Cécile Hilpert

Louis Beranger

Paulo C T Souza

Petteri A Vainikka

Vincent Nieto

Siewert J Marrink

Luca Monticelli

Guillaume Launay
email: guillaume.launay@ibcp.fr

Facilitating CG simulations with MAD: the MArtini Database Server

The MArtini Database (MAD -www.mad.ibcp.fr) is a web server designed for the sharing structures and topologies of molecules parameterized with the Martini coarsegrained (CG) force field. MAD can also convert atomistic structures into CG structures and prepare complex systems (including proteins, lipids etc.) for molecular dynamics (MD) simulations at the CG level. It is dedicated to the generation of input files for Martini 3, the most recent version of this popular CG force field. Specifically, the MAD server currently includes tools to submit or retrieve CG models of a wide range of molecules (lipids, carbohydrates, nanoparticles, etc.), transform atomistic protein structures into CG structures and topologies, with fine control on the process and assemble biomolecules into large systems and deliver all files necessary to start simulations in the GROMACS MD engine.

Introduction

Coarse-grained (CG) force fields allow simulations of macromolecular systems on time and length scales beyond reach for atomistic descriptions. During the past two decades coarsegraining has become a popular solution for the study of a large variety of biological problems [START_REF] Kmiecik | Coarse-Grained Protein Models and Their Applications[END_REF]2 as well as in materials science. [START_REF] Chen | Multiscale modeling and simulations of responsive polymers[END_REF] This created the need for tools to facilitate the preparation and analysis of CG system. These tools are often provided as web services with graphical interfaces that require no installation by the user, while guiding him/her in the choice of the parameters and the validation of the results. Currently available web services are meeting various needs such as: supporting the generation of multiple or specific CG force field parameters, preparing the necessary files for molecular dynamics (MD) simulations, performing quick equilibration of systems or even running MD simulations for simple CG models. [START_REF] Hospital | MDWeb 15 and MDMoby: an integrated web-based platform for molecular dynamics simulations[END_REF][START_REF] Jo | CHARMM-GUI: A web-based graphical user interface for CHARMM[END_REF][START_REF] Kuriata | CABS-flex 2.0: a web server for fast simulations of flexibility of protein structures[END_REF][START_REF] Feig | Local Protein Structure Refinement via Molecular Dynamics Simulations with locPREFMD[END_REF][START_REF] Heo | PREFMD: a web server for protein structure refinement via molecular dynamics simulations[END_REF][START_REF] Czaplewski | UNRES server for physicsbased coarse-grained simulations and prediction of protein structure, dynamics and thermodynamics[END_REF][START_REF] Noel | SMOG 2: A Versatile Software Package for Generating Structure-Based Models[END_REF][START_REF] Kimmett | ProBLM web server: protein and membrane placement and orientation package[END_REF][START_REF] Schneider | Hybrid MM/CG Webserver: Automatic Set Up of Molecular Mechanics/Coarse-Grained Simulations for Human G Protein-Coupled Receptor/Ligand Complexes[END_REF][START_REF] Margreitter | Vienna-PTM web server: a toolkit for MD simulations of protein post-translational modifications[END_REF][START_REF] Damre | MERMAID: dedicated web server to prepare and run coarse-grained membrane protein dynamics[END_REF][START_REF] Marchetto | CGMD Platform: Integrated Web Servers for the Preparation, Running, and Analysis of Coarse-Grained Molecular Dynamics Simulations[END_REF] The Martini force field is one of the most popular choices among the coarse-grained force fields available, [START_REF] Marrink | The MARTINI force field: coarse grained model for biomolecular simulations[END_REF][START_REF] Souza | Martini 3: a general purpose force field for coarse-grained molecular dynamics[END_REF] and offers the possibility to describe molecular interactions in systems containing lipids, [START_REF] Marrink | The MARTINI force field: coarse grained model for biomolecular simulations[END_REF][START_REF] López | Force Field Parameters for Glycolipids[END_REF][START_REF] Wassenaar | Computational Lipidomics with insane: A Versatile Tool for Generating Custom Membranes for Molecular Simulations[END_REF] proteins, [START_REF] Monticelli | The MARTINI coarse-grained force field: extension to proteins[END_REF][START_REF] De Jong | Improved Parameters for the Martini Coarse-Grained Protein Force Field[END_REF] carbohydrates, [START_REF] López | Martini Coarse-Grained Force Field: Extension to Carbohydrates[END_REF][START_REF] Shivgan | Extending the Martini Coarse-Grained Force Field to N-Glycans[END_REF] nucleic acids, [START_REF] Uusitalo | Martini Coarse-Grained Force Field: Extension to DNA[END_REF][START_REF] Uusitalo | Martini Coarse-Grained Force Field: Extension to RNA[END_REF] polymers, [START_REF] Rossi | A MARTINI Coarse-Grained Model of a Thermoset Polyester Coating[END_REF][START_REF] Rossi | Coarse-graining polymers with the MARTINI force-field: polystyrene as a benchmark case[END_REF][START_REF] Grunewald | Transferable MARTINI Model of Poly(ethylene Oxide)[END_REF] nanoparticles, [START_REF] Wong-Ekkabut | Computer simulation study of fullerene translocation through lipid membranes[END_REF][START_REF] Baoukina | Interaction of Pristine and Functionalized Carbon Nanotubes with Lipid Membranes[END_REF] and other molecular systems, recently reviewed in. [START_REF] Marrink | Two decades of Martini: Better beads, broader scope[END_REF][START_REF] Alessandri | The Martini Model in Materials Science[END_REF] With the release of Martini 3, 17 many new systems and applications were also on reach of the model, including drug-like small-molecules, [START_REF] Souza | Protein-ligand binding with the coarsegrained Martini model[END_REF][START_REF] Alessandri | Martini 3 Coarse-Grained Force Field: Small Molecules[END_REF] ionic liquids, [START_REF] Vazquez-Salazar | Martini coarse-grained models of imidazolium-based ionic liquids: from nanostructural organization to liquid-liquid extraction[END_REF] deep eutectic solvents [START_REF] Vainikka | Martini 3 Coarse-Grained Model for Type III Deep Eutectic Solvents: Thermodynamic, Structural, and Extraction Properties[END_REF] and poly-electrolyte coacervates. [START_REF] Tsanai | Coacervate formation studied by explicit solvent coarse-grain molecular dynamics with the Martini model[END_REF] Specific tools for the Martini community already exist, [START_REF] Wassenaar | Computational Lipidomics with insane: A Versatile Tool for Generating Custom Membranes for Molecular Simulations[END_REF][START_REF] De Jong | Improved Parameters for the Martini Coarse-Grained Protein Force Field[END_REF][START_REF] Pezeshkian | Backmapping triangulated surfaces to coarse-grained membrane models[END_REF][START_REF] Vickery | CG2AT2: an Enhanced Fragment-Based Approach for Serial Multi-scale Molecular Dynamics Simulations[END_REF][START_REF] Grünewald | a python suite for facilitating simulations of macromolecules and nanomaterials[END_REF] but they are only partially covering the needs of the users, in particular the available web services. The CGMD/MERMAID webserver [START_REF] Damre | MERMAID: dedicated web server to prepare and run coarse-grained membrane protein dynamics[END_REF][START_REF] Marchetto | CGMD Platform: Integrated Web Servers for the Preparation, Running, and Analysis of Coarse-Grained Molecular Dynamics Simulations[END_REF] is designed the for simulation of membrane or soluble proteins but is currently limited to version 2 of the Martini force field. The CHARMM-GUI server [START_REF] Qi | CHARMM-GUI Martini Maker for Coarse-Grained Simulations with the Martini Force Field[END_REF][START_REF] Hsu | CHARMM-GUI Martini Maker for modeling and simulation of complex bacterial membranes with lipopolysaccharides[END_REF] allows for the preparation of MD input files for versions 2 and 3 of Martini, but offers limited possibility to edit the models.

The CG representation of a molecule in Martini is obtained by combining molecular fragments representing well-defined chemical moieties, modeled by particles or beads. A total of 843 particle types are currently available in Martini 3. 17 Such large number of 2 different particle types can represent with high specificity the polarity, size, and hydrogen bonding capabilities of the building blocks they represent. Determining the most appropriate set of beads and their associated bonded parameters is a critical step in the preparation of a CG molecule. This so-called "parameterization" step often requires expert knowledge of the molecule chemistry associated with a validation protocol to ensure that thermodynamic properties can still be reproduced under the CG model. Thus the creation of accurate CG representation of molecules, while more than ever accessible, remains a daunting crafting task for most users. Although many automatic parameterization protocols have been recently developed, [START_REF] Bereau | Automated Parametrization of the Coarse-Grained Martini Force Field for Small Organic Molecules[END_REF][START_REF] Graham | PyCGTOOL: Automated Generation of Coarse-Grained Molecular Dynamics Models from Atomistic Trajectories[END_REF][START_REF] Empereur-Mot | Swarm-CG: Automatic Parametrization of Bonded Terms in MARTINI-Based Coarse-Grained Models of Simple to Complex Molecules via Fuzzy Self-Tuning Particle Swarm Optimization[END_REF][START_REF] Potter | Automated Coarse-Grained Mapping Algorithm for the Martini Force Field and Benchmarks for Membrane-Water Partitioning[END_REF] there is still a need of an extended and curated database of models that can be used as reference to access their accuracy or even be used for the calibration of such approaches. Furthermore, as the set of Martini models is growing and diversifying , the need of CG molecule repositories is becoming critical resource for the long term availability of molecular models and to make simulations reproducible.

In this context, the MAD server aims at making the set up a CG simulation with Martini accessible to the wide Martini users community, by providing the resources to obtain the CG molecules and preparing the entire system for MD simulation. To achieve this goal, the MAD server is extending the capabilities of the original Martini molecule repository (http://cgmartini.nl/) in three directions. The MAD:Database is storing a large collection of CG molecules readily available for MD simulations. The database is organized in a userfriendly way with modern content browsing and viewing capabilities. For macromolecules such as proteins, not present in the database, the MAD:Molecule Builder tool can produce coarse grained models based on its uploaded all-atom coordinates. Finally, the MAD:System Builder assemble many CG molecules in a simulation box and delivers all the files necessary to start the MD simulation.

Because both MAD:Molecule Builder and MAD:System Builder services make heavy usage of the server computational resources, a login is required to use them. Logged-in user are also granted access to a private storage and a job history with resuming capabilities.

Material and Methods

Overall Organization

The MAD ecosystem can be depicted as a database of CG molecules and CG tools connected together with computational resources and private storage for the user (see Fig. 1). The main point of access to the MAD web server is its welcome page (see Fig. 2), where a left menu hosts links to directly explore the database, access the tools (Molecule Builder and System Builder) or download the supported force-field files. The lower-half of the menu is for users to manage their account: profile, molecule contributed to the database and submitted jobs.

Figure 1: Workflow of the web server. CG models of molecules can be retrieved from or contributed to the MAD:Database, which covers a range of molecule types and force field versions. Alternatively, all-atom structures can be submitted to the Molecule Builder with control over the CG process by the martinize2 program. Every user is granted a private storage which holds a copy of all user generated models. The System Builder interface can use files from the main database and from the user private storage. Submitted structures are processed by insane to obtain the full set of requires files to run MD.

In fact, every registered MAD user is granted a private history module, accessible from this menu under the "my builder history" icon. Here, all his/her previous CG operations 4 performed on the MAD server are reported to the user. Each job is labeled by its input, date of creation, the version of the targeted force field and the type of the operation. From the history panel, the user can visualize a model and eventually resume its modifications. Download links of the corresponding CG files are also available. Models can be deleted from this panel; in any case, their data will not be conserved over 15 days on the MAD file system.

Figure 2: The MAD web server startup page is made of two sections. The left section is a menu dedicated to resources access and user controls. The right side is a table of the currently available CG molecules.

The MAD server is built on a front-end to back-end architecture. The front-end, which is based on the React web component framework, carries most of the steps for the submission, validation, visualisation, and edition of structures (depicted in cyan in Fig. 1). The back-end, which is based on a NodeJS/Express platform, interacts with a slurm scheduler to launch GROMACS, 47 martinize2 and insane 19 jobs on the MAD cluster. The public database of molecule is operated by the NoSQL Couch SGDB. All the molecular visualizations are performed by the NGL [START_REF] Rose | NGL Viewer: a web application for molecular visualization[END_REF] JavaScript library.

5

Database

The database is designed to contain a wide variety of CG molecules which can be expanded through uploaded contributions by users. Each entry of the database corresponds to a particular association of molecule and force field version. The information of an entry is stored as a specific collection of CG files: topology files (.top and itp extension) and coordinate files (gro extension). Currently, GROMACS [START_REF] Abraham | GROMACS: High performance molecular simulations through multi-level parallelism from laptops to supercomputers[END_REF] is the main supported MD engine, but with possible future implementation of Martini in codes as OpenMM [START_REF] Eastman | OpenMM 7: Rapid development of high performance algorithms for molecular dynamics[END_REF] and NAMD, 50 more file formats may be supported by MAD. Force-field conversion tools [START_REF] Rusu | MDWiZ: a platform for the automated translation of molecular dynamics simulations[END_REF][START_REF] Shirts | Lessons learned from comparing molecular dynamics engines on the SAMPL5 dataset[END_REF][53] between different MD engines may be consider by the users, but features of Martini models as virtual sites may not be simply adapted in other codes.

The welcome page of the database (mad.ibcp.fr/explore) displays a top section dedicated to the custom search and a bottom section which gives direct access to the currently stored molecule in table form. Each table row links to a single molecule description page. Alternatively, molecules can be searched in the database with the top section formula of the welcome page. Here, molecules can be searched by force field name (different Martini versions are available), creation methods (manually or by different tools), biochemical category (e.g. carbohydrates, lipids, etc.) and free text searches within name, alias, or whole entry.

The search and the table browsing methods will both link to the MAD description page (Figure 3) of a molecule. The MAD molecule pages are made of four sections. The first one is the General information section which displays the alias, name and category of the molecule along with all the comments section extracted from the corresponding itp file. The comments can provide useful information such as the command line arguments used to generate the CG files. The top-right section of a MAD molecule page is an interactive CG molecular view of the compound, where each sphere is a Martini bead with color and size representing the bead types (Figure 3B). The Details section displays references for the molecules and a download link to bundled topology and structure files.

Molecule Builder

The MAD:Molecule Builder tool generates the CG structure and topology from an all-atom structure. It is currently used to coarse grain proteins, but other types of polymers will be accepted in the future. The Molecule Builder is built on top of our local installation of the vermouth package (https://github.com/marrink-lab/vermouth-martinize) and the martinize2 program. It guides the user in the choice of the input parameters and provides handy edition/post-processing capabilities of the output CG structures. The interface of the MAD:Molecule Builder is centered around an interactive molecular representation of the user molecule. This interactive viewer is accompanied by a left-ended panel which provides the set of control commands appropriate to each molecular building stage.

As a first step, the user uploads the all-atom structure (in PDB format) to be processed.

The uploaded structure is displayed in the molecular viewer, along with CG settings in the 7 [START_REF] Periole | Combining an Elastic Network With a Coarse-Grained Molecular Force Field: Structure, Dynamics, and Intermolecular Recognition[END_REF] and GOMartini. [START_REF] Poma | Combining the MARTINI and Structure-Based Coarse-Grained Approaches for the Molecular Dynamics Studies of Conformational Transitions in Proteins[END_REF] The classic mode will only use pairwise beads bonds, while elastic and Go model options respectively apply additional harmonic and Lennard-Jones potentials. The Go model contact map follows the approach of Wo lek et al., [START_REF] Wo Lek | Determination of contact maps in proteins: A combination of structural and chemical approaches[END_REF] which has been implemented by Moreira et al. [START_REF] Moreira | All atom simulations snapshots and contact maps analysis scripts for SARS-CoV-2002 and SARS-CoV-2 spike proteins with and without ACE2 enzyme[END_REF] The N-terminal and C-terminal fixes modify the protein terminal particles, to improve representations of functional groups charges and geometries, and are activated by default.

The user may activate the "side chain fix", which promotes protein stability and increases the reliability of the structures during MD by the addition of dihedral angles between SC1-BB-BB-SC1 beads; this provides more realistic side chain orientations. 58 The "cystein bridges" options controls the automatic detection of cysteine residues and the application of covalent bonds between cysteine side-chains when the distance between the sulfur atoms is below a threshold of 0.216 nm. In case of long-running computations, the email toggle option makes it possible to be contacted by email upon MAD:Molecule Builder job completion. The email will enclose a link to access and visualize the data, which are privately stored on the server for a period of 15 days.

In case of a job failure, the user would have access to the appropriate logs. Most failures are caused by improper values in the uploaded PDB file, which can easily be traced and fixed using the provided failure logging information.

Once the structure has been coarse-grained, the user can inspect it and, most importantly, can interactively modify the set of constraints that martinize2 created based on the chosen value of the Mode option. This set of constraints defines a network of elastic bond (EB), a mean to preserve the protein structure (intra-domain) while promoting molecular motions of interests (domain-domain interactions). By default, the martinize2 software will deduce an initial network of EB from the all-atom structure. This network often needs to be edited based on (experimental) information on protein structure and dynamics. The EB editor of the MAD:Molecular Builder greatly facilitates this editing process. The creation/deletion of an EB requires a simple click on two beads to add or delete an elastic bond; more sophisticated modifications are achieved by the usage of a selection language to build all EB at once between two selected groups of beads. The MAD:Molecular Builder automatically encodes the EB network in the appropriate CG files. All modifications to the EB network can be reverted through the history section of the MAD:Molecular Builder.

Any stage of molecular editing in the MAD:Molecule Builder can be privately stashed as alternative model into the user private history; the user can access them later to resume the modifications or download the structure and topology files.

System Builder

The MAD:System Builder can combine several CG structures to create large systems ready for use in MD simulations. The following types of structures are allowed: models from the MAD:Database, CG molecules from the user private stash, topology and structure files uploaded by the user.

The first type of system that the MAD:System Builder can produce are phospholipid bilayers in water solution. The builder allows to configure the lipid types and ratio, including the possibility of different compositions for the upper and lower leaflet. Salt concentration and total charge can be set by the user, and polarizable water can be used.

A macromolecule, usually a protein, in water solution constitutes the second type of system available within the MAD:System Builder. The macromolecule files (GRO, TOP, ITP) can be uploaded or imported from the database of compounds or from the user private stash. If no lipids are added during the setup, the resulting simulation box will comprise one 10 instance of the provided macromolecule in a box of water molecules.

The MAD:System Builder can also setup CG MD files for system made of a protein embedded in a lipid bilayer surrounded by water molecules. Similarly to the previous cases, the user is guided through dedicated interface panels to define the protein, lipids and solvent of the system.

The MAD:Molecule Builder makes use of the insane software, [START_REF] Wassenaar | Computational Lipidomics with insane: A Versatile Tool for Generating Custom Membranes for Molecular Simulations[END_REF] a powerful tool for the setup of large macro molecular systems in a simulation box. The MAD:System Builder assists the user with the of insane to deliver the appropriate simulation box. For all three types of systems, the MAD:System Builder features additional common and advanced controls with default values applicable to most cases. Advanced users may override several parameters if so needed: a useful common settings being the geometry of the simulation box. Caution should be exercised when modifications are made to the advanced settings as they could impair the simulation.

Following the computation of the system on the MAD cluster, a view of the computed simulation box is provided. Here, the left-hand panel features visualization options and a download link to the files required to start the simulation (Figure 4). tors that is dedicated to nanoparticles and certain polymers is also included (called Martini 2.2 with CNP). [START_REF] Wong-Ekkabut | Computer simulation study of fullerene translocation through lipid membranes[END_REF][START_REF] Baoukina | Interaction of Pristine and Functionalized Carbon Nanotubes with Lipid Membranes[END_REF][START_REF] Monticelli | On Atomistic and Coarse-Grained Models for C60 Fullerene[END_REF] Because the database is open to submission by users, these numbers are expected to change over time, with Martini 3 CG counts progressively exceeding 2.

Results and discussion

Case study

The elastic network approach consists of a set of harmonic potential added on top of the Martini model to conserve the tertiary structure of proteins. [START_REF] Periole | Combining an Elastic Network With a Coarse-Grained Molecular Force Field: Structure, Dynamics, and Intermolecular Recognition[END_REF] The network is fully dependent on the pdb structure used as reference, with the number of bonds defined by the upper and lower distance cutoff MAD:Molecule Builder parameters. The rigidity of the protein model is defined by the number of EB and by the force constant used. Optimal parameters for the elastic network depend of the studied protein system. It is recommended to use experimental or atomistic simulation data to calibrate the parameters of the elastic network. To illustrate 12 the interest of the elastic network tool in MAD:Molecule Builder, here we show how to build models of T4 Lysozyme, a protein from the bacteriophage T4 (pdb code 181L). [START_REF] Morton | Specificity of ligand binding in a buried nonpolar cavity of T4 lysozyme: Linkage of dynamics and structural plasticity[END_REF] Once the structure is open in the MAD:Molecule Builder, the force field is set to "martini3001".

With "Elastic" activated, the default setup provides for a force constant of 700 kJ/(mol.nm2), the lower and upper cutoffs at 0.5 and 0.9 nm respectively. Finally, neutral termini, auto assignment of cysteine bridges and side-chain fix are applied (Figure 5A). Upon completion of the CG process, the resulting model is displayed (Figure 5B) using the same beads color scales as the database of compounds (Figure 3B). The automatic construction of elastic network from the all-atom structure can lead to artefact EB between CG beads. Within the MAD:Molecule Builder, visual inspection of the EB network of T4 Lysozyme protein superimposed onto its CG model facilitates the identification of such a case. Indeed, an abnormal EB is found to be present between T21 and Q141 (Figure 5C). As the two amino acids are distant in structure, their EB may greatly impair the flexibility of the CG model.

To better prepare the model for subsequent MD simulations it recommended to delete the bond by directly clicking to select and remove (Figure 5C). Alternatively, rigidity can be added to the model in the N-terminal subdomain as exemplified in Figure 5D) where all possible EB between two amino acids selections: D10 (violet color) and D20-T21 (yellow color), are created in a single click.

Conclusion and perspectives

We presented here the MAD server, a new web resource dedicated to the preparation of MD systems with the Martini coarse-grained force field. The MAD server provides a large collection of CG molecules ready to be used. Newly parameterized and published molecules can be uploaded to the MAD:Database. For molecules not yet published, topology and structure files can be provided from the user's computer. Alternatively, all-atom structures can be coarse-grained by the MAD:Molecule Builder. In a final step, CG molecules from 14 any of these sources can be combined by the MAD:System Builder to produce topology and structure files MD-ready. We strongly encourage users to contribute to the repository by uploading their favorite models. For the new users, a tutorial is available at https://mad. ibcp.fr/tutorial/. We plan on expanding the capabilities of the MAD server through the integration of algorithms useful to the MARTINI users community , as Polyply. [START_REF] Grünewald | a python suite for facilitating simulations of macromolecules and nanomaterials[END_REF]

  At the bottom of the page, the Versions section displays the trees version tracking of all 6 the available models of the molecule. Each of these trees corresponds to a specific version of the Martini force field: only force field trees with at least one model in the database are shown. Within a tree, a node represents the specific model (and files) of a molecule stored in the database. The node of the currently displayed database entry is colored in red, the others node in the tree are links to the web page of different models of the same molecule. The children of a molecular model are models which were declared as being derived from this parent model. By clicking on the edit button, the user can effectively submit a new "child" version of the model currently displayed. The database is open, i.e. any registered user can submit a new CG model. Submission requires the Martini files (itp, top) and at least one reference to a publication describing the derivation of the parameters. Submitted molecules will go through a quick curation process by martini developers. If the molecule is new to the database a new entry will be created. In cases where previous versions would exist, the newly submitted version will be added to the corresponding position (molecule type and force field number) in the version tracking tree.

Figure 3 :

 3 Figure 3: A: MAD page of dioleoylphosphatidylcholine (DOPC); B: particle color code used in MAD; the color scale broadly follows particle polarity, with red and blue colors for charged particles.

  Database contentMAD Database currently comprises a total of 283 CG molecules belonging to the following categories: carbohydrates, polymers, amino acids, lipids, ions, phytochemical, solvents, surfactants and synthetic nanoparticles. The largest category is the lipids with currently 213 entries. Molecular entries in the database can be available for the following force fields: Martini 3.001, Martini 2.2 (for protein classic and elnedyn versions) and polarized Martini 2 versions (called 2.2P). The specific Martini 2 version developed by Monticelli and collabora-11

Figure 4 :

 4 Figure 4: A view on the System Builder results: phospholipid bilayer and embed protein(cyan) displayed in a water-filled simulation box.

Figure 5 :

 5 Figure 5: Sequential views of the process of coarse-graining the bacteriophage T4 Lysozyme (pdb code 181L): from the initial settings of parameters on the all-atom structure (A) to the visualization of the beads at a 0.6 scale the CG structure (B), the direct (C) selection of an EB to be removed and (D) the query based selection of amino acids to through newly created EB. Green bonds on panels C and D do not represent covalent bonds but EB between CG beads.

Acknowledgement

We thank all external users for testing MAD during its development. Luca Monticelli thanks the French National Institute of Health and Medical Research (INSERM) for the support. Paulo C. T. Souza acknowledges the supported by French National Center for Scientific Research (CNRS). Further funding of Paulo C. T. Souza and Luca Monticelli came from a research collaboration with PharmCADD. Siewert J. Marrink acknowledges funding from the European Research Council via an Advanced grant Advanced Grant (COMP-O-CELL).


F Jouve
email: florent.jouve@math.u-bordeaux1.fr

THE GEOMETRY OF THE THIRD MOMENT OF EXPONENTIAL SUMS

We give a geometric interpretation (and we deduce an explicit formula) for two types of exponential sums, one of which is the third moment of Kloosterman sums over Fq of type K(ν 2 ; q). We establish a connection between the sums considered and the number of Fq-rational points on explicit smooth projective surfaces, one of which is a K3 surface, whereas the other is a smooth cubic surface. As a consequence, we obtain, applying Grothendieck-Lefschetz theory, a generalized formula for the third moment of Kloosterman sums rst investigated by D. H. and E. Lehmer in the 60's .

Introduction

The problem of estimating exponential sums over nite elds is a quite standard issue in analytic number theory. Indeed, it arises in classical questions such as determining the Fourier coecients of cusp forms or resolving the Waring problem via the circle method. As a consequence of the proof, by Deligne, of the Riemann Hypothesis for varieties over nite elds, and, to an even wider extent, of its vast generalization described in [?], a deep and ecient understanding of fairly general types of such sums was derived (see e.g. [?]). That algebro-geometric method has now become the standard way to estimate exponential sums over nite elds.

In our present work, however, instead of investigating the properties of the -adic sheaves (as the standard attack would suggest) associated to the exponential sum over F q (with q not a power of ) we consider, we exhibit an algebraic surface whose number of F q -rational points is explicitly related to the sum.

Let us rst make precise what are the sums involved. Let p be a prime distinct from 2 and 3, q a power of p and ϕ a nontrivial additive character of F q : ϕ : F q → C × .

For any couple of integers (α, β) such that α -1 = 3(β -1), we consider the sum S(a α , a β ; q) = x∈Fq ϕ(a α x 3 + a β x) , with parameter a ∈ F q .

In [?], Birch was the rst to consider moments of those sums (in the case q = p). In loc. cit. he conjectured that a modular interpretation should exist for a certain type of such moments. In [?] that conjecture was precised by Atkin and Livné nally proved it in [?]. Our rst object of study is what we call the third Birch sum B 3 (q) = a∈Fq S(a α , a β ; q) 3 , and is nothing but the third moment of S(a α , a β ; q) (we point out the fact that a straightforward computation yields the precise value of the rst and second moment of the sums S(a α , a β ; q)).

To evaluate B 3 (q), we establish an explicit relation between the number of F q -rational points on a certain smooth cubic surface and the value of B 3 (q). The arithmetic of such surfaces is well understood thanks to the work of Swinnerton-Dyer who showed that the number of F q -rational points on a smooth projective cubic surface S is entirely determined by the action of the Frobenius morphism on the set of 27 lines lying on S (see [?]). Via that geometric interpretation, we deduce an explicit formula for the value of B 3 (q): Theorem 1. The quantity B 3 (q)/q is precisely the number of F q -rational points on an ane surface, the projective completion of which is a smooth cubic surface. More precisely,

• If p ≡ 1 (mod 3), then there exist integers A p ≡ 1 (mod 3) and B p satisfying 4p = A 2 p + 27B 2 p and such that, if we let ε = 1 if 4 is a cube modulo p and ε = -1 otherwise, we get B 3 (q)q -1 = q 2 + (2 + 2χ q (-1) + ζ δr 6 + ζ6 δr )q -λ r 1 -λ r 2 ,

where q = p r , χ q denotes the Legendre character of F q , ζ 6 and ζ6 are the primitive 6th roots of 1 in C, λ 1 and λ 2 are the complex numbers such that λ 1 λ 2 = p, λ 1 +λ 2 = -A p et δ = (3-χ p (-1)(2ε+1))/2.

• If p ≡ 2 (mod 3), then B 3 (q)q -1 = q 2 + (3 + (-1) r + 2χ q (-1))q -λ r 1 -λ r 2 ,

where λ 1 and λ 2 are the complex numbers such that λ 1 λ 2 = p, λ 1 + λ 2 = 0.

We emphasize the fact that the formula obtained gives the exact value of the sum B 3 (q). We notice here that the investigation of k-th moments for k greater or equal to 4 would surely require other techniques than the study of the variety we would attach to it. Indeed, in the case where k = 4, the variety arising, when using orthogonality relations to simplify the expression of B 4 (q), is 3-dimensional, and there is no a priori analogue of the result of Swinnerton-Dyer (i.e. a totally explicit way to compute the number of F q -rational points) in dimension greater than 2.

Nevertheless, the method we describe enables us to evaluate other types of sums as well. Consider, for instance, the famous Kloosterman sums with parameter λ ∈ F q : K(λ; q) = x∈F × q ϕ(x + λx -1 ) ,

where ϕ still denotes a non-trivial additive character of F q . The moments of order 1, 2 and 3 of those sums were computed by Salié in [?]; however, D. H. and E. Lehmer were the rst to raise the question of the value of the n-th moment indexed by the squares of F q :

σ n (q) = λ∈Fq K(λ 2 ; q) n .

As in the previous case, the computation of the moments σ 1 (q) and σ 2 (q) are straightforward. In [?], the authors obtain an explicit formula for the third moment in the special case where q = p. Their method, though elementary, requires lots of tricks in the transformations of the sums considered.

Noticing that the formula proved in [?] uses the decomposition p = a 2 + 3b 2 for primes p ≡ 1 (mod 3) gives us the intuition that there must exist a geometric interpretation for σ 3 (q) involving the elliptic curve E with Weierstrass model y 2 = x 3 + 1 (or a quadratic twist of that curve). We show that such a link actually exists: Theorem 2. Assume that p = 2, 3. There exits a K3 surface dened over F q and isomorphic, over F q 2 , to the Kummer surface Km(E × E), such that the number of F q -rational points on that surface is explicitly related to the value of σ 3 (q). Moreover, we have the exact formula

σ 3 (q) = ε r q 2 + q(2qχ q (-1) + χ q (-1)(λ r 1 + λ r 2 ) + 2)
where

• if p ≡ -1 (mod 6), then ε = -1 and λ 1 = p, λ 2 = -p,
• if p ≡ 1 (mod 6), then ε = 1 and there exist integers a and b such that p = a 2 + 3b 2 . In that case λ 1 and λ 2 are dened as the reciprocal roots of the polynomial p 2 T 2 -(4a 2 -2p)T + 1.

The Kummer surface Km(E × E) is the smooth model of the surface E × E blown up in its 16 points of order 2 (se [?, page 170]). Thus, the number of F q -rational points on such a surface is explicitly related to |E(F q )|. In the Theorem above, the surface attached to σ 3 (q) is isomorphic to Km(E × E) only when considered as a surface over a eld containing a primitive cube root of 1. That explains why we are naturally led to distinguish the case p ≡ 1 (mod 6) from the case p ≡ -1 (mod 6).

To prove the theorems stated above, we need to compute the number of F q -rational points on two smooth projective surfaces (the rst being a cubic surface and the second being a K3 surface).

To do so we will need standard facts about Zeta functions of varieties over nite elds. We briey review the terminology and results that will be helpful in what follows.

Recall that the Zeta function of a smooth projective variety X over F q is dened by the formal power series

Z(X/F q ; T ) = exp n 1 |X(F q n )| T n n .
Thanks to Dwork's theorem, or, more usefully for us, to the Grothendieck-Lefschetz trace formula (see [?]), we know that this function is rational, and, in the case where X is 2-dimensional:

Z(X/F q ; T ) = P 1 P 3 P 0 P 2 P 4 ,

where

P i = det(1 -T Fr * |H i (X ⊗ F q , Q )), =
p is a prime number and Fr * is induced on the -adic cohomology groups by the geometric Frobenius on X ⊗ F q (base change corresponding to the extension of scalars to a separable closure F q of F q ).

For brevity the -adic cohomology group H i (X ⊗ F q , Q ) will be denoted H i X . For each of the two cases we consider, we need to compute the exact value of the eigenvalues of Fr * acting on the spaces H i X , for 0 i 4. Indeed, to obtain an exact formula for B 3 (q) and σ 3 (q), it is not enough to evaluate both the dimension of the H i X and the modulus of the eigenvalues (which we obtain directly by invoking Deligne's Riemann Hypothesis for varieties over nite elds).

The particular structure of the surfaces appearing in our study will be very helpful in order to obtain such a precise information. Let us recall briey a few facts about surfaces and more specically about K3 surfaces and smooth cubic surfaces.

The classication of algebraic surfaces is much more complex than for algebraic curves. Lots of dierent invariants are involved and we will not give denitions (but only references) for all of these objects as the precise understanding of what they are is not required for the proof of the theorems above. Let us rst state a few facts about smooth cubic surfaces: they are Del Pezzo surfaces of degree 3 (see [?, page 401]) and, in the Enriques-Kodaira classication of surfaces (see [?, page 188]), they appear as surfaces with type 1, that is to say rational minimal surfaces. In particular, the Betti number b

1 = dim H 1 (X, Z) is zero if X is a smooth cubic surface.
Moreover, it is a standard fact that on such a surface lie exactly 27 lines. Computing equations for those lines will be a crucial step in evaluating B 3 (q). Indeed, the cycle classes of these lines span the -adic cohomology space H 2 X , which, concretely, corresponds to the fact that the Galois action on those lines entirely determines |X(F q )| (see [?]). Now we turn to K3 surfaces (surfaces of type 7 in the Enriques-Kodaira classication of [?, Table 10 page 188]), an instance of which will appear explicitly when evaluating σ 3 (q). Such a surface is dened as a geometrically connected surface with trivial canonical sheaf (see [?, page 180]) and Betti number b 1 equal to zero (notice the helpful common point with smooth cubic surfaces). The F. JOUVE arithmetic of such varieties is the object of many recent studies, the fact that they can be seen as 2-dimensional analogues of elliptic curves being quite motivating.

The few information we have just given is enough to give an a priori form for the Zeta function of X/F q if X is either a smooth cubic surface or a K3 surface. Indeed, in both cases, H 1 X = 0, so, by Poincaré duality (see [?]), we have H 3 X = 0 as well. To obtain the expression of the factors P 0 and P 4 of the denominator of Z(X/F q ; T ), we exploit the fact that, X being geometrically irreducible, H 0 X has dimension 1 with a trivial Galois action. In other words H 0 X Q as a Galois module. By Poincaré duality again, we deduce that H 4 X Q (-4), the one dimensional Galois module on which the global geometric Frobenius acts by multiplication by q 2 . From these standard observations we get, for X a smooth cubic surface or a K3 surface, (1)

Z(X/F q ; T ) = 1 (1 -T )(1 -q 2 T )P 2 (T )
,

where the remaining crucial polynomial P 2 satises P 2 (T ) ∈ Z[T ] and P 2 (0) = 1.

To evaluate B 3 (q) and σ 3 (q) we will have to compute the action of Frobenius on H 2 X for a suitable surface X. To do so, we need to exhibit, in each case, the surface involved. To begin with, we discuss the case of B 3 (q).

Evaluation of a Birch sum

Let us recall rst the notations dened in the introduction above: p is a prime number distinct from 2 and 3, q = p r for some integer r 1 and ϕ is an additive character of F q . For simplicity, we consider the sum with parameter a ∈ F q : S(a 4 , a 2 ; q) = x∈Fq ϕ(a 4 x 3 + a 2 x) ; but we keep in mind the fact that the couple of exponents (4, 2) could be replaced by any (α, β) such that α -1 = 3(β -1).

The sum we wish to evaluate is the Birch sum B 3 (q) = a∈Fq S(a 4 , a 2 ; q) 3 .

To prove Theorem 1, the rst step consists in making explicit the smooth cubic surface related to B 3 (q). A straightforward calculation yields B 3 (q) = q 3 + a∈F × q x,y,z∈Fq ϕ(a(x 3 + y 3 + z 3 + x + y + z)) .

That expression naturally leads us to consider the ane cubic surface S dened by the equation:

S : f (x, y, z) = x 3 + y 3 + z 3 + x + y + z = 0 .
Applying orthogonality relations, we deduce B 3 (q) = q|S(F q )| .

We are now reduced to determining the number of F q -rational points on the ane surface S.

In order to feel as comfortable as possible with that problem, we prefer to work with a smooth projective model of S. Indeed, such a setting is particularly well-suited to use the properties of the -adic cohomology groups (where is a prime dierent from p) attached to the variety (see the introduction). Also, working with a smooth projective surface enables us to use the beautiful result of [?] in order to count F q -rational points.

So let S denote the projective compactication of S. In homogeneous coordinates (x : y : z : w), an equation for S can be given by: S : x 3 + y 3 + z 3 + w 2 (x + y + z) = 0 .

A direct application of the Jacobi criterion shows that S is smooth. Hence, it suces to evaluate the number of F q -rational points on the smooth projective cubic surface S to obtain the value of B 3 (q). Indeed, the contribution of points at innity in S(F q ) can be explicitly described as follows:

Proposition 3. The F q -rational points of the complement of the surface S in its projective compactication S are precisely the F q -rational points of the projective curve

C : x 3 + y 3 + z 3 = 0 .
The curve C/F q is an elliptic curve and the number of F q -rational points of C is given by

|C(F q )| = q + 1 -λ r 1 -λ r 2 ,
where λ 1 λ 2 = p and

• if p ≡ 1 (mod 3), then A p = -(λ 1 + λ 2 ) satises 4p = A 2 p + 27B 2 p (for a certain integer B p ) and A p ≡ 1 (mod 3), • if p ≡ 2 (mod 3), then A p = -(λ 1 + λ 2 ) = 0.
Proof. The curve C is a nonsingular cubic curve dened over F q ; (0 : -1 : 1) being a F q -rational point on C, we deduce that C/F q is an elliptic curve. The number of F q -rational points on C is therefore given by (see [?, page 302]):

|C(F q )| = q + 1 -λ r 1 -λ r 2 ,
where λ 1 , λ 2 are the reciprocal roots of the polynomial P (T ) = 1 + A p T + pT 2 , where A p = 0 if p ≡ 2 (mod 3) and, if p ≡ 1 (mod 3), there exists an integer B p such that A p is the unique integer being congruent to 1 modulo 3 and satisfying 4p = A 2 p + 27B 2 p (see [?, Chap. 8.3]).

2.1. The Zeta function of S/F q . To begin with, let us recall briey the link between the set of lines on S and the cohomology space H 2 e S

. We refer the reader to [?, Chap. 4] for the general theory of cubic surfaces. As mentioned in the introduction, smooth cubic surfaces are del Pezzo surfaces of degree 3. In particular (see [?,Th. 24.4]) any such surface can be realized as the blow up of the projective plane P 2 in 6 points, provided they do not all lie on the same conic and that no three of them lie on the same straight line. The surface we obtain contains 27 so called exceptional curves (one for each of the 6 points, one for each of the 15 lines joining two of these points and one for each of the 6 conics passing through 5 of these 6 points). These are precisely the 27 lines on the smooth cubic surface and the cycle classes of these lines generate the Picard group of the surface (see [?, Th. 26.2(i)]). Now, another standard fact about smooth cubic surfaces over nite elds is that their geometric Picard group is isomorphic, as a Galois module, to the -adic cohomology space H 2 attached to the surface once the scalars are extended to a separable closure of the base eld (this can be seen as a way of stating Weil's Theorem [?, Th. 23.1]). Moreover, from a general formula (obtained, e.g. by combining Lemma 24.3.1 and Theorem 24.5 of [?]) valid for any del Pezzo surface we get that the rank of the Picard group for a smooth cubic surface is 7.

In the present context, this implies that

| S(F q )| = q 2 + 7 i=1 η i q + 1 ,
where the η i are roots of unity. The theorem of Swinnerton-Dyer ([?, Table 1]) gives a correspondence between the value of 7 i=1 η i and the type of decomposition in Galois orbits of the 27 lines lying on S.

2.2. The twenty seven lines on S. We now have to nd equations for these 27 lines, so that we can compute the action of Frobenius on that set of lines. Some of them are easy to nd: D z : (t : -t : 0 : w), D z,±i : (t : -t : ±i : w) , D y : (t : 0 : -t : w), D y,±i : (t : ±i : -t : w) ,

D x : (0 : t : -t : w), D x,±i : (±i : t : -t : w) ,
are examples of lines lying on S. However 18 of them remain to be found. To do so, we exploit the method described in [?]. The key observation on which that method is based is that if, by a change of coordinates, we succesively send D x , D y and D z on the z-axis and if we consider, after each transformation, the family of planes with parameter λ, P λ : y = λx, then the intersection of P λ with S is the union of the z-axis and a conic. That conic degenerates in two lines when the parameter λ takes appropriate values. These values correspond to roots of a polynomial Q with degree less than 5.

Applying that method after sending D x on the z-axis, the polynomial we get is

Q(λ) = (λ + 1)(4λ 3 + 1)
.

The root λ = -1 gives us D z and D y ; two lines which we have already found. If λ is a root of Q x (t) = 4t 3 + 1, we nd six new lines:

(λt : 1 2 (t - 2 √ -3λ + 3 3 w) : 1 2 (t + 2 √ -3λ + 3 3 w) : w) , (λt : 1 2 (t + 2 √ -3λ + 3 3 w) : 1 2 (t - 2 √ -3λ + 3 3 w) : w) .
Then sending D y on the z-axis, the polynomial Q(λ) is given by

Q(λ) = λ(λ + 1)(λ 3 + 4) .
For λ = -1 we recover D x and D z , and, for λ = 0, the lines D y,±i . However, if λ is a root of Q y (t) = t 3 + 4, we nd six new lines:

(t(λ + 2 λ 2 ) - √ 3 √ λ + 1 3 w) : t : 2 λ 2 (-t + √ 3 √ λ + 1 3 w) : w) , (t(λ + 2 λ 2 ) + √ 3 √ λ + 1 3 w) : t : 2 λ 2 (-t - √ 3 √ λ + 1 3 w) : w) .
Finally, sending D z on the z-axis, we obtain

Q(λ) = λQ z (λ) ,
where Q z (λ) = 3λ 3 -12λ 2 + 12λ -4. For λ = 0, we get the (already found) lines D z,±i . If λ is a root of Q z , we obtain the six remaining lines:

(t(1 - 1 k(λ) ) + iw k(λ) : ( 1 k(λ) + λ -1)t - i k(λ) w : (1 -λ)t : w) , (t(1 - 1 k(λ) ) - iw k(λ) : ( 1 k(λ) + λ -1)t + i k(λ) w : (1 -λ)t : w) ,
where k(λ) = 3λ + (3/2)λ 2 .

Remark 1. Seeing S as a surface dened over Q, the computations we have just performed show that the smallest Galois extension of Q over which the 27 lines on S are dened is Q(i, ω, 2 1/3 ), where i, ω, are complex numbers, respectively roots of the polynomials X 2 + 1 and X 2 + X + 1.

We can now compute the Galois action on the set of these 27 lines. Using [?, Table 1], we will then be able to deduce the value of | S(F q )|, and eventually, thanks to the arguments given above, to give an exact formula for B 3 (q). We note that it suces, in the computation of the Galois action, to consider S as a surface over the prime eld F p and to determine the eigenvalues of the Frobenius acting on the -adic cohomology of degree 2 of that surface. The eigenvalues for the second cohomology group of S/F p r are then obtained by raising those corresponding to S/F p to the r-th power.

First, if we denote by σ * (following [?]) the restriction of the Frobenius morphism to the set of 27 lines on S, then σ * xes the three lines D x , D y and D z . Let us investigate the action of σ * on the other lines.

If p ≡ 2 (mod 3):

• either p ≡ 1 (mod 4) and then the 6 lines D x,±i , D y,±i and D z,±i as well as the 6 lines corresponding to the roots of Q z , are dened over F p . These 12 lines are exchanged pairwise. We deduce that σ * is an element of the conjugacy class C16 of [?, Table 1].

• or p ≡ 3 (mod 4) and then the 6 lines D x,±i , D y,±i and D z,±i form 3 couples of coplanar lines exchanged pairwise. Moreover, as in the previous case, the roots of Q z give rise to a couple of coplanar lines exchanged pairwise. The 12 remaining lines form 6 couples of skew lines exchanged pairwise. We deduce that σ * is an element of the conjugacy class denoted C17.

If p ≡ 1 (mod 3): either 4 is a cube modulo p and

• either p ≡ 1 (mod 4) and then the 27 lines are dened over F p (and so are all xed by σ * ).

Thus the permutation σ * is an element of the classe C1.

• or p ≡ 3 (mod 4) and the lines which are dierent from D x , D y and D z split in 12 couples of coplanar lines exchanged pairwise. Thus σ * is an element of the class C3.

or 4 is not a cube modulo p and • either p ≡ 1 (mod 4) and then the 6 lines D x,±i , D y,±i and D z,±i are dened over F p . Each of the polynomials Q x , Q y and Q z gives rise to 2 triples of cyclicly permuted lines. Thus, the permutation σ * is an element of the class C6.

• or p ≡ 3 (mod 4) and then D x,±i , D y,±i and D z,±i form 3 couples of lines permuted pairwise and each of the polynomials Q x , Q y and Q z corresponds to an orbit of 6 lines permuted cyclicly. We deduce that σ * is an element of the class C7.

Remark 2. If X is a smooth cubic surface, it is a standard fact that the 27 lines on X can be described through a unique algebraic equation (see [?,Chap. 4] • if p ≡ 2 (mod 3): 1 with multiplicity 4, (-1) r with multiplicity 1, χ q (-1) (where χ q denotes the Legendre character of F q ) with multiplicity 2, F. JOUVE

• if p ≡ 1 (mod 3): 1 with multiplicity 3, χ q (-1) with multiplicity 2, ζ δr 6 with multiplicity 1 and ζ6 δr with multiplicity 1 (using the same notations as in Theorem ??).

We deduce the number of F q -rational points on S:

• if p ≡ 2 (mod 3):

| S(F q )| = q 2 + (4 + (-1) r + 2χ q (-1))q + 1 ,

• if p ≡ 1 (mod 3):

| S(F q )| = q 2 + (3 + 2χ q (-1) + ζ δr 6 + ζ6 δr )q + 1 .

Combining that formula with Proposition ?? and the fact that B 3 (q) = q|S(F q )|, we nally deduce Theorem ??.

The third moment of Kloosterman sums

We work with the same notations and assumptions on q, p, and ϕ as in Section ??. In this section, we are interested in the family of Kloosterman sums with parameter λ dened in the introduction:

K(λ; q) = x∈F × q ϕ(x + λx -1 ) .
Recall that the group morphism

ψ : F q → C × x → e
Tr Fq/Fp (x) p

where e(z) = exp(2iπz), is a non-trivial additive character of F q . Moreover it is well known that every additive character of F q is of the form ψ a : x → ψ(ax), for some a ∈ F q . For µ ∈ F × q , an easy calculation yields K(λµ 2 ; q) = x∈F × q ψ µ (x + λx -1 ) , so varying µ (i.e. the character) for xed λ amounts to varying λ modulo the nonzero squares of F q . In the case where q = p we can also remark that K(λ; p) and K(λµ 2 ; p) are Galois conjugate in the cyclotomic eld Q(e 2iπ/p ). Thus two subfamilies of those sums naturally emerge. Denoting by {1, η} a set of representatives of F × q modulo nonzero squares, these subfamilies give rise to the moments (using notations following [?]):

σ n (q) = ν∈Fq (K(ν 2 ; q)) n and σ n (q) = ν∈Fq (K(ην 2 ; q)) n .

We notice rst that σ n (q) + σ n (q) = 2 λ∈Fq K(λ; q) n and, generalizing Salié's formulae (see [?] and [?, Section 4.4]), we obtain (for p = 2) σ 1 (q) = -σ 1 (q) = χ q (-1)q , σ 2 (q) = q 2 -2q , σ 2 (q) = q 2 , σ 3 (q) + σ 3 (q) = 2(χ q (-3)q 2 + 2q)

where χ q still denotes the Legendre character of F q .

D. H. and E. Lehmer (see also [?] for a simplied proof involving the same type of arguments)

give, in [?], an explicit formula for σ 3 (p). Although the elementary (but very clever) arguments they use would surely yield the exact formula of Theorem ?? for any σ 3 (q) (q being a prime power), most interesting in our method is the geometric interpretation we give for the fact that the decomposition p = a 2 + 3b 2 (provided p ≡ 1 (mod 3)) appears in the formula of [?]. That decompostion, indeed, is strongly related to the so called a p coecient of the CM elliptic curve E/Q with Weierstrass model y 2 = x 3 + 1 (or a quadratic twist of that curve).

Remark 3. It is natural to ask for what can be done to evaluate the fourth moment of Kloosterman sums σ 4 (q). The same easy calculation as the one performed in the next section shows that this problem is equivalent to determining the number of F q -rational points on the threefold given by

Z : x + x -1 + y + y -1 + z + z -1 + t + t -1 = 0 .
In [?], the author, rst resolving the singularities of that variety and then using the Faltings-Serre criterion, shows that the number of F p -rational points on Z is explicitly related to the coecient of the expansion of the cusp form η 4 (2z)η 4 (4z).

In [?], the authors also give an expression relating the number of F p -rational points of Z to the sum of the number of F p -rational points on the Legendre elliptic curves

E λ 2 : y 2 = x(x -1)(x -λ 2 ) ,
where λ ∈ F p \ {0, ±1}.

Another point of view, in order to try and nd the value of σ n (q) for any n greater than 3 would be to apply the standard techniques from algebraic geometry we referred to in the introduction.

A theorem of Deligne (see [?, Th. 4.1.1]) asserts that there exists a lisse -adic sheaf (on G m ) of rank 2 denoted Kl and called Kloosterman sheaf such that the action of the local Frobenius (in any a ∈ F q ) on the bre of Kl over a geometric point ā lying over a satises Tr(Fr a,q | Kl ā) = -K(a; q) .

In our case, those sheaves can be seen as Galois representations over the eld of -adic numbers Q . If we want to use Galois representations to express the moment σ n (q) as well, and if we want those representations to have a dimension that grows as slowly as possible with n, we just need to consider the elevation to the second power [2] : G m → G m and the n-th symmetric power (an operation we denote τ n ) of the pullback [2] * Kl. The sheaf obtained has rank n + 1 and, from standard properties of such -adic sheaves (see e.g. [?, 2.3.3]), the value of σ n (q) is explicitly related to the trace of the global Frobenius acting on the cohomology space with compact support

H 1 c (G m × F q , τ n ([2] * Kl)).
Instead of obtaining a precise value for that trace, which seems very dicult, we only discuss the dimension of the cohomology space involved. Such an information is enough to gain some intuition about the level of diculty of evaluating σ n (q) as n grows.

In [?, proof of Th. 3.1], the decomposition of [2] * τ n (Kl) as a direct sum of Lang sheaves is given. That enables us to see easily that the sheaves τ n ([2] * Kl) and [2] * τ n (Kl) are isomorphic (e.g. via a straightforward computation of the eigenvalues of the local Frobenii acting on those sheaves). The ramication argument given in [?, proof of Th. 3.1] combined with the formula 1.13.

1 of [?] yields dim H 1 c (G m × F q , τ n ([2] * Kl)) = n -2 n 2p , if n is even , n + 1 -2 n 2p + 1 2 otherwise .
so the dimension evaluated increases almost linearly with n (at least in the range 1 n p).

F. JOUVE 3.1. A surface related to σ 3 (q). We start by expanding the formula dening σ 3 (q)

σ 3 (q) = ν∈Fq h∈F × q ψ(h + ν 2 h -1 ) 3 = ν∈Fq x,y,z∈F × q ψ(x + y + z + ν 2 (x -1 + y -1 + z -1 )) = x,y,z∈F × q ψ(x + y + z) + ν∈F × q x,y,z∈F × q ψ(x + y + z + ν 2 (x -1 + y -1 + z -1 )) .
Using orthogonality relations and performing the change of variables x = ν -1 x, y = ν -1 y, z = ν -1 z, we obtain

σ 3 (q) = -1 + ν∈F × q x ,y ,z ∈F × q ψ(ν(x + y + z + x -1 + y -1 + z -1 )) .
This leads us to dene a surface S 0 in G 3 m /F q in the following way .

S 0 : f (x, y, z) = x + y + z + x -1 + y -1 + z -1 = 0 .
The link between σ 3 (q) and the number of F q -rational points of S 0 is given by

σ 3 (q) = -1 + x,y,z∈F × q ν∈F × q ψ(νf (x, y, z)) (2) = -1 + x,y,z∈F × q f (x,y,z) =0 (-1) + x,y,z∈F × q f (x,y,z)=0 (q -1) = -1 -((q -1) 3 -|S 0 (F q )|) + (q -1)|S 0 (F q )| = -(q -1) 3 + q|S 0 (F q )| -1 .
We are now going to see how S 0 is closely related to the K3 surface called C in the paper [?] 1 . First, the equation dening S 0 is obviously equivalent to xyz(x + y + z) + xy + yz + xz = 0 and xyz = 0 .

The surface S 0 is a Zariski open dense subset of the projective surface S 1 ⊂ P 3 Fq (i.e. S 1 is dened over F q ) with homogeneous equation in coordinates (x : y : z : t)

S 1 : xyz(x + y + z) + t 2 (xy + yz + xz) = 0 .
Precisely, S 0 is the open set dened by xyzt = 0. Fixing the value z = 1, we see then that S 0 is isomorphic to the surface S 2 /F q with equation S 2 : xy(x + y + 1) + t 2 (xy + y + x) = 0 and xyt = 0 . x 2 + x + 1 = 0 , y = -x -1 , t = 0 , which happens to be, over F q , the union of the two lines (minus a point)

D ζ 3 = {(ζ 3 ; ζ 2 3 ; t)|t ∈ F q \ {0}} D ζ 2 3 = {(ζ 2 3 ; ζ 3 ; t)|t ∈ F q \ {0}}
We observe that we have D ζ 3 (F q ) = D ζ 2 3 (F q ) = ∅ when F q contains no primitive cube root of 1 (for instance if q = p r with p ≡ 2 (mod 3) and r odd).

So we have the following lemma relating F q -rational points on S 0 and S * 3 :

Lemma 4.

• If ζ 3 ∈ F q then |S 0 (F q )| = |S * 3 (F q )| + 2q -2 . • Otherwise |S 0 (F q )| = |S * 3 (F q )| .
Proof. What we have just observed and the fact that, in any case,

D ζ 3 ∩ D ζ 2 3 = ∅ clearly imply
Lemma ??

We are not going to focus on the computation of |S * 3 (F q )| but instead of that we are going to perform one last transformation on the equation dening S * 3 , so that the ane equation of the From a geometric point of view (i.e. looking at the dierent varieties as dened over a separable closure F q of F q ) the surfaces S and S 3 are isomorphic:

S 3 ⊗ F q S 4 ⊗ F q (x; y; s) → (x; y; is)
where i denotes a square root of -1 in F q .

So the link between the arithmetic properties of S 4 and S 3 can easily be made explicit:

Lemma 5. We have, for all q,

|S 3 (F q )| -q 2 = χ q (-1)(|S(F q )| -q 2 ) .
Proof. We compute

|S 3 (F q )| = x,y∈Fq (1 + χ q (-xy(x + y + 1)(xy + y + x))) = q 2 +
x,y∈Fq χ q (-xy(x + y + 1)(xy + y + x))

= q 2 + χ q (-1)

x,y∈Fq χ q (xy(x + y + 1)(xy + y + x))

= q 2 + χ q (-1)(|S 4 (F q )| -q 2 ) .

We summarize the connections between the dierent surfaces involved in the following diagram where all the arrows are dened over F q S 0 → S 1 S 2 ← S * 3 → S 3 . We also have the following isomorphism dened over F q S 3 ⊗ F q S 4 ⊗ F q .

We have just explained why the problem of evaluating σ 3 (q) can be reduced to determining the number of F q -rational points on S 4 . As in the previous case, we construct the smooth projective model of S 4 , the Zeta function of which we will then make explicit.

3.2. Resolving the singularities of S 4 . In the following geometric study, all the varieties will implicitly be dened over F p (for instance the n-projective space P n Fp will simply be denoted by P n ). However, for the arithmetic application we have in mind, we will keep track of the elds of denition whenever needed.

We are now going to describe briey the construction of the minimal smooth projective model of

S following [?].

We consider the sextic curve in P 2 :

C :

XY Z(X + Y + Z)(XY + Y Z + XZ) = 0 .
We want to construct the smooth model of the double covering of P 2 ramied over C, an open subset of which will happen to be the surface S 4 .

The singularities of C are all simple singularities (see [?] page 61), coming from the intersections between the following rational curves:

D x : X = 0 , D y : Y = 0 , D z : Z = 0 , D x,y,z : X + Y + Z = 0 , C : XY + XZ + Y Z = 0 .
The set of singular points of C consists in:

• 3 triple points:

D x ∩ D y ∩ C = (0 : 0 : 1) , D x ∩ D z ∩ C = (0 : 1 : 0) , D y ∩ D z ∩ C = (1 : 0 : 0) .
• 5 double points:

D x ∩ D x,y,z = (0 : 1 : -1) , D y ∩ D x,y,z = (1 : 0 : -1) , D z ∩ D x,y,z = (1 : -1 : 0) , D x,y,z ∩ C = {(ζ 3 : ζ 2 3 : 1); (ζ 2 3 : ζ 3 : 1)} .
We notice once more that whenever ζ 3 ∈ F q , we have D x,y,z (F q ) ∩ C(F q ) = ∅.

One constructs the smooth model of the double cover of P 2 ramied over C as follows: rst we blow up P 2 at the triple points of C. This gives rise to three exceptional curves which are irreducible components of the total transform of the branch locus C in the blown-up P 2 (the other irreducible components being nothing but the strict transform of the irreducible components of C). The only singularities of the pre-image of the branch locus in the blown-up P 2 are now double points (either coming from the original branch locus C, or from the intersections between the three exceptional curves and the other irreducible components of the pre-image of C). However these double points come from intersections between components of odd multiplicity as summand of the divisor attached to the pre-image of the branch locus, so, as explained in [?], we now have to blow up the pre-image of P 2 in the double points of the pre-image of C.

This gives rise to 14 exceptional curves having all even multiplicity. The only remaining singularities are double points coming from intersections between components of odd and even multiplicity of the total transform of the branch locus. Taking the double cover of the pre-image of P 2 ramied over the components of odd multiplicity of the image of C, we then obtain a smooth surface.

We denote by K4 (adding Kloosterman's name to those of Kähler, Kodaira and Kummer seems fair) the smooth surface we have just constructed (it is the surface called C in [?]). The resolution graphs corresponding to the resolution of singularities we have performed are the following

P 1 D x,y,z C P 1 C D x,y,z D y P 1 D x D x D y D z P 1 P 1 P 1 D z Fig. 1 Fig. 2
On the two dual graphs above, the full points represent components of odd multiplicity whereas empty points represent components of even multiplicity of the total transform of the branch locus (these are all exceptional curves coming from blowing up at the double points of the image of C).

Figure 1 describes the relation existing between the irreducible components of the total transform of the branch locus seen as rational curves over F p . As we mentioned before, it might happen that D x,y,z (F q )∩C(F q ) = ∅. Figure 2 usefully describes the relations between the irreducible components of the image of the branch locus seen as curves dened over such elds F q (that is to say elds containing no primitive cubic root of 1).

The surface K4 is smooth and projective. The following criterion tells us that K4 is a K3 surface (see e.g. [?] page 189 for the proof of a slightly stronger statement): Proposition 6. A surface constructed as the smooth model of the double cover of P 2 ramied over a sextic curve having only double points or triple points as singularities is a K3 surface.

As we have partly seen in the introduction, the remarkable structure of K3 surface that K4 possesses enables us to compute more easily the Galois action on -adic cohomology groups or, equivalently, to obtain a totally explicit form of its Zeta function.

The remaining diculty consists in determining the polynomial P 2 of (??). As K4 is a K3 surface, we know that P 2 has degree 22 (see [?, (6.7), proof of Th. 6]). The problem of determining P 2 happens to be easier to handle when ζ 3 ∈ F q ; as a consequence we are rst going to evaluate |K4(F q )| when ζ 3 ∈ F q .

3.3. Computing the Zeta function of K4 over nite elds containing ζ 3 . We x a nite eld F of characteristic p such that ζ 3 ∈ F. We exploit the proof of Theorem 6 in [?]. The morphism F. JOUVE of P 2 given by

U = ζ 3 Y Z + ζ 3 XZ -ζ 2 3 XZ -ζ 2 3 XY V = ζ 3 Y Z + ζ 3 Y 2 -ζ 2 3 Y 2 -ζ 2 3 XY W = -Y Z + ζ 2 3 XY transforms the curve C : XY Z(X + Y + Z)(XY + Y Z + XZ) = 0 into the curve of equation (U 3 + W 3 )(V 3 + W 3 ) = 0. As ζ 3 ∈ F, the above morphism is dened over F.
As a consequence, K4 is, over F, the smooth model of the double cover of P 2 ramied over the curve (U 3 + W 3 )(V 3 + W 3 ) = 0. That is to say, K4 is isomorphic over F to the Kummer surface Km(E × E) associated to E × E, where we recall that E has Weierstrass model y 2 = x 3 + 1.

We can now exploit the fact that the -adic cohomology of K4 is closely related to that of E. The -adic cohomology of E is well known and it is proved in [?] that, since Km(E × E) is obtained from E × E by rst blowing up its sixteen points of order at most 2 (then quotienting by the involution induced), for some extension F q of F, we have

(5) P 2 (K4/F q ; T ) = (1 -qT ) 16 det(1 -T Fr * |H 2 E×E )
The Künneth formula (see [?]) gives

H 2 E×E H 2 E ⊗ H 0 E ⊕ H 1 E ⊗ H 1 E ⊕ H 0 E ⊗ H 2 E First H 2 E ⊗ H 0 E H 0 E ⊗ H 2 E H 2 E = Q (-2) so det(1 -T Fr * |H 2 E ⊗ H 0 E ⊕ H 0 E ⊗ H 2 E ) = (1 -qT ) 2 .
What's more, we know (see [?, page 301 and 304]) that the Zeta function of E over F q is given by

Z(E/F q ; T ) = (1 -π r T )(1 -πr T ) (1 -T )(1 -qT ) ,
where r is the integer such that q = p r and π is an algebraic integer satisfying ππ = p and |π| = √ p.

An easy calculation then yields

det(1 -T Fr * |H 1 E ⊗ H 1 E ) = (1 -qT ) 2 (1 -π 2r T )(1 -π2r T ) .
Gathering these equalities we eventually obtain

det(1 -T Fr * |H 2 E×E ) = (1 -qT ) 4 (1 -π 2r T )(1 -π2r T ) , thus (6)
P 2 (K4/F q ; T ) = (1 -qT ) 20 (1 -π 2r T )(1 -π2r T ) . Let P 21 (K4/F p r ; T ) = (1 -π 2r T )(1 -π2r T )
The reciprocal roots λ 1 and λ 2 of the polynomial

P 21 (K4/F p ; T ) ∈ Z[T ] satisfy (7)
λ r 1 = π 2r and λ r 2 = π2r . Exploiting (??) and (??), we get the following expression for the Zeta function of K4 over

F q = F p r Z(K4/F q ; T ) = 1 (1 -T )(1 -q 2 T )(1 -qT ) 20 (1 -λ r 1 T )(1 -λ r 2 T ) , which immediately yields (8) |K4(F q )| = 1 + q 2 + 20q + λ r 1 + λ r 2 .
However, the algebraic integer π is not entirely determined by the conditions ππ = p, |π| = √ p.

This means that the expression (??), involving λ 1 and λ 2 , is not completly satisfactory. We are now going to evaluate precisely λ 1 and λ 2 by computing both λ 1 λ 2 and λ 1 + λ 2 .

As the elliptic curve E/Q has complex multiplication by the ring of integers O K of the quadratic eld K = Q(

√

-3), we know that π ∈ O K . Following [?], we focus our attention on whether the prime p splits in O K or not. Indeed, writing π = a + b √ -3, a, b ∈ Z, Beukers and Stienstra show in [?] that, provided p = 2, 3,

• either p splits in O K in which case

λ 1 λ 2 = p 2 λ 1 + λ 2 ∈ {±2(a 2 -3b 2 ), ±(a 2 + 6ab -3b 2 ), ±(a 2 -6ab -3b 2 )} • or λ 1 λ 2 = ±p 2 λ 1 + λ 2 ∈ {0, ±2p}
Remark 4. Recall that a prime number p is ramied in Q( √ -3) if and only if -3 is a square in Z/pZ. Using quadratic reciprocity, this is equivalent, for a prime p 5, to p ≡ 1 (mod 3) (1 is the only nonzero square modulo 3) and thus to p ≡ 1 (mod 6).

Though more precise, the expressions we have obtained for λ 1 and λ 2 remain ambiguous. We are rst going to deal with the case p ≡ 1 (mod 6) (i.e. when p splits in O K ). As sketched in [?], we consider the action of a group of order 6 on K4/F q , q being a power of a prime p ≡ 1 (mod 6).

Let G be the nite group generated by the automorphism γ 1 exchanging the two sheets of the double cover involved in the construction of S and by the automorphism γ 2 induced on K4 by the 3-cycle (X : Y : Z) ∈ P 2 → (Y : Z : X) ∈ P 2 . γ 1 and γ 2 are of respective order 2 and 3, so |G| = 6.

The xed points of γ 1 are the ramication points of K4. Figure 1 describes how many of these points are F q -rational. Indeed, as ζ 3 ∈ F q = F p r , all the ramication points of K4 have coordinates in F q . Each vertex of that dual graph represents a rational curve and, as the empty points have degree 2 with a distinct couple of full points as neighbors, the number of ramication points of K4 over F q is 8(q + 1) + 14(q -1) .

Notice that the other F q -rational points of K4 have a trivial stabilizer under the action of G.

The xed points of γ 2 are the elements of the inverse image of {(ξ : ξ 2 : 1)|ξ 3 = 1} by the nal double cover. If ξ = ζ 3 or ξ = ζ 2 3 , we obtain the two branch points of C ∩ D x,y,z which are also xed points for γ 1 . If ξ = 1, we obtain the two points (x, y, s) = (1, 1, ±3) (provided p = 2, 3) on the ane part S 4 of K4.

Exploiting (??), we nally deduce, in the case where q = p ≡ 1 (mod 6), 1 + p 2 + 20p + λ 1 + λ 2 ≡ 8(p + 1) + 14(p -1) + 2 (mod 6)

thus λ 1 + λ 2 ≡ 2 (mod 6). Looking back at the possible values we obtained for the sum and the product of λ 1 and λ 2 , this yields λ 1 + λ 2 = 2(a 2 -3b 2 ) and λ 1 λ 2 = p 2 .

As p = ππ = a 2 + 3b 2 , we can also write λ 1 + λ 2 = 4a 2 -2p. We can now give the explicit form of the polynomial P 21 :

P 21 (K4/F p ; T ) = p 2 T 2 -(4a 2 -2p)T + 1 .
What's more, if F q = F p r with p ≡ 1 (mod 6), (??) is now an entirely satisfactory expression for |K4(F q )|.

To conclude with the case ζ 3 ∈ F q , we must nd the value of |K4(F q )| when F q is an extension of even degree of F p with p ≡ -1 (mod 6), say q = p 2f , f 1. For that purpose we consider the general situation where X is a smooth projective surface dened over F q and we denote by Div(X)

the abelian group of divisors on X. It is well known that the so called cycle class map once extended to the Q -linear injective map ( 9)

Div(X) ⊗ Z Q → H 2 (X ⊗ F q , Q (1)) Fr-1 nilpotent
is of great importance. For instance, if we denote by m X (1; q) the multiplicity of 1 as eigenvalue of Frobenius acting on H 2 (X ⊗ F q , Q (1)) (which can also be seen as the multiplicity of q as eigenvalue of Fr * acting on H 2 X ) and by ρ(X|F q ) the Picard number of X (i.e. the rank of the Néron-Severi group of X (see [?] page 120)), Tate's conjecture, in the case of nite elds (see e.g. [?]), predicts that ( 10)

ρ(X|F q ) = m X (1; q) .
Notice that this statement is equivalent to the surjectivity of (??) (as it is injective).

However, dealing with the present situation, it is sucient to exploit the injectivity of (??). As assumed earlier, F q is an extension of even degree of F p with p ≡ -1 (mod 6), so ζ 3 ∈ F q and the preceding study shows that we have an isomorphism K4

Km(E × E) dened over F q . As mentioned before, p ≡ -1 (mod 6) implies that p does not split in O K and so (see, e.g. [?, exercice 2.30 page 184]) we conclude that the elliptic curve E/F p is supersingular (see [?, page 137]). Apart from that, we know, by a result of Shioda (see [?]), that if char F q = 2, the Picard number of Km(C × C) is 22 provided the elliptic curve C/F q is supersingular. So, in the present case, we get

ρ(K4|F q ) = ρ(Km(E × E)|F q ) = 22 .
Thus, the injectivity of (??) and the fact that dim Q H 2 K4 = 22 enables us to obtain m K4 (1; q) = 22 , hence the following expressions for the polynomial P 2 (K4/F p 2f ; T ) and for the Zeta function of K4 over F p 2f :

P 2 (K4/F p 2f ; T ) = (1 -p 2f T ) 22 , Z(K4|F p 2f ) = 1 (1 -T )(1 -p 4f T )(1 -p 2f T ) 22 . This immediately yields |K4(F p 2f )| = 1 + p 4f + 22p 2f .
3.4. An elliptic pencil on K4. We now have to compute |K4(F q )| when ζ 3 ∈ F q , in which case K4 is not equipped with the rich structure of Kummer surface. Equivalently, we need to compute the polynomial P 2 (K4/F q ; T ) in the general case. Following [?], we notice that an elliptic bration on K4 can be made explicit. Indeed, let us consider the family of cubics of P 2 parametrized by τ :

E τ : XY Z -τ (X + Y + Z)(XY + Y Z + XZ) = 0 . The discriminant of the generic bre E τ is ∆(τ ) = 1 16 τ 9 (9τ -1)(τ -1) 3 .
We deduce that the singular bres correspond to τ ∈ {∞, 0, 1 9 , 1}. With the same notations as in Section ??, the base points of the family {E τ } τ are

D x ∩ D y ∩ C = (0 : 0 : 1) D x ∩ D z ∩ C = (0 : 1 : 0) D y ∩ D z ∩ C = (1 : 0 : 0) D x ∩ D x,y,z = (0 : -1 : 1) D y ∩ D x,y,z = (-1 : 0 : 1) D z ∩ D x,y,z = (-1 : 1 : 0) .
These are all singular points for the sextic curve C. As noticed by the authors in [?], each curve E τ intersects the branch locus C of K4 exactly in the points at which a blowing up is performed for the construction of K4 (apart from the elements of C ∩ D x,y,z ). So, when blowing up P 2 at the base points of {E τ } τ , the strict transform of the generic cubic intersects no components of odd multiplicity of the total transform of the branch locus.

After blowing up P 2 at the base points, we get an elliptic pencil on a rational surface which gives rise, after double cover ramied over the union of the singular cubics E 0 and E ∞ (exactly corresponding to the branch locus C), to the surface K4. We denote by ϕ the corresponding elliptic bration.

One notices that, taking the nal double cover can be interpreted as performing, in the equation dening E τ the change of variable τ = t 2 so that the generic bre of the elliptic pencil on K4 can be given by ( 11)

XY Z -t 2 (X + Y + Z)(XY + Y Z + XZ) = 0 .
In [?], Beukers and Stienstra, determining the singular bre combination of the elliptic bration on K4 given by (??), prove, using a theorem of Shioda

P 2 (K4/F q ; T ) = (1 -qT ) 19 (1 -ε r qT )P 21 (K4/F q ; T ) , where q = p r , ε = -1 if p ≡ -1 (mod 3), ε = 1 otherwise and P 21 (K4/F q ; T ) ∈ Z[T ], P 21 (K4/F q ; 0) = 1, deg P 21 (K4/F q ) = 2.
>From Section ??, for F q 2 = F q (ζ 3 ), we know that (12) P 21 (K4/F q 2 ; T ) = P 21 (Km(E × E)/F q 2 ; T ) , where we recall that E is the CM elliptic curve given by y 2 = x 3 + 1 .

So, there exist conjugate complex numbers λ 1 and λ 2 such that

Z(K4/F q ; T ) = 1 (1 -T )(1 -q 2 T )(1 -qT ) 19 (1 -ε r qT )(1 -λ r 1 T )(1 -λ r 2 T )
.

From that formula we immediately deduce ( 13)

|K4(F q )| = 1 + q 2 + 19q + ε r q + λ r 1 + λ r 2 ,
where the explicit values of λ 1 and λ 2 are given by the following proposition Proposition 7.

• If p ≡ 1 (mod 6), there exist a, b ∈ Z such that p = a 2 + 3b 2 and λ 1 , λ 2 are the reciprocal roots of

P 21 (K4/F p ; T ) = p 2 T 2 -(4a 2 -2p)T + 1 . • If p ≡ -1 (mod 6) then λ 1 , λ 2 are the reciprocal roots of P 21 (K4/F p ; T ) = -p 2 T 2 + 1 ,
that is to say, λ 1 = p and λ 2 = -p.

F. JOUVE Proof. The case p ≡ 1 (mod 3) has been treated in Section ??.

If p ≡ -1 (mod 6), we consider, as in Section ??, the action of the group G =< γ 1 , γ 2 > on K4

seen as a surface dened over a eld F q such that ζ 3 ∈ F q . Exploiting gure 2, we see that the number of ramication points of K4/F q is 8(q + 1) + 12(q -1) .

In the case where q = p ≡ -1 (mod 6), we get from (??) and the counting of the number of stabilized points on K4 under the action of G performed in Section ??:

1 + p 2 + 18p + λ 1 + λ 2 ≡ 8(p + 1) + 12(p -1) + 2 (mod 6) .

Hence λ 1 + λ 2 ≡ 0 (mod 6). As p is inert in K = Q( √ -3), the study of Section ?? yields λ 1 + λ 2 = 0 and λ 1 λ 2 = ±p 2 . Now, as ζ 3 ∈ F p 2 , the calculation we have just done applied to K4/F p 2 yields |K4(F p 2 )| = 1 + p 4 + 20p 2 + λ 2 1 + λ 2 2 ≡ 8(p 2 + 1) + 14(p 2 -1) + 2 (mod 6) .

Hence λ 2 1 + λ 2 2 ≡ 2 (mod 6) so λ 1 λ 2 ≡ -p 2 (mod 6). We conclude that λ 1 λ 2 = -p 2 and the proof is complete.

Remark 5. In the context of the Inose-Shioda correspondence (see [?]), (??) means that the matrix corresponding to the 2-dimensional lattice of transcendental cycles on K4 is 4 2 2 4 .

By [?], 1.3, this implies that the group of sections of the elliptic bration on K4 is of order 6.

Indeed (??) can be written in Weierstrass form as follows (see [?]):

V 2 W + (1 -3t)U V W -t 4 (t 2 -1)V W 2 = U 3 .

We see immediately that (0 : 0 : 1) is a point of order 3 on the generic bre. Moreover, the ane part Z = 1 of the generic cubic E t can be written, after a suitable change of variables (using, for instance, a computer algebra system) V 2 = f (U ) where In order to nd an element of order 2 on each bre, it is sucient to nd a root of f in the function eld F q (t). The polynomial r(T ) = 1 2 (-1 2 T 4 -T 2 + 1 6 ) fullls this condition.

We are now ready to prove Theorem ??; it suces to recover the number of F q -rational points on the surface S 0 attached to σ 3 (q) from the knowledge of the quantity |K4(F q )|.

3.5. End of the proof of Theorem ??. We now want to obtain an explicit value for σ 3 (q); we rst determine the number of F q -rational points on the ane surface S 4 dened by (??).

Lemma 8. We have |S 4 (F q )| = q 2 + 2q + λ r 1 + λ r 2 ,

where q = p r and λ 1 , λ 2 are dened by Proposition ??.

Proof. From the construction of the surface K4 detailed in Section ??, we have:

|K4(F q )| = |S 4 (F q ) \ {(x, y, s)|s = 0}| + |{ramication points of K4/F q }| Moreover, we claim that |{(x, y, s) ∈ S 4 (F q )|s = 0}| = 4q -7 if ζ 3 ∈ F q , 4q -5 otherwise .

Indeed, the cardinality of the left hand side set in the above equality is the number of F q -rational points on the union of rational ane curves: D x ∪ D y ∪ D x,y ∪ C x,y , where D x : x = 0 D y : y = 0 D x,y : x + y + 1 = 0 C x,y : xy + y + x = 0

The number of F q -rational points of each of the 3 lines D x,y , D x and D y is q . What's more, we can give the following parametrization: C x,y (F q ) = {(λ; -λ λ+1 )|λ ∈ F q \ {-1}}. Hence |C x,y (F q )| = q -1.

We also have C x,y (F q ) ∩ D x (F q )∩D y (F q ) = (0; 0) D x (F q ) ∩ D x,y (F q ) = (0; -1) D y (F q ) ∩ D x,y (F q ) = (-1; 0)

D x,y (F q ) ∩ C x,y (F q ) = {(ζ 3 ; ζ 2 3 ), (ζ 2 3 ; ζ 3 )} , if ζ 3 ∈ F q ∅ otherwise
Gathering these informations, we obtain the wanted formula for |{(x, y, s) ∈ S 4 (F q )|s = 0}|.

Thanks to the calculation of the number of ramication points on K4/F q performed in the proof of Proposition ?? (when ζ 3 ∈ F q ) and in Section ?? (when ζ 3 ∈ F q ), we deduce |K4(F q )| = |S 4 (F q )| -(4q -7) + 8(q + 1) + 14(q -1) if ζ 3 ∈ F q , |S 4 (F q )| -(4q -5) + 8(q + 1) + 12(q -1) otherwise .

= |S 4 (F q )| + 18q + 1 if ζ 3 ∈ F q |S 4 (F q )| + 16q + 1 otherwise Exploiting (??) and noticing that ε r = 1 if and only if ζ 3 ∈ F q = F p r , we see that the proof is complete Combining Lemmas ?? and ??, we get |S 3 (F q )| = q 2 + χ q (-1)(|S 4 (F q )| -q 2 ) = q 2 + χ q (-1)(2q + λ r 1 + λ r 2 ) .

To apply Lemma ?? and deduce |S 0 (F q )|, we must make the relation between |S 3 (F q )| and |S * 3 (F q )| clear. Reasoning as in the proof of Lemma ?? to calculate |S 4 (F q )| from the knowledge of |{(x, y, s) ∈ S 4 (F q )|s = 0}|, we obtain

|S * 3 (F q )| = |S 3 (F q )| -(4q -7) if ζ 3 ∈ F q , |S 3 (F q )| -(4q -5) otherwise .
Hence |S * 3 (F q )| = q 2 + χ q (-1)(2q + λ r 1 + λ r 2 ) -4q + 7 if ζ 3 ∈ F q , q 2 + χ q (-1)(2q + λ r 1 + λ r 2 ) -4q + 5 otherwise .

Finally

  , we dene S 3 /F q by (3) S 3 : s 2 = -xy(x + y + 1)(xy + y + x) , and the Zariski open dense subset S * 3 ⊂ S 3 by xys = 0. 1 The reference [?] was given by N. Katz via E. Kowalski. Then the map S * 3 → S 2 (x, y, s) → (x, y, t = xy(x + y + 1)s -1 ) establishes an isomorphism between S * 3 and S 2 \ (D ζ 3 ∪ D ζ 2 3 ), where ζ 3 is a primitive cube root of unity in F q and D = D ζ 3 ∪ D ζ 2 3 is a degenerate conic in A 3 Fq with equation

  K3 surface C studied by Beukers and Stienstra ([?]) clearly appears. Let us dene (4) S 4 : s 2 = xy(x + y + 1)(xy + y + x) .

f

  (U ) = U 3 + A(T )U + B(T ) ,

  ). Seen over Q, such a polynomial has a splitting eld with Galois group isomorphic to W (E 6 ), the Weyl group of the exceptional algebraic group E 6 . In that context the results of Swinnerton-Dyer ([?]) give us precise information about which subgroups of W (E 6 ) can actually appear as permutation groups of the 27 lines, when working over nite elds.

	Notice now (see [?, page 119]) that, if p ≡ 1 (mod 3), 4 is a cube modulo p if and only if the
	coecient A p dened in Proposition ?? is even. The dierent cases above can then be partly unied.
	Indeed the eigenvalues of the morphism induced on H 2 e S	by the global geometric Frobenius morphism
	on S/F q are:

We can now apply Lemma ??; this yields

Moreover, we get from (??)

σ 3 (q) = -q 3 + 3q 2 -3q + q|S 0 (F q )| .

We nally obtain the following value for σ 3 (q), so that the proof of Theorem ?? is complete:

σ 3 (q) = q 2 + q(2qχ q (-1) + χ q (-1)(λ r 1 + λ r 2 ) + 2) if ζ 3 ∈ F q , -q 2 + q(2qχ q (-1) + χ q (-1)(λ r 1 + λ r 2 ) + 2) otherwise .