Explicit L-functions and a Brauer-Siegel theorem for Hessian elliptic curves

For a finite field $\mathbb{F}_q$ of characteristic $p\geq 5$ and $K=\mathbb{F}_q(t)$, we consider the family of elliptic curves $E_d$ over $K$ given by $y^2+xy - t^dy=x^3$ for all integers $d$ coprime to $q$. We provide an explicit expression for the $L$-functions of these curves in terms of Jacobi sums. Moreover, we deduce from this calculation that the curves $E_d$ satisfy an analogue of the Brauer-Siegel theorem. More precisely, we estimate the asymptotic growth of the product of the order of the Tate-Shafarevich group of $E_d$ (which is known to be finite) by its N\'eron-Tate regulator, in terms of the exponential differential height of $E_d$, as $d\to\infty$.Comment: 16 pages, Comments welcom

On the arithmetic of a family of twisted constant elliptic curves

Let $\mathbb{F}_r$ be a finite field of characteristic $p>3$. For any power $q$ of $p$, consider the elliptic curve $E=E_{q,r}$ defined by $y^2=x^3 + t^q -t$ over $K=\mathbb{F}_r(t)$. We describe several arithmetic invariants of $E$ such as the rank of its Mordell--Weil group $E(K)$, the size of its N\'eron--Tate regulator $\text{Reg}(E)$, and the order of its Tate--Shafarevich group $III(E)$ (which we prove is finite). These invariants have radically different behaviors depending on the congruence class of $p$ modulo 6. For instance $III(E)$ either has trivial $p$-part or is a $p$-group. On the other hand, we show that the product $|III(E)|\text{Reg}(E)$ has size comparable to $r^{q/6}$ as $q\to\infty$, regardless of $p\pmod{6}$. Our approach relies on the BSD conjecture, an explicit expression for the $L$-function of $E$, and a geometric analysis of the N\'eron model of $E$.Comment: 38 pages. v2: minor changes following referee repor

Foresight to reduce asymmetry of information : the experience of the french national research agency

ANR is a public organization devoted to competitive project funding in both fundamental and applied research. ANR was created in 2005 after the vote of a law whose objective was to improve the French Research and Innovation system. ANR is both a result of the evolution of the French research and innovation system and a tool to confirm and stabilize this evolution. The objective of ANR is to promote creativity, to bring more flexibility and, subsequently, reactivity, and to increase competitiveness while keeping a good balance between fundamental research and applied research. ANR addresses both public research institutions and industries with a double mission of producing new knowledge and promoting interaction between public laboratories and industrial laboratories through the development of partnerships. ANR disposes of programs and calls for proposals, but it cannot define on its own the content of these programs. It needs knowledge about the evolution of science at national and international levels, about the evolution of society at national and international levels, about national priorities as well as European ones, about the strategy of research institutions, universities and industries, about competencies, markets¿ and needs¿ analysis. Each year, at the beginning of the programming step, there is asymmetry of information between ANR and scientists from French and European research organisations and universities, scientists from industries and policymakers in different ministries. The agency currently has three broad approaches to define its programs and reduce asymmetry of information: (a) monitoring and Intelligence; (b) expert opinion, i.e. consultations of individuals or panels in the form of an annual epistolary consultation, international consultations and foresight workshops, a Foresight Council as well as sectoral and steering committees; and (c) analysis, i.e. extrapolation of perceived trends through the analyses of proposals and end of programs workshops. The approaches adopted by ANR leave important challenges for the future: ANR must express its underlying paradigms as it has entered the political and the economic arenas; it must stay close to frontier research and reduce the time span between anticipation and action; last but not least, it must drastically increase the number of foreign scientists in its committees. (Résumé d'auteur

Sécurité alimentaire et mondialisation ?

The world food problem becomes more and more a problem of: - Economic and social equity on a global scale.- Environmental policy because the future of the biosphere through the future land is at stake\; - Trade policy: it is hard to imagine that structural adjustment policies adverse to agriculture and trade regime of WTO can persist in this context, including the acceptance of an environmental dumping and social dumping on the part of Agriculture Latin American issues that will come on the agenda of WTO negotiations. - Agricultural policy because it is not clear how great productive effort that is necessary may be possible without an accompanying agricultural policy. - Technological invention because it must combine productivity and ecology.- Education on the one hand to get a large portion of farmers in this poor world to new technologies, and secondly to ensure there is anywhere in the world a true education to food. (Résumé d'auteur