and so on. Since P is non-zero in the upper half-plane H that is simply connected, we can choose on H the determination of √ P that is real and positive on [-1; 0]. This determination can be extended to R and even to the strips below the open intervals bounded by the roots of P(x). Note that it will then be pure imaginary with positive imaginary part on ] -∞; -1], pure imaginary with negative imaginary part on [0; a], real and negative on [a; b], and so on (see also [START_REF] Silhol | The Schottky problem for real genus 3 M-curves[END_REF]Lemma 2.4]).

We construct in the same way the cycles γ j 's on X for j = 1, . . . , 8: the paths ε i are chosen as intervals in R ∪ {∞} for i / ∈ {3, 4, 5}, ε 4 as the interval from i to -i contained in the imaginary axis and ε 3 and ε 5 symmetric to the origin. The orientation is defined by the choice of a determination of √ P that is real and positive on [-b; -a].

The above construction allows to give an expression of the periods of X in terms of the periods of its double cover X.

Lemma 12. Let X be a genus 2 algebraic curve defined by an equation of the form

y 2 = P(x) = x(x + 1)(x -a 2 )(x -b 2 )(x -c 2 )
where a, b and c are complex numbers such that a 2 , b 2 and c 2 are distinct and different from 0 and -1, and X the double cover defined by

w 2 = P(z) = (z 2 + 1)(z 2 -a 2 )(z 2 -b 2 )(z 2 -c 2 ).
Then the period matrix Ẑ of X associated to the basis B of H 1 ( X, Z) has the form

Ẑ =   z 1 z 12 z 13 z 12 z 2 z 12 z 13 z 12 z 1  
and the period matrix Z of X associated to the basis B of H 1 (X, Z) is given by Z = 2z 2 2z 12 2z 12 z 1 + z 13 .

Proof. In order to make the calculation clear, the reader can consider the specialization of the construction of the homology basis B to the case where a, b and c are real numbers such that 0 < a < b < c, see Remark 8 above. The curve X admits the non-hyperelliptic order 2 automorphism defined by ψ 2 : (z, w) → (-z, w).

We note that ψ 2 * (γ 1 ) = -γ 7 as γ 1 and γ 7 have the same orientation, and that ψ 2 * (γ 2 ) = γ 6 since γ 2 and γ 6 have opposite orientations. Moreover, we have ψ 2 * (γ 3 ) = -γ 5 and ψ 2 * (γ 4 ) = -γ 4 . That yields to ψ 2 * ( α1 ) = -α3 , ψ 2 * ( α2 ) = -α2 and ψ 2 * ( α3 ) = -α1 and the same holds for the β j 's. We thus obtain the rational representation of ψ 2 in the symplectic basis B:

t M 2 =         0 0 -1 0 0 0 0 -1 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 -1 0 0 0 0 -1 0 0         = ρ Q (ψ 2 ).
Then the matrix Ẑ is stable under the action of the symplectic matrix M 2 , hence the expression for Ẑ. We denote by .

Noting that ψ 2 * ( ωk ) =ωk for k = 1 and 3, we deduce from the above calculation of M 2 that Â11 = Â31 and Â13 = Â33 . Also, since ψ 2 * ( ω2 ) = ω2 then we have Â12 = -Â32 and Â22 = 0. We obtain identical relations for B, thus we have:

 =   a 1 a 2 a 3 a 21 0 a 23 a 1 -a 2 a 3   and B =   b 1 b 2 b 3 b 21 0 b 23 b 1 -b 2 b 3   .
Then we get Ẑ = Â B-1 under the announced form, with

z 1 = 1 2 a 2 b 2 + 1 2 a 1 b 23 -a 3 b 21 b 1 b 23 -a 3 b 21 , z 12 = a 3 b 1 -a 1 b 3 b 1 b 23 -b 3 b 21 = 1 2 a 21 b 23 -a 23 b 21 b 1 b 23 -b 3 b 21 , z 13 = 1 2 a 1 b 23 -a 3 b 21 b 1 b 23 -b 3 b 21 - 1 2 a 2 b 2 , z 2 = a 23 b 1 -a 21 b 3 b 1 b 23 -b 3 b 21 .
The curve X is the quotient of X by the fixed point free involution τ : (z, w) → (-z, -w), which leaves invariant ω1 and ω3 on X. These differentials then induce holomorphic forms on X, which will be denoted by ω 1 and ω 2 respectively. Let

A = α j ω k j,k and B = β j ω k j,k .
Let p : X → X be the quotient map, then p sends α1 and α3 onto α 2 , and α2 onto α 1 , whereas β1 and β3 are sent onto β 2 , and β2 onto β 1 . Furthermore, the restriction of p is of degree 2 on β2 and bijective on each α j , β j for j = 2. We then have and a direct calculation yields the expression of the period matrix Z = AB -1 of the curve X.

4.3. Characterization of the cover in terms of the periods. This section is devoted to the proof of Theorem A. In the following, we deal with real M-curves in W 9 . This is only to fix ideas because the arguments used here work verbatim for the non-real case: the only difficulty lies in expressing the choice of arcs in P 1 (C) whose pullbacks will provide suitable cycles for the construction of the symplectic bases B and B. The method employed to exhibit the form of the period matrix and identify the corresponding theta characteristic is inspired from [Sil01, Theorem 5.5].

We first establish the expression of a period matrix of X when X is a real M-curve in W 9 .

Proposition 13. Let X be a real M-curve in the family W 9 . Let Ẑ be the period matrix of its double cover X associated to the symplectic basis B of H 1 ( X, Z) described above. Then Ẑ has the form

(12) Ẑ =   iy 1 1 2 iy 1 iy 13 1 2 iy 1 1 2 + 3 4 iy 1 -1 2 iy 13 1 2 iy 1 iy 13 1 2 iy 1 iy 1   with y 1 , y 13 ∈ R.
where z 1 , z 13 ∈ iR are defined by

z 1 = 4 3 a 3 b 1 -a 1 b 3 b 2 1 -b 2 3 z 13 = 1 3 a 3 b 1 -a 1 b 3 + 3(a 1 b 1 -a 3 b 3 ) b 2 1 -b 2 3
hence the expression for Ẑ.

We now compute the period matrix of the double cover of one particular real M-curve in the family W 9 . The natural choice consists in considering the curve defined by the translation surface tiled by three squares: by Proposition 1 this is the only non-singular curve in this family admitting a non-hyperelliptic automorphism. An equation of this curve was given in §3.2. Lemma 14. Let X 1 be the real M-curve defined by the 3-square-tiled translation surface, defined by the equation

y 2 = P(x) = x(x 2 -1)(x -a 2 ) x - 1 a 2 with a = 2 - √ 3
and let X1 be its double cover. Then the period matrix Ẑ1 of X1 associated to B is

Ẑ1 =   4 3 i 2 3 i 1 3 i 2 3 i 1 2 + 5 6 i 2 3 i 1 3 i 2 3 i 4 3 i   .
Proof. The curve X 1 admits an order 4 automorphism induced by x → 1/x. Its double cover X1 is defined by the equation

w 2 = P(z) = (z 4 -1)(z 2 -a 2 ) z 2 - 1 a 2
and ϕ lifts to an order 4 automorphism of X1 , denoted by ψ 4 and defined by

ψ 4 : (z, w) → 1 z , iw z 4 . It is readily checked that ψ 4 * dz w = i z 2 dz w and ψ 4 * z 2 dz w = i dz w .
We also note that ψ 4 * ( α1 ) = -β1 hence, with the notations defined in the proof of Proposition 13, we get B11 = -i Â13 and B13 = -i Â11 . A direct calculation then gives the period matrix Ẑ1 .

Remark 9. By Lemma 12, the period matrix of X 1 associated to the basis B is

Z 1 = 1 + 5 3 i 4 3 i 4 3 i 5 3 i .
As the hyperelliptic curve X1 has many automorphisms, the modular transformation formula will enable us to identify the only even theta characteristic that vanishes for the period matrix Ẑ1 .

Proposition 15. Let X 1 be the real M-curve defined by the 3-square-tiled translation surface, let Ẑ1 be the period matrix of its double cover X1 associated to B. Then

ϑ 1 1 1 1 0 1 ( Ẑ1 ) = 0.
Proof. Let M = ( α β γ δ ) ∈ Sp 6 (R) and p = diag(γ t δ ) and q = diag(α t β ).

Then, following Theorem 10, we have the following transformation formula for theta characteristics:

(14) ϑ 2m 2n M(Z) = φ (M, m, n, Z)ϑ 2m ′ 2n ′ (Z), with (15) m ′ n ′ = t α m -1 2 p + t γ n -1 2 q t β m -1 2 p + t δ n -1 2 q =: M m n
and where φ is a function, depending only on M, on m, n ∈ ( Z / 2Z ) 3 and on Z ∈ S 3 , that never vanishes. By construction, X1 is a genus 3 hyperelliptic curve, hence by Theorem 11, there exists a unique even theta characteristic m n = m 1 m 2 m 3 n 1 n 2 n 3 for which ϑ m n ( Ẑ1 ) = 0. Period matrices of the form (12) are stable under the action of the two symplectic matrices

M 2 =         0 0 -1 0 0 0 0 -1 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 -1 0 0 0 0 -1 0 0         = t ρ Q (ψ 2 ) and M 3 =         1 -2 1 0 1 0 1 -1 0 0 0 -1 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 -1 -2 0 0 0 1 1 1         = t ρ Q (ψ 3 ),
then by equation ( 14) and Remark 6, we must have M k m n ≡ m n (mod 2) for k = 2 and 3. As a consequence, we obtain m 1 ≡ m 2 ≡ m 3 (mod 2) n 1 ≡ m 2 ≡ n 3 (mod 2) which, among the even theta characteristics, reduces the possibilities to the following three: ϑ 0 0 0 0 1 0 , ϑ 1 1 1 1 0 1 and ϑ 0 0 0 0 0 0 the latter never being zero by Riemann theorem on theta divisor (for instance, see [FK92, Theorem VI.2.4]). Furthermore, the period matrix Ẑ1 is stable under the action of

M 4 =        
0 0 0 1 0 0 0 1 0 0 -1 0 0 0 0 0 0 1 -1 0 0 0 0 0 0 2 0 0 -1 0 0 0 -1 0 0 0

        = t ρ Q (ψ 4 ),
hence M 4 m n ≡ m n (mod 2), which provides the extra condition m 2 ≡ 1 (mod 2), hence the conclusion.

Now we can prove the theorem:

Proof of Theorem A. From now on we assume that the period matrices of the double covers are associated to the same symplectic basis B for H 1 ( X, Z), such as these matrices have the form presented in the theorem. The set of isomorphism classes of genus 2 curves defined by an equation of the form ( 16)

y 2 = Q s (x) = x(x + 1)(x -s 2 ) x -f 3 (s) 2 x -f 3 f 3 (s) 2 .
is connected, hence the associated family of curves also form a connected subset in the moduli space of genus 3 hyperelliptic curves. The map associating to the equation of such a double cover its period matrix is continuous, then the set of these matrices is connected. By Theorem 11, exactly one even theta characteristic ϑ m n vanishes for these period matrices. As the set of these period matrices is connected and the set of order 2 theta characteristics is discrete, the same theta characteristic will vanish for the whole family. Since by Proposition 15, the theta characteristic ϑ 1 1 1 1 0 1 is zero for one member of this family, then it also vanishes for the period matrix of any double cover of a curve in the family defined by equation (16). 4.4. Periods of real M-curves from periods of double covers. We now proceed to the proof of the main theorem.

Proof of Theorem B. Let (X 1 , ω 1 ) be the L-shaped translation surface tiled by three squares; if t is a real number such that t > 1, then let (X t , ω t ) = 1 0 0 t • (X 1 , ω 1 ). By Remark 5, Proposition 4 and Lemma 5, every (X t , ω t ) is defined by an equation of the form

y 2 = Q s (x) = x(x + 1)(x -s 2 ) x -f 3 f 3 (s) 2 x -f 3 (s) 2 (17) with ω t = λ s dx y + xdx y
for some real number s such that 0 < s < √ 3/3 and some constant λ s ∈ C * . We construct from equation (17) a symplectic basis B = ( α1 , α2 , β1 , β2 ) for H 1 (X t , Z) as described in p. 7: with the notations defined in Figure 6, the point P 1 corresponds to (-1, 0), P 2 to (0, 0), P 3 to s 2 , 0 , P 4 to f 3 f 3 (s)

2 , 0 , P 5 to f 3 (s) 2 , 0 and P 6 to the point at infinity. For all t ≥ 1, we choose the constant λ s such that β1 ω t = 1. For t being fixed, let η 1 = ω t and let η 2 be the unique holomorphic 1-form on X t such that β1 η 2 = 0 and β2 η 2 = 1. As X t is a real M-curve, and by construction of the homology Then the period matrix of X t associated to B is

Z t = t N(A t B -1 ) = 1 + i(

FIGURE 1 .

 1 FIGURE 1. The L-shaped translation surface tiled by three squares According to[START_REF]Teichmüller curves in genus two: discriminant and spin[END_REF] Example in §6], this family is the Teichmüller curve of discriminant 9 and, following McMullen's notation, will be denoted by W 9 (comprehensive overviews on Teichmüller curves can be found e.g. in[START_REF] Curtis | Billiards and Teichmüller curves on Hilbert modular surfaces[END_REF] and[START_REF] Lochak | On arithmetic curves in the moduli spaces of curves[END_REF]). We will sometimes denote by W M 9 the Teichmüller geodesic generated by the 3-square-tiled surface, that is, its orbit under the diagonal subgroup 1 {( t 0 0 t -1 )} t>0 . The 3-square-tiled surface admits an order 4 automorphism; period matrices of Riemann surfaces defined by such translation surfaces in H (2) were computed by R. Silhol in[START_REF]Genus 2 translation surfaces with an order 4 automorphism, The geometry of Riemann surfaces and abelian varieties[END_REF] §3]. We will first show the following.

  FIGURE 3. A L-shaped translation surface defines a real M-curve

  for the holomorphic 1-forms on X, and we introduce the matrices  =

  The basis B = (α 1 , α 2 , β 1 , β 2 ) for H 1 (X t , Z) defined in p. 16 is obtained from B by the change of basis given by the symplectic matrix Sp 4 (Z).

  ω)

						P 1
	t	t	t	P	t	t	t
	t	t	t	t	t	t
				P 2
	t	t		t	t
			✗	t			t
			t				t
		t	t			t	t
					❞
	t			t		t
		t			t

  Stretching the 3-square-tiled surface basis B, the period matrix A t B -1 must satisfy ℜe(A t B -1 ) = 0. A direct calculation then shows that there exist two unique real numbers y t and y ′ t such that

							t		t	❞	t
				(X 1 , ω 1 )			❞	β2 ✛	❞
	t			t	❞		t
		P 4	❞	P 1	❞	P 3	t	α2 ❄	t	✲ β1 ❞	t
	t			t	❞		t
						P 2	❞	❞	❞	(X t , ω t )
	❞	P 5	❞	❞	P 6				❄ α1
	t			t			t		t
		FIGURE 6. A t := α j η k j,k =	-it iy t -2it iy ′

t and B :=

  whose imaginary part is positive definite if, and only if y t > 2t/3. Hence, following the notations of Theorem A, we havez 1 = iy t z 13 = i(ty t /2)from which we deduce the expression of the period matrix associated to the basis B of the double cover Xt :

			2y t -t) iy t	iy t i(y t /2 + t)
	Ẑt =	  i(t -y t /2) iy t iy t /2	iy t /2 2 + i(y t -t/2) 1 iy t /2	i(t -y t /2) iy t /2 iy t



 .

Proof. Let f 3 be the Möbius transformation defined in §3.1, then X admits the equation

where a, b and c are real numbers satisfying the conditions of Lemma 5. The double cover X is defined by w 2 = (z 2 + 1)(z 2a 2 )(z 2b 2 )(z 2c 2 ), and f 3 induces an order 3 automorphism on X, denoted by ψ 3 , defined by

Let ( α j , β j ) ( j = 1, 2, 3), ( ω1 , ω2 , ω3 ) and Â, B be as in p. 16. From the proof of Lemma 12, we know that Â11 = Â31 , Â13 = Â33 , Â12 = -Â32 and Â22 = 0 on the one side, and B11 = B31 , B13 = B33 , B12 = -B32 and B22 = 0 on the other side.

A direct calculation gives

and, noting that

we find Â23 = 5 4 Â13 -3 4 Â11 + 1 2 B23 and in a similar way, we finally obtain B23 = 3 2 B11 -5 2 B13 .

We have shown that  and B have the form

with a 1 , a 3 ∈ iR and b 1 , b 3 ∈ R (by definition of the α j 's and β j 's for j = 1, 3 and Remark 8). Then the period matrix of X associated to the symplectic basis B for H 1 ( X, Z) is

Consider the symplectic basis obtained from B by the change of basis given by

then the period matrix associated to this new basis is

fo which we can prove, by the same arguments as those used in the proof of Theorem A, that the corresponding vanishing even theta characteristic is ϑ 1 1 1 0 0 0 . Developping, we get

hence equation (1).


G Vincenzi

G Deligeorgis

F Coccetti

M Dragoman

L Pierantoni

D Mencarelli

R Plana

Extending ballistic graphene FET lumped element models to diffusive devices

In this work, a modified, lumped element graphene field effect device model is presented. The model is based on the "Top-of-the-barrier" approach which is usually valid only for ballistic graphene nanotransistors. Proper modifications are introduced to extend the model's validity so that it accurately describes both ballistic and diffusive graphene devices. The model is compared to data already presented in the literature. It is shown that a good agreement is obtained for both nano-sized and large area graphene based channels. Accurate prediction of Drain current and transconductance for both cases is obtained.

Introduction

Graphene's electronic band structure has been the subject of theoretical investigation more than 60 years ago [START_REF] Wallace | The Band Theory of Graphite[END_REF][START_REF] Mcclure | Band Structure of Graphite and de Haas-van Alphen Effect[END_REF][START_REF] Slonczewski | Band Structure of Graphite[END_REF][START_REF] Neto | The electronic properties of graphene[END_REF]. However, the isolation of graphene in 2004 [START_REF] Novoselov | Electric field effect in atomically thin carbon films[END_REF] allowed measuring its properties and verified the expected high electron mobility and reduced scattering [START_REF] Novoselov | Two-dimensional gas of massless Dirac fermions in graphene[END_REF]. Quantum models [START_REF] Neto | The electronic properties of graphene[END_REF], [START_REF] Reich | Tight-binding description of graphene[END_REF] were readily accessible to the theoretical modeling of these properties at the time. These were also used for the treatment of Graphene Nanoribbon (GNR) devices [START_REF] Son | Energy Gaps in Graphene Nanoribbons[END_REF][START_REF] Yang | Quasiparticle Energies and Band Gaps in Graphene Nanoribbons[END_REF][START_REF] Mucciolo | Conductance quantization and transport gaps in disordered graphene nanoribbons[END_REF]. GNRs are graphene stripes with a width in the nanometer scale. In field effect transistors based on such systems (GNRFETs), the drain current has been computed in the ballistic limit through Tight Binding (TB) models, solved using Non-Equilibrium Green's Functions (NEGF) formalism [START_REF] Ouyang | Comparison of performance limits for carbon nanoribbon and carbon nanotube transistors[END_REF][START_REF] Fiori | Simulation of Graphene Nanoribbon Field-Effect Transistors[END_REF][START_REF] Ouyang | Analysis of ballistic monolayer and bilayer graphene field-effect transistors[END_REF] or the scattering matrix approach [START_REF] Mencarelli | Coherent carrier transport and scattering by lattice defects in single-and multibranch carbon nanoribbons[END_REF], [START_REF] Mencarelli | Scattering matrix approach to multichannel transport in many lead graphene nanoribbons[END_REF]. However, the severe computational load of such models -which scales with the dimensions of the device -limited their applicability to graphene device modeling. On the other hand, "Top-of-the-barrier" semiclassical models [START_REF] Rahman | Theory of ballistic nanotransistors[END_REF] have been applied to GNRFET for ballistic [START_REF] Ouyang | Comparison of performance limits for carbon nanoribbon and carbon nanotube transistors[END_REF], [START_REF] Liang | Performance Projections for Ballistic Graphene Nanoribbon Field-Effect Transistors[END_REF] and tunneling/thermionic [START_REF] Jiménez | A current-voltage model for Schottky-barrier graphene-based transistors[END_REF] carrier transport, bypassing the computational load limit. Unfortunately, the latter are valid only for shortchannel devices which limits their use.

Most experiments on graphene conductivity properties were based on Graphene FET (GFET) devices with gate length ranging from hundreds of nm to several microns. Graphene of those dimensions is called large-area graphene because such systems are characterized by carrier diffusion [START_REF] Chen | Intrinsic and extrinsic performance limits of graphene devices on SiO2[END_REF], [START_REF] Dorgan | Mobility and saturation velocity in graphene on SiO[sub 2[END_REF] rather than ballistic transport. Semiclassical models that are applied to GFET structures followed different approaches such as drift-diffusion semi-analytical models [START_REF] Meric | Current saturation in zero-bandgap, top-gated graphene field-effect transistors[END_REF][START_REF] Meric | RF performance of top-gated, zero-bandgap graphene field-effect transistors[END_REF][START_REF] Thiele | Modeling of graphene metal-oxide-semiconductor field-effect transistors with gapless large-area graphene channels[END_REF][START_REF] Thiele | Modeling of the steady state characteristics of large-area graphene field-effect transistors[END_REF], the virtual-source semi-empirical model [START_REF] Wang | Compact Virtual-Source Current-Voltage Model for Top-and Back-Gated Graphene Field-Effect Transistors[END_REF] and analytical models for both GFET [START_REF] Ryzhii | Device model for graphene bilayer field-effect transistor[END_REF] and GNRFET [START_REF] Ryzhii | High-frequency properties of a graphene nanoribbon field-effect transistor[END_REF], [START_REF] Ryzhii | Current-voltage characteristics of a graphene-nanoribbon fieldeffect transistor[END_REF]. The first two model types were validated using DC measurements of GFETs of different dimensions, but they failed to correctly predict the behavior of a ballistic GNR FET device. The last family instead has not been experimentally validated to our knowledge.

In this work a model that can correctly predict both ballistic as well as diffusive transport and thus describe all graphene field effect devices is presented and validated. It is based upon the "Top-of-thebarrier" model for ballistic graphene nanotransistors [START_REF] Rahman | Theory of ballistic nanotransistors[END_REF], [START_REF] Liang | Performance Projections for Ballistic Graphene Nanoribbon Field-Effect Transistors[END_REF], which is extended to correctly predict the behavior of long-channel diffusive graphene FETs. Section 2 presents the "Top-of-the-barrier" model and details the modifications necessary to extend its validity to large-area graphene. Section 3 presents model validation comparing modeling and experimental results for two FET devices selected from literature.

Theoretical model

"Top-of-the-barrier" models [START_REF] Rahman | Theory of ballistic nanotransistors[END_REF], [START_REF] Liang | Performance Projections for Ballistic Graphene Nanoribbon Field-Effect Transistors[END_REF] have been successfully used to model ballistic GFET's. The structure of a typical GNR FET is shown in Fig 1 . According to this approach, the drain current is given by: [START_REF] Wallace | The Band Theory of Graphite[END_REF] where the I n and I p correspond to electron and hole current respectively. A similar convention for all variables is followed throughout this work. In a ballistic nanotransistor, mobile carriers are injected directly from the contacts to the channel without undergoing an energy relaxation process into the channel's statistics [START_REF] Datta | Electronic Transport in Mesoscopic Systems[END_REF]. The net current is given by the difference of the injected electron and hole currents. The electron current is given by the Landauer equation of conduction [START_REF] Datta | Electronic Transport in Mesoscopic Systems[END_REF], [START_REF] Datta | Quantum Transport: from Atom to Transistor[END_REF]:

(2) where is the carrier velocity, D(E) is the density of states and f S and f D are the Fermi functions for the two contacts. As already described, for ballistic transport, injected carriers are described by the Fermi potential of their respective contact. The velocity of carriers , is calculated by the carrier dispersion relation of the channel material. In the case of a GNR, this is obtained as described in [START_REF] Liang | Performance Projections for Ballistic Graphene Nanoribbon Field-Effect Transistors[END_REF].

The hole current is computed in a similar fashion. Differentiating the dispersion relation and averaging over all available subbands and k-space yields the band velocity:

(3

)
where k is the wavenumber, e n (k) is the energy of the n-th sub-band of the material and N is the number of unit cells. In [START_REF] Rahman | Theory of ballistic nanotransistors[END_REF] the carrier velocity is expressed as function of energy; carriers are injected at the top of the barrier where their velocity is at their lowest and cannot accelerate along the channel. This underestimates their overall velocity in the Top of the barrier model. The use of an averaged value of vx as in Eq. ( 3) is then a more convenient choice for both accuracy and simplicity. The density of states D is analytically derived from the bandstructure [START_REF] Fang | Carrier Statistics and Quantum Capacitance of Graphene Sheets and Ribbons[END_REF]. To account for impurities and for the gate metal work function, the neutrality point is shifted by the Dirac Voltage not shown here. The Dirac voltage is a free parameter in the model [START_REF] Rahman | Theory of ballistic nanotransistors[END_REF]. Assuming the source contact potential is set as the reference, the drain potential is given by -qV ds , where q is the electron charge and V ds is the drain-source voltage difference. For this case, the expressions of the Fermi Dirac statistic f S for the source and f D for the drain become: [START_REF] Neto | The electronic properties of graphene[END_REF] where f FD is the Fermi-Dirac statistics equation.

To take into account the electrostatics of the problem, all the previous quantities are shifted in energy by the electrostatic potential U. This latter is the sum of two terms [START_REF] Datta | Quantum Transport: from Atom to Transistor[END_REF], the charge-less potential U L and the mobile charge potential U P . The expression of the total potential U is thus:

(5)
The electrostatic part is computed using a lumped element model. A capacitor network consisting of a gate capacitance C G , a source C S and drain C D capacitance [START_REF] Rahman | Theory of ballistic nanotransistors[END_REF], [START_REF] Datta | Quantum Transport: from Atom to Transistor[END_REF] are taken into account. For simplicity it is assumed that C S =C D . The source and drain capacitance is a fit parameter [START_REF] Rahman | Theory of ballistic nanotransistors[END_REF]. Taking into account that V S =0, the expression of the Laplace potential U L becomes: [START_REF] Novoselov | Two-dimensional gas of massless Dirac fermions in graphene[END_REF] U P is calculated using the linearized Poisson equation, where the potential is proportional to the charge unbalance in the channel [START_REF] Datta | Electronic Transport in Mesoscopic Systems[END_REF]. For ambipolar graphene it is: [START_REF] Reich | Tight-binding description of graphene[END_REF] where C Σ is the sum of the three capacitors described above, N and N 0 are the number of mobile and fixed electrons in the channel respectively [START_REF] Datta | Quantum Transport: from Atom to Transistor[END_REF], P and P 0 are the number of holes. The inter-dependence between potential and charge concentration is solved self-consistently. As described in [START_REF] Rahman | Theory of ballistic nanotransistors[END_REF], the mobile charge concentration is computed taking the average of the two Fermi statistics that correspond to the contacts as these are described by eq.( 4). Equations ( 5), ( 7) and ( 8) are iteratively computed. Finally equations ( 3) and ( 4) are used to compute the current for a given bias taking into account equations ( 2) and (1).

The model described so far, is unable to correctly account for carrier scattering. Thus it breaks down for large channel length where diffusion transport becomes dominant. In order to extend the validity of this model to large-area GFET, we propose the following modification.

Assuming the distribution of scatterers is uniform across the channel, the transmission probability T of ballistic carriers scales inversely with gate length [START_REF] Datta | Electronic Transport in Mesoscopic Systems[END_REF]. When the gate length is small compared to the mean free path, i.e. T~1, carriers that originate from the source contact are all described by the source pseudo-Fermi potential. Respectively those that originate from the drain are all described from the drain pseudo-Fermi potential. As the gate length increases, the value of T declines towards 0. The phenomenon is better understood looking into the carriers that move in the channel. The portion of carriers that ballistically traverse the channel decrease in number, while it is assumed that those that are scattered regain the thermodynamic equilibrium. In this work the elastic and inelastic scattering processes are considered to be intimately related, avoiding any effort to conceptually divide those two processes. Each elastic scattering event is considered to be followed by thermalization, which is thought as inevitably present as assumed in [START_REF] Datta | Electronic Transport in Mesoscopic Systems[END_REF]. Hence carriers that have scattered should be described by the channel Fermi level. a fraction T that is transmitted and a fraction (1-T) that is scattered. To restore the apparent reduction of carriers introduced into the ballistic model by using the T factor, a modified, effective pseudo-Fermi potential ϕ is defined. As described in [START_REF] Datta | Electronic Transport in Mesoscopic Systems[END_REF], ϕ is the weighted average of the contact pseudo-Fermi and the channel Fermi level. The weight factor is merely the transmission probability T. This way, all carriers originating from the drain contact are described by the effective level ϕ. A parameter λ related to the mean free path is introduced to describe the transmission factor T. The relation between T and the length coordinate x [START_REF] Datta | Electronic Transport in Mesoscopic Systems[END_REF] is given by: [START_REF] Yang | Quasiparticle Energies and Band Gaps in Graphene Nanoribbons[END_REF] In order to obtain a value for ϕ that can be used in a lumped model the average of T over the entire length of the channel is used:

(10)
The level ϕ depends linearly from the parameter λ , and scales inversely with the gate length. This modification to the "Top-of-the-barrier" models allows simulating GFETs with gate lengths larger than the mean fee path. In the case of short channel compared to the mean free path, this model reduces to the "top of the barrier" representation.

Model Validation

The proposed model is validated against two devices presented in the literature. A narrow channel graphene nanoribbon transistor described in [START_REF] Fiori | Simulation of Graphene Nanoribbon Field-Effect Transistors[END_REF] and a large area wide channel graphene FET described in [START_REF] Meric | Current saturation in zero-bandgap, top-gated graphene field-effect transistors[END_REF]. The two devices are shortly presented in Table 1.

The first device simulated is a GNR nanotransistor [START_REF] Fiori | Simulation of Graphene Nanoribbon Field-Effect Transistors[END_REF] with gate length L=15 nm, based upon a semiconducting nanoribbon. For clarity, the device cross section is sketched in Fig. 1. The parameters used by the model are calculated using the procedure described in Section 2. The optimum results are obtained for a λ value of 21 nm. The current flowing through the device for two distinct drain voltages is depicted in Fig 3 .   The quality of the agreement to the simulated currents from [START_REF] Fiori | Simulation of Graphene Nanoribbon Field-Effect Transistors[END_REF] is excellent for the low drain bias curve, while it's sub-optimum for the high drain bias one. The cause of this discrepancy is that for ballistic devices the act of referring the source contact to the 0 eV potential throughout the transistor's dynamic range is a condition too strong. After [START_REF] Rahman | Theory of ballistic nanotransistors[END_REF], the solution to this issue is to empirically identify a reference potential for each bias point. This procedure has not been endorsed by the authors of this work, in order to contain the model complexity at the expense of its accuracy. Moreover, the limited presence of scattering in the simulation of this ballistic transistor has been proven to be not a concern to the model accuracy. Finally, table 2 shows an excellent agreement of simulated g m with that presented in [START_REF] Fiori | Simulation of Graphene Nanoribbon Field-Effect Transistors[END_REF].

The second device is a large-area graphene FET [START_REF] Meric | Current saturation in zero-bandgap, top-gated graphene field-effect transistors[END_REF] with gate length L=3 µm and width W=2.1 µm (see Table 1). The structure of the FET (shown in Figure 4) is slightly more complex because of the back-gate electrode. This can be taken into account by adding a backgate capacitance term in equation ( 7) which thus becomes: [START_REF] Ouyang | Comparison of performance limits for carbon nanoribbon and carbon nanotube transistors[END_REF] where V b is the potential of the back-gate electrode and V g is the potential of the top-gate electrode. The values of the capacitors C G and C B have been computed using a finite elements simulator. The use of large-area graphene also implies the use of the related density of states equation, as reported in [START_REF] Tan | Measurement of Scattering Rate and Minimum Conductivity in Graphene[END_REF].

The parameters for the large-area graphene case remain the same compared to the GNR model, only the procedure to extract them is different. First, the Dirac voltage is empirically found by matching the position of the minimum conductivity point in the I d (V gs ) transfer characteristic. According to the discussion in reference [START_REF] Meric | Current saturation in zero-bandgap, top-gated graphene field-effect transistors[END_REF], the gate voltage -which corresponds to minimum conduction -is V gs =2.38V thus the computed Dirac voltage has a value of V Dirac =0.213 V.

The λ parameter is calculated using the I d (V ds ) for low V DS . In effect, the λ parameter will have a pronounced effect on the onset of the saturation effects in the I d (V gs ) curve. In this case the calculated optimum λ=380nm which is consistent with the value found in literature [START_REF] Tan | Measurement of Scattering Rate and Minimum Conductivity in Graphene[END_REF].

Finally, the value of the average carrier velocity is found empirically by matching the magnitude of the current. The best match was obtained at =3.25e4 ms -1 . A direct comparison with literature is not possible, since drift-diffusion models use a saturation model, where the drift velocity is an empirical function of the longitudinal electric field [START_REF] Khakifirooz | A Simple Semiempirical Short-Channel MOSFET Current-Voltage Model Continuous Across All Regions of Operation and Employing Only Physical Parameters[END_REF]. The value of carrier velocity used in this model is however consistent with the drift velocity that may be computed in the literature for similar structures [START_REF] Meric | Current saturation in zero-bandgap, top-gated graphene field-effect transistors[END_REF], [START_REF] Thiele | Modeling of graphene metal-oxide-semiconductor field-effect transistors with gapless large-area graphene channels[END_REF].

The output current I d (V sd ) is plotted in Fig. 5. The points correspond to DC current measurement taken from ref [START_REF] Meric | Current saturation in zero-bandgap, top-gated graphene field-effect transistors[END_REF], whereas the family of solid lines is the predicted values using this model. It should be pointed out that although the model is based on a lumped element approach, unlike models described in ref [START_REF] Meric | Current saturation in zero-bandgap, top-gated graphene field-effect transistors[END_REF], [START_REF] Thiele | Modeling of graphene metal-oxide-semiconductor field-effect transistors with gapless large-area graphene channels[END_REF] and [START_REF] Thiele | Modeling of the steady state characteristics of large-area graphene field-effect transistors[END_REF], it is able to correctly predict the presence of saturation effects in the currentvoltage characteristics of the device. To our knowledge the only lumped element model presented so far with ability to predict the saturation and second linear characteristics of GFET's was based on different equations for each region. Finally, Table3 presents the peak measured and simulated g m for two different gate biases. The agreement is not excellent but this could be attributed to the g m calculation method used in [START_REF] Meric | Current saturation in zero-bandgap, top-gated graphene field-effect transistors[END_REF] to remove the effect of the contact on the device behavior.

In summary, the described model uses four parameters to fit the experimental results. Those are presented in Table 4, for the two simulated devices, with the exception of the velocity of FET1 which is analytically derived using (3).

Conclusions

A simple modification to the "top of the barrier" model that enables accurate simulation of a broad range of graphene based transistors was presented. The model retains the simplicity of a lumped element approach and is able to correctly describe the I-V characteristics of both ballistic and diffusive devices. Furthermore, it is able to correctly predict the behavior of both large-area as well as graphene nanoribbon based field effect devices. Finally the presented model achieves a new level of generic graphene device modeling with a simplicity that makes it easier to use compared to complex detailed models based on NEGF or Tight Binding thus it is a favorable solution for use in graphene enabled circuit simulation tools.

( 8 )

 8 Fig 2(a) depicts the channel under the injection of carriers from the source and the drain contact, the corresponding pseudo-Fermi levels are labeled "S" and "D" respectively. The source injects carriers -in this case holes -which are described by its pseudo-Fermi potential and move to the right; similarly, the drain injects carriers moving to the left.

Fig 2 (

 2 Fig 2(b) shows a simplified case in which the charge scattering process breaks the drain flux in two parts: a fraction T that is transmitted and a fraction (1-T) that is scattered. To restore the apparent reduction of carriers introduced into the ballistic model by using the T factor, a modified, effective pseudo-Fermi potential ϕ is defined. As described in[START_REF] Datta | Electronic Transport in Mesoscopic Systems[END_REF], ϕ is the weighted average of the contact pseudo-Fermi and the channel Fermi level. The weight factor is merely the transmission probability T. This way, all carriers

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