FERMILAB-CONF-09-675-E

Search for a Low Mass Standard Model Higgs Boson at DØ in ppbar Collisions at

√ s = 1.96TeV

Murilo Rangel * † Laboratoire de l'Accélérateur Linéaire E-mail: rangel@fnal.gov

We present combined searches for the Low Mass Standard Model Higgs boson at a center-of-mass energy of √ s =1.96 TeV, using up to 5 fb -1 of data collected with the D0 detector at the Fermilab Tevatron collider. The major contributing processes are associated production (W H → lνbb, ZH → ννbb, ZH → llbb). The significant improvements across the full mass range resulting from the larger data sets and improved analyses as well as future prospects are discussed. The standard model (SM) of particle physics is a succefull theory despite from the fact that the electroweak symmetry breaking mechanism is still unknown. The simplest proposed mechanism involves the introduction of a complex doublet of scalar fields that generate the masses of elementary particles via their mutual interactions. After accounting for longitudinal polarizations for the electroweak bosons, this so-called Higgs mechanism also gives rise to a single scalar boson (SM Higgs) with an unpredicted mass.

XXth Hadron Collider Physics

We combined results of different direct searches for Low Mass SM Higgs (mass less than 135 GeV) recorded by the DØ experiment [1]. The analyses serch for a signal of Higgs bosons produced in association with vector bosons (q q → W /ZH), through gluon-gluon fusion (GGF) (gg → H), through vector boson fusion (VBF) (q q → q ′ q′ H), and in association with top quarks (t t → t tH). The analyses utilize data corresponding to integrated luminosities ranging from 2.1 to 5.4 fb -1 , collected during the period 2002-2009. The Higgs boson decay modes studied are H → b b, H → W W , H → τ τ and H → γγ. The analyses were designed to be mutually exclusive after analysis selections.

Since the most recent DØ SM combined Higgs boson search results [2], from the analyses sensitive to Low Mass Higgs, the ZH → ν νb b was updated [3].

The main Higgs decay at Low Mass is H → b b, therefore, an algorithm for identifying jets consistent with the decay of a heavy-flavor quark is applied to each jet (b-tagging). Several kinematic variables sensitive to displaced jet vertices and jet tracks with large transverse impact parameters relative to the hard-scatter vertices are combined in a neural network (NN) discriminant trained to identify heavy-flavor quark decays and reject jets arising from light-flavor quarks or gluons [4].

We combine results using the CL s method with a negative log-likelihood ratio (LLR) test statistic. The value of CL s is defined as CL s = CL s+b /CL b where CL s+b and CL b are the confidence levels for the signal-plus-background hypothesis and the background-only hypothesis, respectively. The confidence levels are evaluated by integrating corresponding LLR distributions populated by simulating outcomes via Poisson statistics. Separate channels and bins are combined by summing LLR values over all bins and channels. Systematics are treated as Gaussian uncertainties on the expected number of signal and background events, not the outcomes of the limit calculations. The CL s approach used in this combination uses binned final-variable distributions rather than a singlebin (fully integrated) value for each contributing analysis. The exclusion criteria are determined by increasing the signal cross section until CL s = 1 -α, which defines a signal cross section excluded at 95% confidence level for α = 0.95.

In order to minimize the degrading effects of systematics on the search sensitivity, the individual background contributions are fitted to the data observation by maximizing a likelihood function for each hypothesis [5]. The likelihood is a joint Poisson probability over the number of bins in the calculation and is a function of the nuisance parameters in the system and their associated uncertainties, which are given an additional Gaussian constraint associated with their prior predictions. The maximization of the likelihood function is performed over the nuisance parameters. A fit is performed to both the background-only (b) and signal-plus-background (s+b) hypothesis separately for each Poisson MC trial.

We derive limits on SM Higgs boson production cross section times the SM Higgs branch ratio. The limits are derived at 95% C.L. To facilitate model transparency and to accommodate analyses with different degrees of sensitivity, we present our results in terms of the ratio of 95% C.L. upper cross section limits to the SM predicted cross section as a function of Higgs boson mass. The SM prediction for Higgs boson production would therefore be considered excluded at 95% C.L. when this limit ratio falls below unity.

The expected and observed 95% C.L. cross section limit ratios to the SM cross sections for all analyses combined over the probed mass region (100 ≤ m H ≤ 135 GeV/c 2 ) are presented in Table 1. These limits and the LLR distributions for the full combination are shown in Fig. 1. We have presented upper limits on standard model Higgs boson production derived from 60 Higgs search analyses including data corresponding to 2.1-5.4 fb -1 . We have combined these analyses and form new limits more sensitive than each individual limit. The observed (expected) 95% C.L. upper limit ratios to the SM Higgs boson production cross sections are 4.0 (2.8) at m H = 115 GeV/c2

Figure 1 :

 1 Figure 1: Left: Expected (median) and observed 95% C.L. cross section upper limit ratios for the combined Higgs analyses over the 100 ≤ m H ≤ 200 GeV/c 2 mass range. Right: Log-likelihood ratio distribution for the combined analyses over the 100 ≤ m H ≤ 200 GeV/c 2 mass range.

Table 1 :

 1 Combined 95% C.L. limits for SM Higgs boson production. The limits are reported in units of the SM production cross section times branching fraction.

	m H (GeV/c 2 ) 100 105 110 115 120 125 130 135
	Expected:	2.35 2.40 2.85 2.80 3.25 3.31 3.30 3.35
	Observed:	3.53 3.40 3.47 4.05 4.03 4.19 4.53 5.58


Shavkat Rakhmatullaev
email: rakhmatullaev@rambler.ru

Frédéric Huneau
email: frederic.huneau@u-bordeaux1.fr

Philippe Le Coustumer

Mikeal Motelica- Heino

Masharif Bakiev
email: bakiev1947@rambler.ru

Facts and Perspectives of Water Reservoirs in Central Asia: A Special Focus on Uzbekistan

Keywords: transboundary waters, dam management, water reservoirs, irrigation, hydropower, sedimentation, water policy

The political transformation of the Central Asian region has induced the implosion of the interconnected physical hydraulic infrastructure and its institutional management systems. Land-locked Central Asian countries with their climatic conditions and transboundary water resources have been striving to meet their food security, to increase the agricultural production, to sustain the energy sectors and to protect the environment. The existing water reservoirs are strategic infrastructures for irrigation and hydropower generation. The upstream countries (Tajikistan and Kyrgyzstan) favor the reservoirs operation for energy supply while the downstream countries (Uzbekistan, Turkmenistan and Kazakhstan) push on the irrigation use. This paper overviews the current challenges and perspectives (technical, institutional and legal regulations) of man-made water reservoirs in Central Asia with special focus on Uzbekistan.

Introduction

Transboundary water management and governance has been a hot topic in Central Asia for the last two decades after the disintegration of the former Soviet Union [5; 7; 21; 26]. In fact the United Nations Development Programme (UNDP) [START_REF]Central Asia human development report. Bringing down barriers: regional cooperation for human development and human security; UNDP Regional Bureau for Europe and the Commonwealth of Independent States[END_REF] reports that the Central Asian region loses $1.7 billion per year i.e. three per cent of the region's GDP (Gross Domestic Product) from the poor water management that lowers the agricultural yields. Moreover some 22 million people depend directly or indirectly on irrigated agriculture in these countries [START_REF]Water and energy nexus in Central Asia: Improving regional cooperation in the Syr Darya Basin[END_REF]. The sustainability of irrigated agriculture is one of the main platforms for food security, employment, livelihoods and environment protection in Central Asia [START_REF] Rakhmatullaev | Groundwater resources use and management in Amu Darya river basin (Central Asia)[END_REF].

The Central Asia countries (Tajikistan, Kyrgyzstan, Kazakhstan, Turkmenistan and Uzbekistan) have inherited an interconnected and sophisticated hydraulic infrastructure from the Soviet era. As described by Libert et al [START_REF] Libert | Water management in Central Asia and the activities of UNECE[END_REF] and Rakhmatullaev et al [START_REF] Rakhmatullaev | Reservoir Sedimentation in Uzbekistan: A case study of Akdarya reservoir[END_REF] the hydraulic mission of the soviet administration was set up from an engineering perspective based on the construction of large dams and water reservoirs in the mountain areas, upstream countries (Tajikistan and Kyrgyzstan) due to the attractiveness of natural conditions, i.e., more water accumulation per unit area in comparison to plains (lowland) conditions, downstream countries (Uzbekistan, Kazakhstan and Turkmenistan). On the other hand lowlands were suitable for practicing irrigated agriculture and growing water intensive agricultural crops (cotton, rice and wheat).

The recent pivotal area of discussions is hydropower versus irrigation mode for the operation of the reservoirs and dams in Central Asia e.g., the Toktogul reservoir in Kyrgyzstan (Syr Darya River Basin) and the construction of the Rogun hydropower station in Tajikistan (Amu Darya River Basin).

In fact various paramount technical, operational and biophysical aspects impact the sustainable operation and management of dams and water reservoirs in the region which do not have transboundary issues such as sedimentation, improper operation and over-use of hydraulic infrastructures against the designed operational regimes, lack of national legal and institutional frameworks on dam safety. These acute issues are to be seriously addressed at national and regional levels.

Some experts argue that the region will face global warming effects which will have impact on the formation of water resources in the mountain systems of the Tian Shan and Pamir-Alay due to the decrease of the ice cover [4; 10].

The World Commission on Dams (WCD) [START_REF] Wcd | Dams and Development: A New Framework for Decision-Making[END_REF] outlines that large hydraulic infrastructures such as dams and its associated water reservoirs have played an important role in the regional development in many parts of the world. In the Central Asian region, man-made water reservoirs play a particularly paramount role where natural precipitation is erratic or seasonal with uneven spatial and temporal water resources distribution [START_REF] White | A review of current knowledge[END_REF]. In fact water reservoirs store water during wet periods to make it available during dry periods for regulating floods, generating hydropower, and irrigation. Thus it is important to discuss technical, management aspects of dams and its associated water reservoirs.

The primary objective of this paper is to overview the situation on dams and water reservoirs in terms of technical, legal and institutional aspects in the new geopolitical realities in the region with a special focus on the Uzbekistan experience with a large hydraulic infrastructure safety framework. Perspectives and concrete recommendations are given for improving the regional dialogue on sustainable hydraulic infrastructure operation and management.

Water reservoirs

The first water reservoirs in the Central Asian region were constructed as early as in the X-XI centuries [1; 12]. For the regulation of the region's rivers hundreds of man-made water reservoirs were constructed to mitigate natural shortage of water resources (Figure 1). The era of massive water reservoir construction was documented in the late 1900s. From 1950 to 1980) more than 60 large reservoirs have been constructed in the region (Figure 2). The gross reservoir volume has increased by 50 folds from 0.17 to 49.3km 3 as a part of unprecedented increase of water storage capacities around the globe. Most of the dams and water reservoirs in the region have been constructed for irrigation purposes with a few for hydropower generation. As a result over a period of 90 years the areas under irrigation have increased by 3 fold in Central Asia, e.g. from 4.51 million ha in 1960 to 6.92 million ha in 1980 and to 7.85 million ha in 2000 [13; 25]. The peak of dam construction is dated to the soviet period when its policy was aiming at expanding irrigated lands for the mass production of cotton. At the moment there are more than 290 water reservoirs in Central Asia with a total volume capacity over 163 km 3 that regulate more than 50% of the monthly regions river flow and the area occupied by reservoirs constitutes roughly 6% of the Central Asian countries irrigated areas (Table 1) [2; 20]. For example, on average about 30% of irrigation water is delivered from reservoirs in the region, ranging as high as 54% in Turkmenistan and as low as 13% in Kyrgyzstan. There are 45 large-scale hydropower stations with a gross capacity of 36.7 GWh/year on large water reservoirs in the Central Asian region [START_REF] Peyrouse | The hydroelectric sector in Central Asia and the growing role of China[END_REF]. However the hydropower constitutes only 11% in Uzbekistan whereas more than 90% is produced in Tajikistan [START_REF]Study on intra-regional cooperation on water and power for efficient resources management in Central Asia[END_REF]. Kyrgyzstan and Tajikistan have about 78% of the total hydropower potential of the region but utilize only 10% of its potential.

Reservoirs in Uzbekistan

The total number of man-made water reservoirs in Uzbekistan is 55 (Table 2) [17; 23]. The total gross volume capacity of all reservoirs is about 19 km 3 and the useful volume capacity is about 14.5 km 3 whereas the total surface area of reservoirs is estimated at ca. 1450 km 2 in Uzbekistan. The total reservoir volume capacity is defined as maximum water amount stored, whereas the useful volume capacity is actual available water storage. The arithmetic difference makes the dead storage of a reservoir. In other words the dead volume storage in a reservoir is determined as the storage volume between the stream bed and the lowest elevation from which water can be withdrawn by gravity [START_REF] Mahmood | Reservoir sedimentation. Impact, extent and mitigation[END_REF]. The difference observed between total capacity and useful capacity is an indicator of the efficiency of the dam and water management. The 4.5 km 3 difference represents a loss percentage of 26% due to topographic peculiarities, i.e., most of water reservoirs are constructed in lowlands in Uzbekistan.

The nine major reservoirs constitute about 86% (16.8 km 3 ) of the total reservoir volume capacity and about 94% (1362 km 2 ) of the total surface area (Table 3). Almost 90% of all dams are used for irrigation purposes and only two for hydropower (Andijan and Charvak) [START_REF] Rakhmatullaev | Geostatistical approach in estimation of sedimentation: A case study of Akdarya reservoir[END_REF]. About 75% of dams were constructed in the 30-years period from 1961 to 1990 with 93% of a total reservoir capacity known as the "soviet period". At the moment, 3 water reservoirs are being constructed in the Namangan, Jizzak and Samarkand provinces of Uzbekistan for increasing the water storage capacities.

All dams and their associated water reservoirs are aging and numerous biophysical, technical and management aspects need to be addressed for the sustainable management of water resources and hydraulic infrastructures. These aspects should be addressed at the national levels and in particular for hydraulic structures which have transboundary importance.

Threats to water reservoirs

Almost all hydraulic infrastructures in particular dams and their associated water reservoirs are aging (constructed during the 1930-40 period) [START_REF] Libert | Water and energy crisis in Central Asia[END_REF]. Thus there is some urgency for the safe operation on such structures in order to avoid devastating emergency situations in the event of technical operational failures or natural phenomena such as earthquakes. For example, Civil engineering, in term of life cycle and survey, but also water practices and legal aspects are linked and supposed to plan management of the structure and resources at different steps and scale. This is, actually, a key point discuss at national level and very often problematic at international one (i.e. case of the transboundary rivers).

Reservoir sedimentation

A reduced reservoir volume capacity due to sedimentation triggers operation and maintenance issues coupled with economic feasibility of the project, environmental concerns and social aspects [11; 28]. Sedimentation is a natural geomorphologic process but human interference increases its rates. Sedimentation reduces the main reservoir asset i.e. its volume capacity over time due to the feeding rivers [START_REF]ICOLD (International Commission on Large Dams[END_REF]. It is reported that an annual average 0.5-1% loss of volume capacities of small and large reservoirs is observed due to sedimentation in the world. Palmieri et al. [START_REF] Palmieri | Reservoir conservation: Volume I. The RESCON approach: economic and engineering evaluation of alternative strategies for managing sedimentation in storage reservoirs[END_REF] reports that the loss in volume capacity requires an annual replacement cost of 13 billion US$.

The most recent reports indicate that an average annual reservoir volume loss is estimated at 0.5% in Uzbekistan (Table 3) [START_REF] Rakhmatullaev | Geostatistical approach in estimation of sedimentation: A case study of Akdarya reservoir[END_REF]. As it can be seen the total volume capacities of major reservoirs have decreased by about 20% and its dead storage capacities have decreased by 55%. For example, the dead volume capacities of 7 reservoirs decreased by more than 75%. This is an alarming signal that in the future the sedimentation rates can be unprecedented with several operation and maintenance problems. * Tuyamuin is composed of four reservoirs: 1) Ruslovoy -2.34km 3 ; 2) Sultansanjar -2.69km 3 ; 3) Kaparas -0.96km 3 ; 4) Koshbulak -1.81km 3 .

Reservoir sedimentation is an acute issue in Central Asia due to naturally high turbidity of the watercourses. In fact most of the reservoirs have been silted to a great extent. Sedimentation impacts the guaranteed water supply for the different water users (irrigation, industry and hydropower) at the national and regional levels.

For example, the direct loss of reservoir volume capacity is in the form of less hydropower production capacity available for sale, less irrigated land to produce food and a reduced flood routing capacity. Moreover the deterioration of water quality in reservoirs can be a major issue with increasing levels of various contaminants from agriculture, industry, and natural sources whether organic (pesticides, PCBs, PAHs) or inorganic (trace metals) [START_REF] Rubinova | Water reservoirs of Aral Sea basin and their impact on water quantity and quality[END_REF]. Dam cavitation and abrasion of conduits, valves, sluice gates and hydropower turbines dramatically impact hydraulic facilities and structures [START_REF] Trussart | Hydropower projects: a review of most effective mitigation measures[END_REF]. Moreover social aspects can also be the unattractiveness for tourism and the loss of recreation opportunities.

The common engineering practice advocates for "design for life" approach for dam and reservoir conception. The design for life approach assumes a finite project life which means that future generations should bear all occurring costs and treats [START_REF]ICOLD (International Commission on Large Dams[END_REF].

Palmieri et al [START_REF] Palmieri | Reservoir conservation: Volume I. The RESCON approach: economic and engineering evaluation of alternative strategies for managing sedimentation in storage reservoirs[END_REF] proposed a new "life cycle management" approach for the sustainable management and use of hydraulic infrastructures. The ultimate goal of this approach is the sustainable use, where the major functions of the dam are maintained, through good management and maintenance. It means that these hydraulic infrastructures should be considered as economic assets and by proper mitigations measures there should be long-term generating benefits to local populations.

Water reservoirs operation guidelines

It is known that all large scale hydraulic infrastructures in particular dams and water reservoirs are designed and constructed according to a design operation regime which safeguards the durability of the operations. However, the real situation is quite different. The weighted decision for the operation of water reservoirs lies in the hand of local government authorities (hokimiyats, i.e., governor) who are responsible for the agricultural production of cotton and wheat. The water management organizations are left with only a technical assistance role for the water supply matters. In most cases, the design operation regime is violated and poses severe circumstances for hydraulic infrastructures. Infrastructure fatigue puts additional pressure on proper management of scare water resources.

Institutional management of water reservoirs

The hydraulic infrastructures in particular dams and water reservoirs are managed, rehabilitated and operated by the Ministry of Agriculture and Water Resources of Uzbekistan. In addition the State Joint Stock Company UzbekEnergo is a responsible authority for the operation and management of hydropower plants in Uzbekistan. For example, about 11.5% of power generation is produced through 29 hydropower plants and two large dams (Charvak and Andijan). There is an overlap of responsibilities of the two state agencies on the management of dams and its associated water reservoir.

Gosvodkhoznadzor is a state inspection agency for the safety of large hydraulic infrastructures in Uzbekistan [START_REF]Gosvodkhoznadzor (State Inspection for Large Hydraulic Infrastructures Safety of Uzbekistan)[END_REF]. There are almost 273 large hydraulic infrastructures (55 water reservoirs, 35 pumping stations, 29 hydropower plants, 60 irrigation main canals, 64 water-intake schemes, 24 main collectors and 7 riverbank protection structures). Gosvodkhoznadzor inspects the safety of large hydraulic infrastructures and sanctions the Ministry of Agriculture and Water Resources of Uzbekistan for proper actions and mitigation measures. Unfortunately, in most cases the recommendations are in form of suggestions.

Institutional and Legal frameworks

The main bottleneck for dam safety monitoring and control is lack of national legal frameworks and appropriate institutions in all Central Asian countries except Uzbekistan [START_REF] Libert | Water and energy crisis in Central Asia[END_REF]. After the dissolution of the former Soviet Union, large hydraulic infrastructures have been left without proper attention from the Central Asian countries. For example, in Kazakhstan, the hydropower plants are managed by semiprivate entities and the reservoirs are still operated by government water management authorities. There is always a dispute for the coordinated operation and maintenance of hydropower plant and dam.

In Uzbekistan, the national Law on the safety of hydraulic structures was adopted in 1999. There is an established institutional framework for monitoring of large hydraulic infrastructures namely the State Inspection (Gosvodkhoznadzor) for control and monitoring of technical conditions and safety operations of large-scale hydraulic infrastructures under the Cabinet of Ministries of the Republic of Uzbekistan.

UNECE (United Nations Economic Commission for Europe) is actively participating in the development of regional agreement on cooperation on dam safety in particular the information exchange and notification of other countries in case of accidents with dams [START_REF] Libert | Water management in Central Asia and the activities of UNECE[END_REF].

Transboundary issues

Uneven spatial and temporal water resources and a Soviet inherited unified hydraulic infrastructure have raised transboundary reservoir management issues over the water resources allocation among the countries in the region such as Kyrgyzstan, Tajikistan, Kazakhstan, Uzbekistan and Turkmenistan. The rivers such as Syr Darya and Amu Darya are already regulated by more than 78% and 94% respectively and attempts for new reservoir projects upstream raises increased concerns of the downstream countries (e.g. the Rogun hydropower station in Tajikistan and the Toktogul reservoir in Kyrgyzstan) (Figure 3).

Competition for water resources is prioritized by the downstream countries (Uzbekistan, Turkmenistan and Kazakhstan) for irrigation whereas upstream countries (Tajikistan and Kyrgyzstan) use them mainly for hydropower generation [5; 13]. There has been a serious conflict related to the water resource management of some hydraulic infrastructures like the Toktogul reservoir located in Kyrgyzstan in the recent post-soviet period. The surplus of hydropower generation by Toktogul dam in summer was transmitted into the former Central Asian Power System Grid for the usage by downstream countries of Kazakhstan and Uzbekistan. For compensation Kazakhstan supplied fossil fuels and Uzbekistan provided electricity to Kyrgyzstan for electricity needs in winter months.

However the situation has significantly changed due to the end of the former Soviet Union. During 1990 to 2000 summer releases declined to 45 per cent and winter releases increased to 55 per cent of the annual discharges [START_REF]Central Asia human development report. Bringing down barriers: regional cooperation for human development and human security; UNDP Regional Bureau for Europe and the Commonwealth of Independent States[END_REF].

Perspectives

The following key perspectives are addressed in the article for a sustainable management of the water reservoirs and dams.

Mini Hydropower schemes

Small hydropower schemes should be advocated in CA countries for their financial, technical, management and environmental feasibility aspects. These mini hydropower schemes would not significantly alter the hydrological regimes of main transboundary watercourses for different water uses. These schemes are financially attractive for international donor communities and less environmentally impact-oriented for upstream and downstream riparian ecosystems.

Institutional framework

For a regional cooperation on dams and water reservoirs management in Central Asia, the authors recommend the creation of a regional chapter of the International Commission on Large Dams (ICOLD) which can play as a neutral key regional institution on dam safety. Under the umbrella of the Central Asia ICOLD, national responsible agencies can cooperate and discuss various topicalities. The mandates of such organization could be harmonization of cooperation of the Central Asian states on dam safety, monitoring procedures, information exchange and development of comprehensive database. This can be a first step to create a regional platform for discussion of current issues on management, operation, maintenance of large hydraulic structures.

Legal framework

In order to synchronize the cooperation on dams and its associated water reservoirs management each country should develop its own laws on safety of hydraulic infrastructures. Only Uzbekistan has adopted such a law in 1999. Only after each country has its own law then regional legal framework can be developed.

Mitigation measures

For a regional cooperation on dams and water reservoirs safety, proper technical and management activities should take place at the national level in order to sustainably operate and manage these sophisticated infrastructures. It is time to overlook and assess the present technical conditions of all large infrastructures for each country.

The best practices (mitigation measures) can then be shared and duplicated among other regional countries. For example, in Uzbekistan a new GIS (Geographic Information System) in combination with depth measurement systems are being used for the operative estimation of reservoir sedimentation volumes [START_REF] Rakhmatullaev | Geostatistical approach in estimation of sedimentation: A case study of Akdarya reservoir[END_REF].

As the infrastructure fatigue will be reached in this century, appropriate planning programs should be designed and implemented such as a retirement fund for dam and water reservoir [START_REF] Palmieri | Reservoir conservation: Volume I. The RESCON approach: economic and engineering evaluation of alternative strategies for managing sedimentation in storage reservoirs[END_REF]. The retirement fund is intended to accumulate annual contributions made during the life of the dam to pay for any actions required at its retirement. For example, a change of purpose (e.g. recreation, farming, environment creation) from its present irrigation or hydropower generation and in extreme cases dam retirement using partial or complete removal of the dam. However, the partial or full decommissioning of dam would not be economically feasible in the region due to the high financial costs and the existence of a real local expertise. The practical approach is the change of dam or reservoir purpose which still can be attractive to local authorities.

Conclusions

Large hydraulic infrastructures play an important role for the Central Asian region development in particular dams and its associated water reservoirs. The main purpose of dams and water reservoirs is hydropower generation, water supply for irrigation and industry. However, these hydraulic structures face numerous problematic issues such as infrastructure fatigue, biophysical issues, operation and maintenance on national levels.

With new geopolitical realms in the region after the collapse of the former Soviet Union and emergence of new independent states regional and transboundary aspects have arisen for cooperative management of water resources and cost sharing of operation and maintenance. In the domain of large hydraulic infrastructures safety transparent legal and institutional frameworks should be developed and adopted reflecting transnational good spirit of cooperation.

A regional platform should be developed such as a branch of the ICOLD which will serve as professional union for the harmonized management and operation of dams and water reservoirs on transboundary watercourses.

Figure 1 .

 1 Figure 1. Schematic location of main water reservoirs and their volume capacities in Central Asia (excluding reservoirs in central and northern region of Kazakhstan).

Figure 2 .:

 2 Figure 2.: Number of reservoirs and cumulative volume in Central Asia for the last century.

Figure 3 .

 3 Figure 3. Major transboundary rivers in Central Asia

Table 1 : Distribution of water reservoirs by total volume and numbers in Central Asia countries (Food and Agriculture Organization of United Nations, 2009) Country Total reservoir volume capacity (km 3 ) Number of reservoirs

 1

	Irrigation water
	from reservoirs
	(%)

*The number of reservoirs represents for whole Kazakhstan.

Table 2 .

 2 Distribution of water reservoirs in Uzbekistan by administrative boundaries (United Nations

			Development Program, 2007)
	Province	Number	Useful Volume	Province	Number	Useful Volume
			Capacity			Capacity
			(km 3 )			(km 3 )
	Amu Darya River Basin	Syr Darya River Basin
	Khorezm	1	4.5	Andijan	3	1.7
	Kashkadarya	14	2.3	Tashkent	5	1.9
	Samarkand	7	1.1	Fergana	4	0.25
	Surkhandarya	4	0.9	Namangan	7	0.23
	Navoi	2	0.8	Jizzak	4	0.18
	Bukhara	2	0.4	Syrdarya	2	0.01
	Total	30	10.1		25	4.45

Table 3 .

 3 Loss of reservoir volume (total and dead) of selected major reservoirs in Uzbekistan

	Reservoir	River Basin	Total Volume	Silted	Started to
			(Mm 3 )	volume	operate
				(%)	(Year)
	Talimarjan	Amu Darya	1525	3.9	1985
	Janubiy	Surkhan Darya	800	37	1967
	Surkhan
	Kuyimazar	Amu Darya	310	11.2	1958
	Tudakul	Amu Darya	1200	13.7	1983
	Andijan	Kara Darya	1900	13.4	1970
	Kattakurgan	Zerafshan	900	22.5	1953
	Chimkurgan	Kashka Darya	500	22.7	1963
	Ruslovoy*	Amu Darya	2340	44.9	1980
	Kaparas*	Amu Darya	960	1.9	1983
	Average			19

Acknowledgements

The authors would like to thank the French Embassy in Tashkent for supporting the French-Uzbek cooperation in the field of water sciences. This study has also been supported by INTAS fellowship Nr. 04-83-3665 and by the French Ministry of Foreign Affairs via the Eiffel fellowship program No. 530909C. However, the views expressed in this paper do not necessarily reflect those of the funding agencies, and no official endorsement should be inferred from it.


Safia Haloui

THE MINIMUM AND MAXIMUM NUMBER OF RATIONAL POINTS ON JACOBIAN SURFACES OVER FINITE FIELDS

Keywords: Mathematics Subject Classification. 14H40, 14G15, 14K15, 11G10, 11G25 Abelian varieties over finite fields, jacobians, number of rational points

We give some bounds on the numbers of rational points on abelian varieties and jacobians varieties over finite fields. The main result is that we determine the maximum and minimum number of rational points on jacobians varieties of dimension 2.

Introduction

We are interested in the number of rational points over finite felds on particular abelian varieties, namely jacobian varieties. In particular, for given integers q = p e (p prime and e ≥ 1) and g ≥ 1, we consider the quantities J q (g) = max X #J X (F q ) and j q (g) = min X #J X (F q ) where X runs over the set of (absolutely irreducible, smooth, projective, algebraic) curves of genus g defined over the finite field F q and J X is the jacobian of X.

After giving some general bounds for these quantities, we determine the values of J q (2) and j q (2).

Let A be an abelian variety defined over F q of dimension g. The characteristic polynomial f A (t) of A is defined to be the characteristic polynomial of its Frobenius endomorphism. This is a polynomial with integer coefficients and its set of roots has the form {ω 1 , ω 1 , . . . , ω g , ω g } with |ω i | = √ q (Riemann Hypthesis for abelian varieties proved by Weil). For 1 ≤ i ≤ g, we set x i = -(ω i + ω i ) and we refer the sum g i=1 (ω i + ω i ) = -g i=1 x i as the trace of A.

The data of the x i 's is equivalent to the data of the isogeny class of A (by the Honda-Tate Theorem); we will say that A has type [x 1 , . . . , x g ]. The type of an absolutely irreducible, smooth, projective, algebraic curve over F q (as defined in [START_REF] Lauter | The maximum or minimum number of rational points on genus three curves over finite fields[END_REF]) is the type of its jacobian.

The number of rational points on A is given by #A(F q ) = f A (1), and since

f A (t) = g i=1 (t -ω i )(t -ω i ) = g i=1 (t 2 + x i t + q),
we obtain #A(F q ) = g i=1 (q + 1 + x i ). (1)

Abelian varieties

Let A be an abelian variety over F q of dimension g and of type [x 1 , . . . , x g ]. Since each x i has an absolute value less than or equal to 2 √ q by the Riemann Hypothesis, we obtain from (1) the well-known Weil bounds for abelian varieties:

(q + 1 -2 √ q) g ≤ #A(F q ) ≤ (q + 1 + 2 √ q) g .
We can derive an upper bound for #A(F q ) which depends on the trace of A. Indeed, recall that, if c 1 , . . . , c n are non negative real numbers, the arithmeticgeometric mean states that

n √ c 1 . . . c n ≤ 1 n (c 1 + • • • + c n )
with equality if and only if the c i 's are equal. Applying this inequality to (1), we get the following proposition, independently shown by Quebbemann [START_REF] Quebbemann | Lattices from curves over finite fields[END_REF] and Perret [START_REF] Perret | Number of points of Prym varieties over finite fields[END_REF]:

Proposition 1 (Perret-Quebbemann). We have

#A(F q ) ≤ q + 1 + g i=1 x i g g
with equality if and only if the x i 's are equal.

We set m = [2 √ q]. Using the arithmetic-geometric mean, Serre [START_REF] Serre | Sur le nombre des points rationnels d'une courbe algébrique sur un corps fini[END_REF] proved that g i=1

x i ≤ gm, (2)
and so Proposition 1 implies that #A(F q ) ≤ (q + 1 + m) g .

It is natural to ask whether #A(F q ) has a lower bound of this kind, and the answer turns out to be yes. Theorem 2. Let A be an abelian variety defined over F q of dimension g. Then

(q + 1 -m) g ≤ #A(F q ) ≤ (q + 1 + m) g .
The first inequality is an equality if and only if the x i 's are equal to -m and the second if and only if they are equal to m.

Proof. Only the first inequality has to be proved (note the similarity of the following proof with the one of ( 2)).

For k = 0, . . . , g let t k be the kth symmetric function of the (m + 1

+ x i )'s, for i = 1, . . . , g (i.e. g i=1 (t + (m + 1 + x i )) = g k=0 t k t g-k ). Let k ∈ {1, . .

. , g} and define the quantity

T k = {i1,...,i k }⊆{1,...,g} k j=1 (m + 1 + x ij ).
T k is a non zero integer (since it is an algebraic integer invariant by Q-automorphisms of C). Thus

T k ≥ 1. (3)
On the other hand, using the arithmetic-geometric mean, we obtain

T 1/( g k ) k ≤ 1 g k {i1,...,i k }⊆{1,...,g} k j=1 (m + 1 + x ij ) = 1 g k t k . (4)
Combining ( 3) and (4), we find

g k ≤ t k . (5)
Moreover, (5) remains true for k = 0.

Multiplying both sides of ( 5) by (qm) g-k and adding the inequalities obtained for k = 0, . . . , g we obtain

g k=0 g k (q -m) g-k ≤ g k=0 t k (q -m) g-k
from which the result follows.

Jacobians

Since jacobians are particular cases of abelian varieties, the bounds of the previous section can be applied. Using Theorem 2, we have:

Corollary 3.
(q + 1m) g ≤ j q (g) ≤ J q (g) ≤ (q + 1 + m) g .

Since the number of rational points on a curve X of genus g and of type [x 1 , . . . , x g ] is related to the trace of its Frobenius by

#X(F q ) = q + 1 + g i=1 x i , Proposition 1 gives us #J X (F q ) ≤ q + 1 + #X(F q ) -(q + 1) g g (6)
and we find again the bound proved by Quebbemann [START_REF] Quebbemann | Lattices from curves over finite fields[END_REF] and Perret [START_REF] Perret | Number of points of Prym varieties over finite fields[END_REF]. These bounds should be compared with those of Lachaud and Martin-Deschamps [START_REF] Lachaud | Nombre de points des jacobiennes sur un corps fini[END_REF]:

( √ q -1) 2 q g-1 -1 g #X(F q ) + q -1 q -1 ≤ #J X (F q ).
Moreover if X admits a map of degree d onto P 1 , we have

#J X (F q ) ≤ e(2g √ e) d-1 q g
where e is the base of the natural logarithm. The bound [START_REF] Mumford | Abelian varieties[END_REF] is better than the Lachaud-Martin-Deschamps bound when g is large and q is fixed, except when X has a small gonality with respect to the genus.

Let N q (g) = max X #X(F q ) where X runs over the set of (absolutely irreducible, smooth, projective, algebraic) curves of genus g defined over F q . Then the bound (6) can be applied to give the following proposition.

Proposition 4. We have J q (g) ≤ q + 1 + N q (g) -(q + 1) g g with equality if and only if there exists an optimal curve (i.e. with N q (g) points) with only one Frobenius eigenvalue (with its conjugate).

Remarks.

1. A curve that reaches the Serre-Weil bound, i.e. such that #X(F q ) = q + 1 + m must have x i = m for all i (see [START_REF] Lauter | Geometric methods for improving the upper bounds on the number of rational points on algebraic curves over finite fields[END_REF]). Thus, by Corollary 3, its jacobian is optimal i.e. has J q (g) points.

2. When g = 2, a jacobian with J q (g) points is the jacobian of an optimal curve (i.e. with N q (g) points) but the jacobian of an optimal curve doesn't always have J q (g) points. For example, when q = 3, there exist genus 2 curves with N 3 (2) = 8 points whose jacobian have 36, 35, 34 and 33 points (it follows from the results of [START_REF] Howe | Jacobians in isogeny classes of abelian surfaces over finite fields[END_REF]).

The case of jacobians of dimension 1 corresponds to elliptic curves. The values of J q (1) and j q (1) where calculated by Waterhouse [START_REF] Waterhouse | Abelian varieties over finite fields[END_REF].

Recall that q = p e .

Proposition 5.

(1) J q (1) is equal to

• (q + 1 + m) if e = 1, e is even or p |m • (q + m) otherwise. (2) j q (1) is equal to • (q + 1 -m) if e = 1, e is even or p |m • (q + 2 -m) otherwise.

Abelian surfaces over finite fields

In order to determine J q (2), we focus on abelian varieties of dimension 2. Let A be an abelian surface over

F q of type [x 1 , x 2 ].
Its characteristic polynomial f A (t) has the form

f A (t) = t 4 + a 1 t 3 + a 2 t 2 + qa 1 t + q 2
with a 1 and a 2 integers. We consider the real Weil polynomial of A:

fA (t) = (t + x 1 )(t + x 2 ) = t 2 + a 1 t + a 2 -2q.
By elementary computations, we can show that (see [START_REF] Rück | Abelian surfaces and Jacobian varieties over finite fields[END_REF] or [START_REF] Maisner | Abelian surfaces over finite fields as jacobians[END_REF]) the fact that the roots of f A (t) are q-Weil numbers (i.e. algebraic integers such that their images under every complex embedding have absolute value √ q) is equivalent to

|a 1 | ≤ 2m and 2|a 1 | √ q -2q ≤ a 2 ≤ a 2 1 4 + 2q. (7) Moreover, #A(F q ) = q 2 + 1 + (q + 1)a 1 + a 2 . ( 8
)
Table 1 gives all the possibilities for (a 1 , a 2 ) such that a 1 ≥ 2m -2.

The numbers of points are classified in decreasing order and an abelian variety with (a 1 , a 2 ) not in the table has a number of points strictely less than the values of the table. Indeed, if -2m ≤ a 1 < 2m -2, we have 1. Couples (a 1 , a 2 ) maximizing the number of points on A

a 1 a 2 Type Nb of pts 2m m 2 + 2q [m, m] (q + 1 + m) 2 2m -1 m 2 -m + 2q [m, m -1] (q + 1 + m)(q + m) m 2 -m -1 + 2q [m + -1+ √ 5 2 , m + -1- √ 5 2 ] (q + 1 + m + -1+ √ 5 2 )(q + 1 + m + -1- √ 5 2 ) 2m -2 m 2 -2m + 1 + 2q [m -1, m -1] (q + m) 2 m 2 -2m + 2q [m, m -2] (q + 1 + m)(q -1 + m) m 2 -2m -1 + 2q [m -1 + √ 2, m -1 - √ 2] (q + m + √ 2)(q + m - √ 2) m 2 -2m -2 + 2q [m -1 + √ 3, m -1 - √ 3] (q + m + √ 3)(q + m - √ 3) Table
(q + 1)a 1 + a 2 ≤ [(q + 1)a 1 + a 1 2 4 + 2q] ≤ [(q + 1)(2m -3) + (2m -3) 2 4 + 2q] = (q + 1)(2m -3) + m 2 -3m + 2 + 2q = (q + 1)(2m -2) + (m 2 -2m -2 + 2q) + (3 -(q + m)) < (q + 1)(2m -2) + (m 2 -2m -2 + 2q).
For the second inequality, note that the function x → (q + 1)x + x 2 4 is increasing on the interval [-2m, 2m -3].

In the same way, we build the table of couples (a 1 , a 2 ) with a 1 ≤ -2m + 2 (note that the extremities of the interval containing a 2 given by ( 7) depend only on the value of a 1 , hence the a 2 's are the same as in the previous table ).

a 1 a 2 Type Nb of pts -2m m 2 + 2q [-m, -m] (q + 1 -m) 2 -2m + 1 m 2 -m -1 + 2q [-m + 1+ √ 5 2 , -m + 1- √ 5 2 ] (q + 1 -m + 1+ √ 5
2

)(q + 1 -m + 1- √ 5
2 ) 2. Couples (a 1 , a 2 ) minimizing the number of points on A Again, the numbers of points are classified in increasing order and an abelian variety with (a 1 , a 2 ) not in the following table has a number of points strictly greater than the values of the table. Indeed, if -2m + 2 < a 1 ≤ 2m, we have

m 2 -m + 2q [-m, -m + 1] (q + 1 -m)(q + 2 -m) -2m + 2 m 2 -2m -2 + 2q [-m + 1 + √ 3, -m + 1 - √ 3] (q + 2 -m + √ 3)(q + 2 -m - √ 3) m 2 -2m -1 + 2q [-m + 1 + √ 2, -m + 1 - √ 2] (q + 2 -m + √ 2)(q + 2 -m - √ 2) m 2 -2m + 2q [-m, -m + 2] (q + 1 -m)(q + 3 -m) m 2 -2m + 1 + 2q [-m + 1, -m + 1] (q + 2 -m) 2 Table
(q + 1)a 1 + a 2 ≥ (q + 1)a 1 + 2|a 1 | √ q -2q ≥ (q + 1)(-2m + 3) + 2(2m -3) √ q -2q = (q + 1)(-2m + 2) + 2(2m -3) √ q -q + 1 = (q + 1)(-2m + 2) + (m 2 -2m + 1 + 2q) -(m 2 -2m + 1 + 2q) + 2(2m -3) √ q -q + 1 = (q + 1)(-2m + 2) + (m 2 -2m + 1 + 2q) -(m -1) 2 + 4(m -1) √ q -4q + q + 2 √ q + 1 = (q + 1)(-2m + 2) + (m 2 -2m + 1 + 2q) -(m -1 -2 √ q) 2 + ( √ q + 1) 2 > (q + 1)(-2m + 2) + (m 2 -2m + 1 + 2q).
For the second inequality, note that the function x → (q + 1)x+ 2|x| √ q is increasing on [-2m + 3, 2m] .

Jacobians of dimension 2

In this section, we determine J q (2) and j q (2). Theorems 6 and 7 will be proved in the following way:

(1) We look at the highest row of Table 1 or 2 (depending on the theorem being proved). ( 2) Then we check if the corresponding polynomial is the characteristic polynomial of an abelian variety.

(3) When it is the case, we check if this abelian variety is isogenous to a jacobian variety. (4) When it is not the case, we look at the following row and we come back to the second step. For the second step, we use the results of Rück [START_REF] Rück | Abelian surfaces and Jacobian varieties over finite fields[END_REF] and Maisner-Nart-Howe [START_REF] Maisner | Abelian surfaces over finite fields as jacobians[END_REF] who solved the problem of describing characteristic polynomials of abelian surfaces, in particular the fact that if (a 1 , a 2 ) satisfy ( 7) and p does not divide a 2 then the corresponding polynomial is the characteristic polynomial of an abelian surface.

For the third step, we use [START_REF] Howe | Jacobians in isogeny classes of abelian surfaces over finite fields[END_REF] where we can find a characterization of isogeny classes of abelian surfaces containing a jacobian.

The determination of J q (2) in Theorem 6 is closely related to that of N q (2) as was done by Serre [START_REF] Serre | Rational points on curves over finite fields[END_REF].

In order to simplify the proof of Theorem 7, we use the fact that given a curve of genus 2, if we denote by (a 1 , a 2 ) the coefficients of its characteristic polynomial, there exists a curve (its quadratic twist) whose coefficients are (-a 1 , a 2 ). This allows us to adapt the proof of Theorem 6.

Let us recall the definition of special numbers introduced by Serre. An odd power q of a prime number p is special if one of the following conditions is satisfied (recall that m = [2 √ q]): (1) m is divisible by p, (2) there exists x ∈ Z such that q = x 2 + 1, (3) there exists x ∈ Z such that q = x 2 + x + 1, (4) there exists x ∈ Z such that q = x 2 + x + 2. In [START_REF] Serre | Nombre de points des courbes algébriques sur Fq[END_REF], Serre asserts that if q is prime then the only possible conditions are conditions (2) and (3). When q is not a prime then condition (2) is impossible, condition (3) is possible only if q = 7 3 and condition (4) is possible only if q = 2 3 , 2 5 or 2 13 . Moreover, using basic arithmetic, it can be shown (see [START_REF] Lauter | The maximum or minimum number of rational points on genus three curves over finite fields[END_REF] for more details) that conditions (2), ( 3) and ( 4) are respectively equivalent to m 2 -4q = -4, -3 and -7.

The complete set of possible values of J q (2) is given in the following theorem. Theorem 6.

a) If q is a square, then J q (2) equals:

• (q + 1 + m) 2 if q = 4, 9 • 55 if q = 4 • 225 if q = 9 b) If q is not
a square, then J q (2) equals:

• (q + 1 + m) 2 if q is not special • (q+1+m+ -1+ √ 5 2 )(q+1+m+ -1- √ 5 2 ) if q is special and {2 √ q} ≥ √ 5-1 2 • (q + m) 2 if q is special, {2 √ q} < √ 5-1 2 , p = 2 or p|m • (q + 1 + m)(q -1 + m) otherwise.
Proof. a) If q is a square.

• Let q = 4, 9. Since N q (2) is the Serre-Weil bound (see [START_REF] Serre | Nombre de points des courbes algébriques sur Fq[END_REF]), by Proposition 4 a curve reaching this bound has type [m, m] (see [START_REF] Serre | Rational points on curves over finite fields[END_REF] for an explicit construction).

• Let q = 4. First we prove that J 4 (2) ≤ 55. We have m = 4 and q + 1 + 2m = 13. Every curve X of genus 2 over F q is hyperelliptic and therefore the number of rational points is at most 2(q + 1) = 10 = 13 -3. Since #X(F q ) = q + 1 + a 1 , we deduce that a jacobian of dimension 2 over F 4 must have a 1 ≤ 2m -3 = 5.

If a 1 = 5 then by [START_REF] Perret | Number of points of Prym varieties over finite fields[END_REF] we have: a 2 ≤ 14. The real Weil polynomial fA (t) of an abelian variety A over F 4 with a 1 = 5 and a 2 = 14 is fA

(t) = t 2 + a 1 t + a 2 -2q = t 2 + 5t + 6 = (t + 3)(t + 2) = (t + (m -1))(t + (m -2)).
This kind of polynomial is never the real Weil polynomial of a jacobian (because (m -1) -(m -2) = 1, see [START_REF] Howe | Jacobians in isogeny classes of abelian surfaces over finite fields[END_REF]). Thus we have a 2 ≤ 13 and a jacobian surface over F 4 with a 1 = 5 has at most q 2 + 1 + 5(q + 1) + 13 = 55 points by relation [START_REF] Quebbemann | Lattices from curves over finite fields[END_REF]. If a 1 < 5, we have

q 2 + 1 + (q + 1)a 1 + a 2 ≤ q 2 + 1 + (q + 1)a 1 + a 1 2 4 + 2q = 25 + 5a 1 + a 1 2 4 ≤ 25 + 5 × 4 + 4 2 4 = 49
(for the third row, note that the function x → 5x + x 2 4 is increasing on [-8, 4] (and a 1 ≥ -8)). Thus an abelian surface over F 4 with a 1 < 5 has less than 55 points, hence J 4 (2) ≤ 55.

It remains to prove that J 4 (2) ≥ 55. It is enough to prove that there exists a jacobian with (a 1 , a 2 ) = (5, 13) (because such a jacobian will have q 2 +1+5(q +1)+ 13 = 55 points). There exists an abelian surface with (a 1 , a 2 ) = (5, 13) (because p = 2 does not divide 13). Its real Weil polynomial is fA (t) = t 2 + 5t + 13 -2q = t 2 + 5t + 5

= (t + 5 + √ 5 2 )(t + 5 - √ 5 2 ) = (t + (m - 3 + √ 5 2
)

)(t + (m - 3 - √ 5 2
)).

Therefore its x i 's are not integers and thus this abelian surface is simple. Finally, using [START_REF] Howe | Jacobians in isogeny classes of abelian surfaces over finite fields[END_REF], we find that it is isogenous to a jacobian.

• If q = 9, we have m = 6 and q + 1 + 2m = 22. Moreover, 2(q + 1) = 20 = 22 -2.

Hence, we must have a 1 ≤ 2m-2. The highest row of Table 1 such that a 1 = 2m-2 is that with type [m-1, m-1]. By [START_REF] Waterhouse | Abelian varieties over finite fields[END_REF] there exists an elliptic curve of trace -(m-1) and by [START_REF] Howe | Jacobians in isogeny classes of abelian surfaces over finite fields[END_REF] the product of two copies of this curve is isogenous to a jacobian and has (q + m) 2 = 225 points.

b) If q is not a square.

In [START_REF] Serre | Rational points on curves over finite fields[END_REF], Serre proved the following facts:

• There exists a jacobian of type [m, m] if and only if q is not special.

• An abelian surface of type [m, m -1] is never a jacobian.

• If q is special, then there exists a jacobian of type

[m + -1+ √ 5 2 , m + -1- √ 5 2 ] if and only if {2 √ q} ≥ √ 5-1 2 . Note that {2 √ q} ≥ √ 5-1 2 is equivalent to m+ -1+ √ 5 2 ≤ 2
√ q, thus it is obvious that this condition is necessary.

• If q is special, {2 √ q} < √ 5-1
2 , p = 2 or p|m, then there exists a jacobian of type

[m -1, m -1]. • If q is special, {2 √ q} < √ 5-1
2 , p = 2 and p |m, that is, q = 2 5 or 2 13 (for

q = 2 3 , we have {2 √ q} ≥ √ 5- 1
2 ), then there exists a jacobian of type [m, m -2].

It remains to prove that for q = 2 5 and 2 13 , there does not exist a jacobian of type [m -1, m -1]. In fact, when q = 2 5 and 2 13 , an abelian variety with all the x i 's equal to (m -1) must have a dimension respectively multiple of 5 and 13 (see [START_REF] Maisner | Abelian surfaces over finite fields as jacobians[END_REF], Prop. 2.5).

The complete set of possible values of j q (2) is given in the following theorem.

Theorem 7.

a) If q is a square, j q (2) equals:

• (q + 1 -m) 2 if q = 4, 9 • 5 if q = 4
• 25 if q = 9 b) If q is not a square, j q (2) equals:

• (q + 1m) 2 if q is not special

• (q + 1 -m + 1+ √ 5
2

)(q + 1 -m + 1- √ 5
2 ) if q is special and {2 √ q} ≥ √ 5-1 2

• (q + 2 -m - √ 2)(q + 2 -m + √ 2) if q is special and √ 2 -1 ≤ {2 √ q} < √ 5-1 2 • (q + 1 -m)(q + 3 -m) if q is special, {2 √ q} < √ 2 -1, p |m and q = 7 3 • (q + 2 -m) 2 otherwise.
Proof. a) If q is a square.

• Using twisting arguments and the proof of Theorem 6, we see that if q = 4, 9, there exists a jacobian of type [-m, -m].

• Let q = 4 (and m = 4). First we prove that j 4 (2) ≥ 5. We have a 1 ≥ -5 since the quadratic twist of a curve with a 1 < -5 would have a 1 > 5 which is not possible (see the proof of Theorem 6).

If a 1 = -5 then by [START_REF] Perret | Number of points of Prym varieties over finite fields[END_REF] we have a 2 ≥ 12. The real Weil polynomial of an abelian variety A over F 4 with a 1 = -5 and a 2 = 12 is

fA (t) = t 2 + a 1 t + a 2 -2q = t 2 -5t + 4 = (t -4)(t -1) = (t -m)(t -(m -3)).
This is not the real Weil polynomial of a jacobian (it is one of an almost ordinary abelian surface, m 2 = 4q and m -(m -3) is squarefree, see [START_REF] Howe | Jacobians in isogeny classes of abelian surfaces over finite fields[END_REF]). Thus we have a 2 ≥ 13 and a jacobian surface with a 1 = -5 has at least q 2 + 1 -5(q + 1) + 13 = 5 points.

If a 1 > -5, we have

q 2 + 1 + (q + 1)a 1 + a 2 ≥ q 2 + 1(q + 1)a 1 + 2|a 1 | √ q -2q = 9 + 5a 1 + 4|a 1 | ≥ 9 + 5 × (-4) + 4 × 4 = 5
(for the third row, note that the function x → 5x + 4|x| is increasing on [-4, 8]).

Thus an abelian surface with a 1 > -5 has more than 5 points and our result is proved.

It remains to prove that j 4 (2) ≤ 5. There exists a jacobian with (a 1 , a 2 ) = (-5, 13): this is the jacobian of the quadratic twist of the curve with (a 1 , a 2 ) = [START_REF] Maisner | Abelian surfaces over finite fields as jacobians[END_REF][START_REF] Vlȃdut | An exhaustion bound for algebro-geometric "modular" codes[END_REF] in the proof of Theorem 6. This jacobian has q 2 + 1 -5(q + 1) + 13 = 5 points.

• If q = 9 (and m = 6), using the same argument as in the last step, we must have a 1 ≥ -2m + 2. We look at the rows of Table 2, beginning by the rows on the top, for which a 1 = -2m + 2. The first two can be ignored since {2 √ q} = 0 is less than √ 3 -1 and less than √ 2 -1. An abelian surface of type [-m, -m + 2] is not a jacobian (this is an almost ordinary abelian surface, m 2 = 4q and m -(m -2) is squarefree, see [START_REF] Howe | Jacobians in isogeny classes of abelian surfaces over finite fields[END_REF]). The product of two copies of an elliptic curve of trace (m -1) is isogenous to a jacobian (such a curve exists since 3 |(m -1)).

b) If q is not a square.

Using twisting arguments and the proof of Theorem 6, we see that:

• There exists a jacobian of type [-m, -m] if and only if q is not special.

• If q is special, there exists a jacobian of type [-m +

-m+1+ √ 3, -m+1- √ 3], it is necessary to have m-1+ √ 3 ≤ 2 √ q which is equivalent to {2 √ q} ≥ √ 3-1. When {2 √ q} < √ 5-1
2 , this condition is never satisfied (since

√ 5-1 2 < √ 3 -1)
.

• In order to have the existence of an abelian surface of type

[-m + 1 + √ 2, -m + 1 - √ 2], it is necessary to have {2 √ q} ≥ √ 2 -1.
Suppose that this condition holds; we will show that there exists an abelian surface of type

[-m + 1 + √ 2, -m + 1 - √ 2]
. We use the same kind of argument as Serre used in [START_REF] Serre | Rational points on curves over finite fields[END_REF]. If p|m, we are done since p

|a 2 = m 2 -2m -1 + 2q. Otherwise, (m -2 √ q)(m + 2 √ q) = m 2 -4q ∈ {-3, -4, -7}, hence {2 √ q} = 2 √ q -m = 4q-m 2 m+2 √ q ≤ 7 2m and if m ≥ 9, 7 2m < √ 2 -1.
It remains to consider by hand the powers of primes of the form x 2 + 1, x 2 + x + 1 and x 2 + x + 2 with m < 9 (i.e. q < 21). These powers of primes are precisely 2, 3, 4, 5, 7, 8, 13 and 17. For q = 2, 8, we have {2 √ q} ≥ √ 5-1 2 . For q = 3, we have p|m. For q = 4, 7, 13, 17, we have {2 √ q} < √ 2 -1. For q = 5, m = 4 and p = 5 do not divide a 2 = m 2 -2m -1 + 2q = 17. So we are done.

Finally, using [START_REF] Howe | Jacobians in isogeny classes of abelian surfaces over finite fields[END_REF], we conclude that this abelian surface is isogenous to a jacobian.

• We suppose that q is special, {2 √ q} < √ 2-1, p |m and q = 7 3 . Then p |(m-2). To see this, we take p = 2 (if p = 2, it is obvious) and we use the remark after the definition of "special". Suppose that p divides (m -2), then it also divides m 2 -4 -4q = (m + 2)(m -2) -4q. Since p = 2, we must have m 2 -4q ∈ {-3, -4}. If m 2 -4q = -3, p divides -3 -4 = -7 thus p = 7; q is not prime (since for q = 7, p |(m -2) = 5) therefore we must have q = 7 3 and this case is excluded. If m 2 -4q = -4, p divides -4 -4 = -8 thus p = 2 which contradicts our assumption. Thus the result is proved.

Finally, by [START_REF] Waterhouse | Abelian varieties over finite fields[END_REF] there exist elliptic curves of trace m and (m -2) and by [START_REF] Howe | Jacobians in isogeny classes of abelian surfaces over finite fields[END_REF] their product is isogenous to a jacobian.

• We suppose that q is special, {2 √ q} < √ 2 -1 and p|m, or q = 7 3 . By [START_REF] Waterhouse | Abelian varieties over finite fields[END_REF], if p|m, there does not exist an elliptic curve of trace m (q = 2 and 3 are excluded since in those cases, {2 √ q} ≥ √ 2 -1). If q = 7 3 (thus (m -2) = 35) there does not exist an elliptic curve of trace (m -2). Therefore, in both cases, an abelian surface of type [-m, -m + 2] cannot exist.

• If q is special, {2 √ q} < √ 2 -1 and p|m or q = 7 3 , then there exists a jacobian of type [-m + 1, -m + 1]: this is the jacobian of the quadratic twist of the curve of type [m -1, m -1] in the proof of Theorem 6.

Asymptotic parameters for jacobians

Let us consider the following aymptotic quantities: J q (∞) = lim sup g→∞ (#J q (g)) 1/g and j q (∞) = lim inf g→∞ (#J q (g)) 1/g . Proposition 8. We have q ≤ j q (∞) ≤ J q (∞) ≤ q + √ q ≤ q + 2 √ q + 1.

Proof. The first inequality comes from the lower bound of Lachaud-Martin-Deschamps, the third one from Proposition 1 and the last one is the Weil bound.

For q square, we have the following bound proved by Vlȃdut ¸ [START_REF] Vlȃdut | An exhaustion bound for algebro-geometric "modular" codes[END_REF].

Proposition 9. If q is a square, q( q q -1 ) √ q-1 ≤ J q (∞).

However, for q ≫ 0, we have q( q q -1 ) √ q-1 = q + √ q -1 2 + o(1).

Hence, for q large q + √ q -1 2 + o(1) ≤ J q (∞) ≤ q + √ q, which raises to the following question.

Question. What are the values between q + √ q -1 2 and q + √ q which are attained by a sequence of curves reaching the Drinfeld-Vlȃdut ¸bound?

  In order to have the existence of an abelian surface of type [

	1-2 , -m + 1+ √ 5 2 ] if and √ 5 • An abelian surface of type [-m, -m + 1] is never a jacobian. only if {2 √ √ 5-1 2 . q} ≥ Now suppose that q is special and {2 √ q} < √ 5-1 2 .
	•

Acknowledgements

I would like to thank my thesis advisor, Yves Aubry, for his help and encouragement. I am also grateful to Hamish Ivey-Law for his careful reading of this paper.


Audrey Rudel

Hélène Mazaleyrat

Construction du sens en contexte Polysèmes et items multi-facettes dans les syntagmes nominaux Construction du sens en contexte Polysèmes et items multi-facettes dans les syntagmes nominaux 1. Représentation conceptuelle de livre 1.1 Postulat 1.2 Livre : un nom à facettes 1.3 Modélisation sémantique 2. Construction du sens 2.1 Un phénomène complexe 2.2 Combinaison de livre avec l'adjectif cher 2.3 Combinaison de livre avec l'adjectif curieux

Conclusion

1.Fig. 2

 2 Fig.1 Schéma et élaborations


Simon Denier
email: simon.denier@emn.fr

Traits Programming with AspectJ

Keywords: traits, aspects, reusability, composition, Java, AspectJ. MOTS-CLÉS : traits, aspects, réutilisabilité, composition, Java, AspectJ

Traits as defined by Schärli, Ducasse et al. allow for the explicit handling of a meaningful set of methods. This article presents an attempt to map the trait model from Smalltalk to the Java language. We use AspectJ introduction mechanism to do this. Thus we enlighten purposes of locality and reusability shared by traits and structural AOP. RÉSUMÉ. Les traits tels que définis par Schärli, Ducasse et al. permettent la manipulation explicite d'un ensemble de méthodes sémantiquement corrélées. Cet article présente une tentative de portage du modèle des traits sous Smalltalk vers le langage Java. Nous réalisons ce portage à l'aide du mécanisme d'introduction d'AspectJ. Ainsi nous illustrons les objectifs de localité et de réutilisabilité communs aux traits et aux aspects structurels.

Introduction

Traits for class-based languages, as introduced in Squeak by [START_REF] Schärli | Traits: Composable Units of Behavior[END_REF], are lightweight entities which serve as basic building blocks for classes, primitive units of code reuse and elements of class structuration [START_REF] Ducasse | Traits: A Mechanism for finegrained Reuse[END_REF]. Traits come as sets of methods without state. The purpose of traits is reusability: they address some limits of inheritance in object-oriented languages. In particular they are well-suitable in single-inheritance language like Smalltalk, where they avoid code duplication. The compositional model supporting traits seamlessly integrates methods into classes: it allows a fine, explicit control on the composition of traits by means of operators. Programming style and practice [START_REF] Black | Applying Traits to the Smalltalk Collection Classes[END_REF] suggests that traits best fit as small sets of related methods.

As another single inheritance language, Java does not have such a mechanism. Taking a look at AspectJ -a language experimenting aspects in Java [START_REF] Kiczales | An overview of AspectJ[END_REF]) -we notice one special feature: inter-type declaration, formerly known as introduction. The inter-type declaration mechanism offers a subset of structural reflection, which allows for extension/modification of classes (with methods, fields, interfaces, etc. ) at compile-time. Inter-type declaration is described as a means to modularize crosscutting structure. In particular, basic examples (available on ❤tt♣✿✴✴❡❝•✐♣s❡✳♦r❣✴❛s♣❡❝t•✴ ) seem to pursue quite the same goals as traits: to provide general, possibly reusable, features (i.e. states and methods) in a modular way. One purpose of our work is to explore the relationship between traits and a structural subset of AOP.

The core section of this article describes our experiment to build a trait-like entity in Java thanks to AspectJ, with respect to the Squeak traits model. Starting with the need for structuration of trait behaviors, we will see how the reusability concern is handled, how we can express requirement and how we build a class from a composition of traits. The goal of this experiment however is not to propose an implementation for traits in Java, but rather to push existing mechanisms to an end to get a view of their limits. So we will finally see how conflicts are partially resolved by class delegation.

The remainder of this article is structured as follows. We give an overview of Smalltalk traits in section 2. Section 3 exposes our experiment to emulate traits properties in Java with AspectJ introduction. We then discuss in section 4 some questions on delegation, conflicts and static typing. Section 5 concludes with some thoughts on modules and aspects.

Overview of Smalltalk Traits

Inheritance is considered as a fundamental way to promote reusability in objectoriented languages. However the properties and constraints introduced by current mechanisms do not allow to achieve full capabilities (see [START_REF] Schärli | Traits: Composable Units of Behavior[END_REF] for a discussion on failures of different inheritance models).

In fact inheritance and in particular multiple inheritance suffers from a dual conception of class: as a generator of instances, a class must be completely descriptive; but as a unit of code reuse, it should stay small, focused on its subject matter. There is a discontinuity between the level at which we actually express and allow reuse (class, mixin) and the level at which we really want reuse (methods).

So traits propose to address part of the reusability concern at a finer-grained level, by providing sets of behaviors -each set being soundful and intended for generic use -a flattening yet seamless compositional model and some mechanisms for conflict resolution. Traits are well-suited for single inheritance languages, for which they avoid code duplication. However traits currently do not address state reuse, as no general solution is known for state conflict resolution1 .

Anatomy of a Trait

A trait is a stateless entity which provides a set of methods. Provided methods may be parameterized by some other methods, which are referred to as required. Required methods are only declared in the trait. In particular, required methods can be used to specify accessors for states/properties that the trait needs but does not define. Figure 1 shows a schematic view of a trait, as introduced by Schärli et al.

Using Traits

Class definition in Smalltalk has been enhanced with a ✉s❡ field, which allows for the declaration of expected traits. The ✉s❡ field cardinality is multiple, as a class may use many traits; the order of composition is irrelevant. A class eager to use a trait must provide implementations for the required methods by local means or with the help of other traits. Traits can use other traits in the same way a class does: they are called composite traits. The flattening property, one of the most distinguishing property of traits, happens at the class level: trait methods behave as if they were directly defined in the class. It follows that a class enhanced with traits can also be seen without the traits syntax. The use feature and the flattening property allow for the primitive composition mechanism, referred to as a sum. The equation below gives the point:

Class = Superclass + States + T raits + Glue
"Glue" is a vocable primarely covering methods that fulfill requirements from traits. Glue code is needed for a soundful composition of class definitions with traits. In Figure 1 using the trait and providing a state and the required method. Equivalent Smalltalk code is given

Conflicts Resolution

Smalltalk traits may conflict at composition time when providing identically named methods that do not originate from the same trait 2 . Two mechanisms help to resolve most of conflicts, whether in a class or in a composite trait: first methods defined by the compound entity (class or trait) have the property to override methods from used traits; second the use field can contain method excluding statements to suppress some trait methods from the compound. Another mechanism, aliasing, allows the access of overriden or deleted method by another name. Together these handle conflicts between provided methods by simply selecting one implementation among others, or by merging the implementations into a new one (see 3.4.1 for an illustration in Java).

Guidelines for Traits Implementation

From traits we retain the following ideas. These are the guidelines for our emulation in Java:

-ability to localize a set of (related) methods in a dedicated entity (and possibly reify a trait as a language element); 2. Then the diamond case is not considered a conflict.

Traits Programming with AspectJ 73 -reusability, provided by modularity, independance from class and easiness of use;

-ability to express requirement (as abstract methods for example); -ability to compose clearly and cleanly with classes; -ability to have composite traits -create traits from traits; -ability to detect and resolve conflicts in clean and meaningful ways.

Our Experiment: Emulating Traits in Java with AspectJ

Through this section, we target a ❈✐r❝•❡ class which we want to enhance with a color property in a modular fashion, aside from the inheritance tree. We will gradually introduce AspectJ notions and their roles in trait simulation, then examine the example in consequence. Finally we will see how we enhance our model with some kinds of physics property. Introducing this property will cause a conflict with the color property which we will then resolve.

Providing Methods in a Reusable Way

Traits provide behaviors independently from the traditional hierarchy of classes: methods definition is localized in a separate entity targeting a concern to promote sharing and maintenance. Its usage must be independent from the primary inheritance mechanism, which first expresses a type family.

Placing Methods in a Container

We need to place the code in a container apart from the class hierarchy, and then make accessible the implementation inside the container in an existing class. We do this by means of AspectJ introduction mechanism: an aspect module can contain some identifiers and method code which will be introduced at weaving time in the chosen classes. Then, at runtime, a modified class behaves as if introduced methods were directly defined in it. So we have some flattening property as in Smalltalk traits. As shown in Figure 2, the aspect module ❚❈♦•♦r|❡♣♦s✐t♦r② introduce methods ❣❡t❍✉❡✭✮ and ❣❡t|❡❞✭✮ in class ❈✐r❝•❡, as denoted by the special notation ❈✐r❝•❡✳❣❡t❍✉❡✭✮ and ❈✐r❝•❡✳❣❡t|❡❞✭✮.

In case the class already provides by itself a definition for a method, the compiler will not introduce the one from the aspect. Methods from class seem to have priority over introduced methods in AspectJ 3 . This is the same behavior as overriding trait methods in Smalltalk, which allows for the resolution of those conflict cases in a natural way. On the contrary, the AspectJ compiler is not able to dismiss the case when a method is both inherited from the superclass and introduced by an aspect: it declares this as a conflict. This conflict can still be resolved in the same way others are (see 3.4). This is a divergence from the Smalltalk flattening property, whose definition implies inherited methods are overriden by traits methods. We will discuss this in 4.2.

From the programmer point of view, the aspect acts as a container for the trait implementation but does not hold the trait identity. That is we can use the color trait unaware of the ❚❈♦•♦r|❡♣♦s✐t♦r② module. By convention, we call the aspect a repository and distinguish it with a name suffix. So our trait-like entities will each have their own ❚<♣r♦♣❡rt②>|❡♣♦s✐t♦r②.

❛s♣❡❝t ❚❈♦•♦r|❡♣♦s✐t♦r② ④ ♣✉❜•✐❝ ❢•♦❛t ❈✐r❝•❡✳❣❡t❍✉❡✭✮ ④ ❢•♦❛t❬( ❤s❜ ❂ ❈♦•♦r✳|•❇t♦❍❙❇✭❣❡t|❡❞✭✮✱ ❣❡t•r❡❡♥✭✮✱ ❣❡t❇•✉❡✭✮✱ ♥✉••✮❀ r❡t✉r♥ ❤s❜❬✵(❀ ⑥ ♣✉❜•✐❝ ✐♥t ❈✐r❝•❡✳❣❡t|❡❞✭✮ ④ r❡t✉r♥ t❤✐s✳❣❡t|•❇✭✮✳❣❡t|❡❞✭✮❀ ⑥ ✴✴ ❣❡t|•❇✭✮ ✐s ❛ r❡q✉✐r❡♠❡♥t ✳✳✳ ✴✴s❛♠❡ ❢♦r ❣❡t•r❡❡♥✭✮✱ ❣❡t❇•✉❡✭✮ ⑥ Figure 2.
An aspect module as a repository for behavior implementation 3.1.2. Using and Reusing Methods from the Container So far we have localized the implementation but we need more to be able to reuse the methods code: we should not have any reference to classes that use the trait. Currently this is not the case as we do have a reference to ❈✐r❝•❡ within ❚❈♦•♦r|❡♣♦s✐t♦r②. Suppose we want to use our trait in a |❡❝t❛♥❣•❡ class: we have to introduce the reference in ❚❈♦•♦r|❡♣♦s✐t♦r② and, even worse, to duplicate the code as this syntax does not allow to introduce the same portion in multiple classes.

In fact, the problem happens because we implicitly instantiate the use relation between ❈✐r❝•❡ and the trait inside ❚❈♦•♦r|❡♣♦s✐t♦r② (notice we have not changed the ❈✐r❝•❡ class for the time being). But it is rather a responsibility of the class to explicitly express its need for the trait with a use relation. Then we also need a way to express the use relation in a class.

Fortunately, AspectJ's introduction allows introduction in combination with an interface. This means that AspectJ can declare a method as part of an interface and link it to the implementation in an aspect module. In Figure 3, we use the interface ❚❈♦•♦r as an identifier for the color property. The interface declares its methods signatures: although it is not mandatory in AspectJ introduction, we gain in clarity by doing so. Then the ❚❈♦•♦r|❡♣♦s✐t♦r② module gives an implementation for each method in the ❚❈♦•♦r interface.

By this we throw away the reference to the ❈✐r❝•❡ class and replace it with a kind of abtract reference, in total independance with the existing hierarchy. But this interface usage also gives the use relationship we need: any class which implements a trait interface is provided with the implementation in the repository, thanks to AspectJ introduction (Figure 4). As the implement relation coexists in Java with the extend relation, we can use a trait interface anywhere in a class hierarchy. So our trait is reusable. The key relation is that the interface stands for the trait identity.

✐♥t❡r❢❛❝❡ ❚❈♦•♦r ④ ✴✴ ❚r❛✐t ✐♥t❡r❢❛❝❡ ♣✉❜•✐❝ ✐♥t ❣❡t|❡❞✭✮❀ ♣✉❜•✐❝ ❢•♦❛t ❣❡t❍✉❡✭✮❀ ✳✳✳ ⑥ ❛s♣❡❝t ❚❈♦•♦r|❡♣♦s✐t♦r② ④ ✴✴ ❇❡❤❛✈✐♦rs ✐♠♣•❡♠❡♥t❛t✐♦♥ ♣✉❜•✐❝ ✐♥t ❚❈♦•♦r✳❣❡t|❡❞✭✮ ④ ✳✳✳⑥ ✴✴ ❝❤❛♥❣❡ ❈✐r❝•❡ t♦ ❚❈♦•♦r ♣✉❜•✐❝ ❢•♦❛t ❚❈♦•♦r✳❣❡t❍✉❡✭✮ ④ ✳✳✳ ⑥ ✴✴ ❝❤❛♥❣❡ ❈✐r❝•❡ t♦ ❚❈♦•♦r ✳✳✳ ⑥

Expressing Requirements

Traits need to integrate their behaviors with classes they are applied to: they need some requirements from classes to interact with. Moreover, since states are not allowed in traits, classes have to provide the necessary structure to support traits. These requirements take the form of methods signatures, particularly accessors, which classes have to implement jointly with some states.

A Java interface, referred to as a requirement interface (Figure 4), is a natural way to express those: required methods take the form of declared methods in the interface which are simply not implemented by the aspect module. A class eager to use a trait then fullfills these requirements with "glue code": it provides some implementation for the interface as well as the underlying structure (states) implied. In our emulation we make the trait interface extends the requirement interface 4 . Moreover we have the compiler to check the implementation of required methods.

✐♥t❡r❢❛❝❡ ❚❈♦•♦r|❡q✉✐r❡♠❡♥t ④ ✴✴ ❚r❛✐t r❡q✉✐r❡♠❡♥t ♣✉❜•✐❝ ❈♦•♦r ❣❡t|•❇✭✮❀ ✳✳✳ ⑥ ✐♥t❡r❢❛❝❡ ❚❈♦•♦r ❡①t❡♥❞s ❚❈♦•♦r|❡q✉✐r❡♠❡♥t ④ ✴✴ ❚r❛✐t ✐♥t❡r❢❛❝❡ ♣✉❜•✐❝ ✐♥t ❣❡t|❡❞✭✮❀ ♣✉❜•✐❝ ❢•♦❛t ❣❡t❍✉❡✭✮❀ ✳✳✳ ⑥ ❛s♣❡❝t ❚❈♦•♦r|❡♣♦s✐t♦r② ④ ✴✴ ❇❡❤❛✈✐♦rs ✐♠♣•❡♠❡♥t❛t✐♦♥ ✳✳✳ ⑥ ❝•❛ss ❈✐r❝•❡ ✐♠♣•❡♠❡♥ts ❚❈♦•♦r ④ ♣r✐✈❛t❡ ❈♦•♦r ❝♦•♦r❀ ✴✴ ❙t❛t❡ ❢♦r r❡q✉✐r❡♠❡♥t ♣✉❜•✐❝ ❈♦•♦r ❣❡t|•❇✭✮④ r❡t✉r♥ ❝♦•♦r❀ ⑥ ✳✳✳ ⑥ Figure 4.
The complete ❚❈♦•♦r trait example with repository, interface and requirement. Then the ❈✐r❝•❡ class provides some state and the required method ❣❡t|•❇✭✮ so as to implement the ❚❈♦•♦r trait and benefit from it

Composing Traits

Traits target the reusability concern in a way similar to multiple inheritance. That is, a class should be able to use behaviors from multiple origins -its own corpus, from parent classes and from traits. So composition is a major feature of traits implementation. It should be carefully looked at as it is also a source of conflicts.

Composing a class from multiple traits is straightforward since interfaces in Java are not constrained by the single inheritance rule. So a class just has to declare implementing as many trait interfaces it wants, as long as this does not induce some conflicts.

4. The application of the introduction mechanism to interfaces is disturbing as it breaks the programmer's view of Java interfaces as a mean to enforce signatures. With AspectJ the role is reversed as interface becomes a mean to provide implementation. That is why we separate both concerns -contract enforcement by required methods and reusable code by provided methods -in two different interfaces.

Composing a trait from multiple traits -producing composite traits -is also straightforward as an interface could extend some others. Then a trait eager to reuse another one must extend the other's interface. By this, it can also provide some implementations for the used trait requirements; unprovided requirements are still being propagated by the interface mechanism.

By propagation of types, the AspectJ weaver recognizes all traits interfaces and introduces methods from the right repositories. In Figure 5, trait ❚❈♦•♦r is enhanced with trait ❚❊q✉❛•✐t② which provides some identity predicates. However the syntax seems confusing as the same mechanism (extension of interface) allows for the propagation of requirement and for the composition of trait. Conflicts ask for a meaningful composition dealing with the class purpose. There is many ways to do this: it could be by separating, prioritizing, excluding, merging, encapsulating or cross-referencing features [START_REF] Bracha | Modularity meets inheritance[END_REF]. Some of these solutions imply combinations of aliasing and exclusion as in Smalltalk while others need some kind of scope or module.

✐♥t❡r❢❛❝❡ ❚❊q✉❛•✐t② ❡①t❡♥❞s ❚❊q✉❛•✐t②|❡q✉✐r❡♠❡♥t ④ ✳✳✳ ⑥ ✐♥t❡r❢❛❝❡ ❚❈♦•♦r ❡①t❡♥❞s ❚❈♦•♦r|❡q✉✐r❡♠❡♥t✱ ❚❊q✉❛•✐t② ④ ✳✳✳ ⑥
AspectJ can not manipulate methods at the grain aliasing and excluding operators ask for. To work around conflicts, we have to come back to delegation which supports many features -although in a verbose manner.

Using Delegate Classes

Class delegation is a way for a complex class to deal with its multiple behaviors in a modular way. Delegating means that these behaviors will actually be provided by smaller agregated classes. For the purpose of traits, a delegate class can be an intermediary level between a trait and the compound class: it comes with the minimal structure and requirement needed by the trait so as to be self-sufficient, but it does not realize the integration with others traits. It is still the class responsibility to do so.

There is many ways a class can work with delegates, according to its purpose: it can enable state sharing between delegates or, on the contrary, ask for a strict separation of state. It satisfies message sends to trait methods by itself or by forwarding to its delegates. Indeed, thanks to the class over trait priority rule (see 3.1.1), its self- -selecting one behavior among others by delegating to it;

-merging delegate behaviors in the new delegating method;

-aliasing a behavior thanks to an indirection to a delegate.

In Figure 6, we enhance the ❈✐r❝•❡ class with ❚❈♦•♦r and ❚P❤②s✐❝s traits. As both traits provide an ❡q✉❛•s✭❖❜•❡❝t✮ method, each one targeting its own property, conflict arises. In Figure 7, we choose to merge both behaviors. We could also have excluded the color equality property from the default ❡q✉❛•s method, but still provided it via an alias, like in Figure 8.

Here are the guidelines for composing conflicting traits. First, each trait should be provided with a delegate class which implements the trait interface and satisfies structure and requirement. Then, for each trait, the compound class comes with a private instance variable of the associate delegate class. A delegate class can be designed to be sufficiently generic and reusable, or to simply fit the compound class purpose. The compound class can also implement all trait interfaces and, for each provided or required methods, must provide either a delegating method or one of the solution above. -resolve name conflicts by merging, excluding, aliasing; -provide a meaningful, carefully selected, interface for the compound class; -restrain the execution scope of a trait to a delegate, thus disabling possible semantic mischieves (see 4.2); -create either specific or generic delegate class, thus enabling for a set of reusable "glue" classes for traits.

❝•❛ss ❈♦•♦r ✐♠♣•❡♠❡♥ts ❚❈♦•♦r ④ ✴✴ ❉❡•❡❣❛t❡ ❝•❛ss ♣r✐✈❛t❡ •❛✈❛✳❛✇t✳❈♦•♦r ❝♦•♦r❀ ♣✉❜•✐❝ •❛✈❛✳❛✇t✳❈♦•♦r ❣❡t|•❇✭✮ ④ ✳✳✳ ⑥ ✳✳✳ ⑥ ❝•❛ss P❤②s✐❝s ✐♠♣•❡♠❡♥ts ❚P❤②s✐❝s ④ ✳✳✳ ⑥ ✴✴ ❉❡•❡❣❛t❡ ❝•❛ss ✴✴ ❯s✐♥❣ ❝•❛ss ❝•❛ss ❈✐r❝•❡ ✐♠♣•❡♠❡♥ts ❚❈♦•♦r✱ ❚P❤②s✐❝s ④ ♣r✐✈❛t❡ ❈♦•♦r ❝♦•♦r❀ ✴✴ tr❛✐t ❚❈♦•♦r ❞❡•❡❣❛t❡ ♣r✐✈❛t❡ P❤②s✐❝s ♣❤②s✐❝s❀ ✴✴ tr❛✐t ❚P❤②s✐❝s ❞❡•❡❣❛t❡ ✴✴ ❞❡•❡❣❛t✐♥❣ ♠❡t❤♦❞ ♣✉❜•✐❝ ✐♥t ❣❡t❍✉❡✭✮ ④ t❤✐s✳❝♦•♦r✳❣❡t❍✉❡✭✮❀ ⑥ ✳✳✳ ✴✴ ♠❡r❣✐♥❣ ❡q✉❛•s ❜❡❤❛✈✐♦r ♣✉❜•✐❝ ❜♦♦•❡❛♥ ❡q✉❛•s✭❖❜•❡❝t ❛♥❖❜•❡❝t✮ ④ r❡t✉r♥ ♣❤②s✐❝s✳❡q✉❛•s✭❛♥❖❜•❡❝t✮ ✫✫ ❝♦•♦r✳❡q✉❛•s✭❛♥❖❜•❡❝t✮❀ ⑥ ⑥
But there is also some drawbacks from the use of delegation in our design. There is the cost of implementing the framework -delegate class and delegating behaviors -although most IDEs nowadays can do this in an automatic way. There is the cost of one indirection at runtime. However the main drawback comes from the breaking of flattening property. We discuss this issue in section 4.1.

✴✴ ❯s✐♥❣ ❝•❛ss ❝•❛ss ❈✐r❝•❡ ✐♠♣•❡♠❡♥ts ❚❈♦•♦r✱ ❚P❤②s✐❝s ④ ♣r✐✈❛t❡ ❈♦•♦r ❝♦•♦r❀ ♣r✐✈❛t❡ P❤②s✐❝s ♣❤②s✐❝s❀ ✳✳✳ ✴✴ ❡①❝•✉❞✐♥❣ ❚❈♦•♦r✳❡q✉❛•s✭✮ ♣✉❜•✐❝ ❜♦♦•❡❛♥ ❡q✉❛•s✭❖❜•❡❝t ❛♥❖❜•❡❝t✮ ④ r❡t✉r♥ ♣❤②s✐❝s✳❡q✉❛•s✭❛♥❖❜•❡❝t✮❀ ⑥ ✴✴ ❛•✐❛s✐♥❣ ❚❈♦•♦r✳❡q✉❛•s✭✮ ♣✉❜•✐❝ ❜♦♦•❡❛♥ ❝♦•♦r❊q✉❛•s✭❖❜•❡❝t ❛♥❖❜•❡❝t✮ ④ r❡t✉r♥ ❝♦•♦r✳❡q✉❛•s✭❛♥❖❜•❡❝t✮❀ ⑥ ⑥ Figure 8
. An alternative to merging: first the ❚P❤②s✐❝s ❡q✉❛•s behavior is preferred to the ❚❈♦•♦r one as default behavior (or, the ❚❈♦•♦r ❡q✉❛•s is excluded); second, the ❚❈♦•♦r ❡q✉❛•s behavior is provided via an alias

Conflicts in Composite Traits

Creating a composite trait actually means composing abstract methods in interfaces. So a conflict in a composite trait will not be detected until a class actually uses it and the weaver tries to introduce the different implementations. Thus the resolution of conflicts in a composite trait is postponed to classes using it. Moreover this began to be cumbersome when we have an hierarchy of composite traits, each possibly coming with conflicts.

The AspectJ compiler allows overriding in composite traits: methods provided by the composite override methods from inherited traits. But as we have no aliasing, we can not reuse the overriden methods: this clearly limits the practical interest for this use of overriding. One solution is to provide for each composite a generic delegate class. But then, dealing with both an hiearchy of traits and a set of delegate classes to reuse is cumbersome. Using delegates is definitely not as simple as traits are.

To conclude, we can not express all the semantics of composite traits in a modular way: resolving conflicts in composite traits is the point where our emulation fails.

Discussions

This section presents some thoughts on conceptual and practical issues dealing with the traits model as well as with our emulation. We highlight some questions con-cerning delegation, conflicts or types -sometimes to give a provocative look towards Smalltalk model choices.

Delegation: Drawbacks and Enhancement

An obvious criticism of delegation is the cost and the noise introduced by delegating methods, which produce runtime overcost and confuse the programmer about class capabilities. However the main drawback, which is a conceptual one, is that the scope introduced by delegation breaks the flattening property. The scope allows for a clean separation of context and obvious semantics of behaviors (see 4.2.2). The flattening property allows for a quick and seamless integration of traits: for example one can fulfill a requirement from another in an oblivious way. Doing such a thing with delegation means having a reference in the delegate outside its scope. This implies a link to the current implementation and so a more fragile structure.

On the other hand, delegates offer the means to provide generic, reusable stateful traits; although they will not fit well everywhere, they are direct to implement, easy to understand and have obvious semantics. Notice that we are not arguing that traits should be delegate classes, but that each trait could come with its delegate class. This benefits the reusability concern.

Cases for Conflicts

Generally speaking a conflict arises when features with the same identity are combined along different paths. The identity is primarely defined by a name and could be refined by type or signature for a method. The path distinction is prone to discussion depending on whether or not a same origin make the distinction (diamond case).

Conflicts Between Traits

When composing multiple traits in the same class, the obvious case for conflict arises when provided methods with same name are picked up along different paths (case A in Figure 9). Since the order is irrelevant, the language can not choose between different implementations. However, if the origin is the same, then we could easily consider there is no conflict; if not, then a choice need to be made by the programmer, depending on the purpose of its class.

The problem is a bit different when it comes about required methods combined along different paths (case B in Figure 9). Due to a lack of semantics on requirements, the common hypothesis is to consider those identical as the same: then one implementation should fit for all. The right question is: does the concerned traits share the same requirements? By example, should they share a state or not? In case they should, how do we check that range constraints on type are compatible? Among solutions we can think of a scope notion or some kind of aliasing for requirement.

A last case (C in Figure 9) does not have to be considered as a conflict but rather as a potential pitfall for programmer. When a provided method from a trait matches a required method from another one, the first will silently satisfy the second at composition time. The operation could be oblivious to the programmer -this is different from the case where the programmer gives a "glue" method to fulfill a requirement. However, as both traits could have been developped in independance, nothing tells us that the implementation really fits the requirement. Thus a warning could be issued by the language to inform the programmer.

Stress between Classes and Traits

The overriding property of class methods over trait methods allows one to resolve conflicts in a nice and simple way. However, its lack of scope can also break the implementation, as the overriding code will impact on the whole trait hierarchy used. Suppose a method provided by a trait is called in the trait subhierarchy. Then, if a class using this trait hierarchy overrides this method with its own implementation, the last one will prevail, even in place where the previous was appropriate.

Finally we consider the rule for conflicts between traits and the superclass. In the Smalltalk model, trait methods override ones from the superclass. Although this is a natural consequence of the flattening property, deciding which methods from a superclass or from a trait to choose depends on the class purpose. A priori, one could argue that superclass possibly best fits the particular concern addressed by the class whereas a trait targets a generic yet broad concern. In AspectJ the choice is to declare this as a conflict.

On Conflicts, Scope and Modules

The use of scope to solve conflicts is a fundamental of module systems. However this notion operates at large scale and can be favourably enhanced by fine-grained operations like aliasing or excluding: this is the proposal from Bracha in its Jigsaw model [START_REF] Bracha | Modularity meets inheritance[END_REF]) -but we must then give up some simplicity. A more recent alternative comes from Classboxes [START_REF] Bergel | Classboxes: Controlling Visibility of Class Extensions[END_REF], a module system where local rebinding allow to narrow the impact of changes to local features -possibly helping with the problem above.

Static Types Impact

The implementation of traits in a statically typed language like Java raises new questions, on the way traits are intrinsically influenced by types but also on the way they impact on the existing type system. At first sight types can reduce the number of conflicts by distinguishing between signatures, in the same way they allowed overloading. However one should be aware of a pitfall already present in the type system of Java: when considering concrete methods, the language makes no distinction between those who only differ by their return type; but this difference exists when it comes about abstract methods. So one can be trapped in a pathological case where requirements differing by their return types can not be satisfied, as implementations conflict.

Another, more practical, question is if user-defined types are allowed in traits definition. For example, a method parameter could be defined by a dedicated interface. Thus the interface becomes part of the requirements: it is the responsibility of the trait user to define a class satisfying the parameter interface. It can enhance the expressivness of traits as well as disturbing its usability by asking "too much" requirements. Also the design of reusable traits could entail the use of the ❖❜•❡❝t class as a generic type, which implies casting or exception handling. We can expect that generics in Java 5 will ease this situation.

At last, a fundamental problem arises when thinking of a trait itself as a type. Without trait typing, using a trait in Java would mean that trait methods are transformed to be plain members of their compound class and loose connection with the trait identity. For example a binary method ❢♦♦✭❚✮ defined by a trait ❚ will be later refined as ❢♦♦✭❇✮ as a class ❇ is using it. Clearly we lost some genericity in doing so, as a instance of class ❈ using ❚ could not be passed to ❢♦♦✭❇✮.

In our experiment, the trait identity is assimilated to an interface: so using a trait for a class means being of the trait type, in the same way as implementing an interface or extending a class. Therefore a r❡t✉r♥ t❤✐s statement in a trait method simply returns an object of the trait type. This also enables the direct definition of ❢♦♦✭❚✮ in class ❇ and so an instance of class ❈ can be passed: this is consistent with what we expect from polymorphism in Java.

Yet we have another problem considering the exclusion operator. Take a class C excluding from its interface a method of trait T: if we call this method on a T instance (T being the static type) which resolves in a C instance at runtime, the Java virtual machine will stop by throwing ◆♦❙✉❝❤▼❡t❤♦❞❊rr♦r. Indeed one purpose of the static type system is to guarantee at compilation time that any method call is correct. Our first assumption combined with the exclusion definition breaks the safety of the type system. The use of interfaces as trait identifier in our emulation, coming from AspectJ, is misleading.

We can think of two solutions to work around this problem. The first and easy one is simply to throw away the excluding operation. Indeed we can still resolve conflicts by overriding them 5 . But we lost some capabilities and ease of use by doing so. The second solution is to change the semantics of the exclusion: it defines a new type which is not any more compatible with the original trait type and its hierarchy. This can be interpreted as the fact that it defines a new anonymous trait before composition without the subtyping relation 6 . So there is a cost in using the exclusion operator which the programmer should be aware of. To push this thought to an end, we observe the same kind of duality which classes suffer with inheritance: classes play a role in implementing methods but they also play a role in defining a type, which primarely guarantees the interface.

Conclusion

The benefits in reusability, maintenance and readibility expected from traits are of high value for a mainstream language like Java. We enlighten in this article some key ideas for an implementation of traits. We illustrate those by an experiment with AspectJ introduction mechanism and interfaces, emulating some traits properties. However we did not achieve the full capabilities of traits regarding conflicts resolution, as we lack fine-grained operators. Even so we were able to get a good emulation by using delegation, providing both drawbacks and enhancements.

We also extend our thoughts on conflicts by showing how the flattening property can sometimes lead to a non-desirable behavior, and how a scope notion is useful in those cases. Finally we discuss with some thought experiments in Java the way static types impact the trait model and reciprocally. In particular we show that a naive interpretation of exclusion breaks the safety of static typing. We propose some solutions to preserve this safety, either with or without exclusion.

A key idea underlying this article is that there is a thin connection between traits and a subset of AOP. Structural AOP purpose is to localize scattered, related struc-5. We still need the aliasing operator to get good reusability. 6. A trait containing aliasing also defines a new type but does not break the subtyping relation.

tures in a single entity: so it shares the localization concern with traits. But the reusability concern, although often hidden in AOP, is also of great value -as shown by the AspectJ introduction properties. However traits rely on an explicit mechanism whereas aspects make much more use of implicit properties, thanks to pointcuts. We show here that this process can be made partly explicit.

The structural part of AspectJ is first conceived as a support for behavioral aspects. It lacks some features to deal with conflicts. In particular, the notion of obliviousness in AOP [START_REF] Filman | Aspect-Oriented Programming is Quantification and Obliviousness[END_REF] is often discussed as it exposes to the same pitfalls that follow from lack of scope in traits model -as shown in [START_REF] Aldrich | Open Modules: Modular Reasoning in Aspect-Oriented Programming[END_REF]. As underlined in [START_REF] Douence | Composition, Reuse and Interaction Analysis of Stateful Aspects[END_REF], interaction detection and composition (that is, conflict resolution) of advices in behavioral AOP is a key feature for aspects. Then the idea of composition and conflict resolution in structural AOP seems relevant: we could pay attention to traits operators in order to improve static programming and weaving. It is noticeable that HyperJ [START_REF] Ossher | Multi-Dimensional Separation of Concerns and The Hyperspace Approach[END_REF], which focuses on structure, proposes some similar operators. Future work involves implementing operators for conflicts resolution in an AspectJ-like extensible weaver: one idea could be to use the Reflex framework [START_REF] Tanter | Versatile Kernels for Aspect-Oriented Programming[END_REF] extended for structural reflection.

Figure 1 .

 1 Figure 1. The trait ❚❈♦•♦r provides some methods shown in left column and requires the method in right column. The ❈✐r❝•❡ class is enhanced with ❚❈♦•♦r behaviors by

Figure 3 .

 3 Figure 3. A basic trait example. Notice how the name changes from ❈✐r❝•❡ to ❚❈♦•♦r in the introduction syntax since Figure 2

Figure 5 .

 5 Figure 5. An example of trait composition: trait ❚❈♦•♦r reuses ❚❊q✉❛•✐t②

Figure 6 .

 6 Figure6. Using simultaneously ❚P❤②s✐❝s and ❚❈♦•♦r traits is a case for conflict, as both provides ❡q✉❛•s method with their own meaning. For ❈✐r❝•❡ class, the problem can be solved by containing each implementation in a delegate class and providing a meaningful interface, either delegating, merging, excluding behaviors from delegates

Figure 7 .

 7 Figure 7. A complete example of conflict resolution by delegation and merging, from Figure 6. ❣❡t❍✉❡ is a single behavior of ❚❈♦•♦r and hence needs only delegating to the ❈♦•♦r instance. ❡q✉❛•s behaviors from ❚❈♦•♦r and ❚P❤②s✐❝s are merged

Figure 9 .

 9 Figure9. Three cases for conflicts in trait composition. When combining together ❚•r❛♣❤ and ❚▲♦❝❛t✐♦♥, the programmer should dismiss conflicts and meaningless links by carefully examining the situation. (A) Which behavior to choose for ❡q✉❛•s? One of both or a merge of both or nothing? (B) Does ❚▲♦❝❛t✐♦♥ and ❚•r❛♣❤ share the same requirement when asking for ❣❡t❳? (C) Is the ❣❡t( implementation from ❚•r❛♣❤ convenient for the requirement of ❚▲♦❝❛t✐♦♥?

We used version 0.5 of the traits prototype for Squeak during our experiments.

This is an experimental result. Although not described in the AspectJ documentation, this is an intuitive behavior.