Richard Griffon
Postdoc in Number Theory
Universität Basel - Departement Mathematik und Informatik Spiegelgasse 1 CH-4051 Basel, Switzerland E-mail : richard.griffon /at/ unibas.ch |
Since September 2018, I am a postdoctoral researcher at the University of Basel (Switzerland), in the research group of Pierre Le Boudec. Before that, I was a postdoc at Leiden University (the Netherlands) for two years.
I obtained my PhD degree at Université Paris Diderot in July 2016, working under the supervision of Marc Hindry.
My main field of research is the arithmetic of elliptic curves over global fields.
I mostly study them over function fields in positive characteristic, and I usually take an analytic approach (by using their $L$-function).
The main topic of my research is the asymptotic behaviour of the so-called Brauer-Siegel ratio of elliptic curves: this invariant is related to the difficulty of computing their Mordell-Weil groups.
It can be controlled by estimating the size of the special values of their $L$-functions.
In some cases, one can prove that the Brauer-Siegel ratio has a limit.
In all known cases, this limit is $1$ (i.e. the Mordell-Weil groups of the elliptic curves are indeed complicated to compute).
I have also worked with surfaces over a finite field: for some families, I have proved very precise bounds on some of their geometric invariants.
Keywords: Elliptic curves over global fields, L-functions of elliptic curves (and explicit computations of those), Birch and Swinnerton-Dyer conjecture, Estimates of special values of L-functions at s=1, Surfaces over finite fields, Estimates of their special value at s=1, Artin-Tate conjecture, Brauer-Siegel theorem and analogues.
Let $K$ be the function field of a smooth projective algebraic curve.
We first describe the relation between the heights of two isogenous elliptic curves defined over $K$ (for various notions of height: modular height, differential height, stable height). In contrast to the case of elliptic curves over number fields the height is preserved under isogeny, unless the characteristic of $K$ divides the degree of the isogeny.
Secondly, we prove an "isogeny theorem" for elliptic curves over $K$ (in the vein of results of Masser-Wüstholz and Gaudron-Rémond concerning elliptic curves over a number field). In particular, if $K$ has characteristic $0$, this theorem says that, between two isogenous elliptic curves over $K$, there is an isogeny of degree $\leq c(K)$, where $c(K)$ is a constant depending at most on $K$.
Let $\mathbb{F}_q$ be a finite field of odd characteristic $p$.
We exhibit elliptic curves over the rational function field $K = \mathbb{F}_q(t)$ whose Tate-Shafarevich groups are large.
More precisely, we consider certain infinite sequences of explicit elliptic curves $E$, for which we prove that their Tate-Shafarevich group $Ш(E)$ is finite and satisfies $|Ш(E)| = H(E)^{1+o(1)}$ as $H(E)\to\infty$, where $H(E)$ denotes the exponential differential height of $E$.
The elliptic curves in these sequences are pairwise neither isogenous nor geometrically isomorphic.
We further show that the $p$-primary part of their Tate-Shafarevich group is trivial.
The proof involves explicitly computing the $L$-functions of these elliptic curves, proving the BSD conjecture for them, and obtaining estimates on the size of the central value of their $L$-function.
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 Néron-Tate regulator $\mathrm{Reg}(E)$, and the order of its Tate-Shafarevich group $Ш(E)$ (which we prove is finite).
These invariants have radically different behaviours depending on the congruence class of $p$ modulo $6$.
For instance $Ш(E)$ either has trivial $p$-part or is a $p$-group.
On the other hand, we show that the product $|Ш(E)|\mathrm{Reg}(E)$ has size comparable to $r^{q/6}$ as $q\to\infty$, regardless of $p\,\mathrm{mod}\,6$.
Our approach relies on the BSD conjecture, an explicit expression for the $L$-function of $E$, and a geometric analysis of the Néron model of $E$.
Let $\mathbb{F}_q$ be a finite field of odd characteristic and $K= \mathbb{F}_q(t)$. For any integer $d\geq 1$, consider the elliptic curve $E_d$ over $K$ defined by $y^2=x\cdot\big(x^2+t^{2d}\cdot x-4t^{2d}\big)$. The curves $E_d$ satisfy the BSD conjecture, so that their rank equals the order of vanishing of their $L$-function at the central point. We provide an explicit expression for the $L$-function of $E_d$, and use it to study this order of vanishing in terms of $d$. We deduce that the rank of the Mordell-Weil group $E_d(K)$ is unbounded as $d$ varies (this answers a question of L. Berger). We also prove that the average Mordell-Weil rank is unbounded.
We study the Artin-Schreier family of elliptic curves over $\mathbb{F}_q(t)$ defined by $y^2=x\big(x+1\big)\big(x+(t^{q^a}-t)^2\big)$ for all $a\geq 1$.
We prove an asymptotic estimate on the size of the special value of their $L$-function in terms of the degree of their conductor; loosely speaking, we show that the special values are "asymptotically as large as possible".
We also provide an explicit expression for the $L$-function of the elliptic curves in the family.
The proof of the main result uses this expression and a detailed study of the distribution of some character sums related to Kloosterman sums.
Via the BSD conjecture (proved by Pries and Ulmer in this case), the main result translates into an analogue of the Brauer-Siegel theorem for these 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 Néron-Tate regulator, in terms of the exponential differential height of $E_d$, as $d\to\infty$.
We prove an analogue of the Brauer-Siegel theorem for the Legendre elliptic curves over $\mathbb{F}_q(t)$. More precisely, if $d \ge 1$ is an integer, we denote by $E_d$ the elliptic curve with model $y^2=x(x+1)(x+t^d)$ over $K=\mathbb{F}_q(t)$. We give an asymptotic estimate of the product of the order of the Tate-Shafarevich group of $E_d$ (which is known to be finite) with its Néron-Tate regulator, in terms of the exponential differential height of $E_d$, as $d\to\infty$.
We study Fermat surfaces over a finite field. Let $\mathbb{F}_q$ be a finite field and, for all $d$ coprime to $d$, let $F_d\subset \mathbb{P}^3$ be the Fermat surface of degree $d$ over $\mathbb{F}_q$, defined by $x_0^d+x_1^d+x_2^d+x_3^d=0$. We show that $$ \log\big( |\mathrm{Br}(F_d)|\cdot \mathrm{Reg}(F_d)\big) \sim \log q^{p_g(F_d)}\qquad \text{as }d\to\infty,$$ where $p_g(F_d)$ is the geometric genus of $F_d$, $\mathrm{Br}(F_d)$ denotes the Brauer group of $F_d/\mathbb{F}_q$ (which is known to be finite) and $\mathrm{Reg}(F_d)$ is the regulator of $F_d/\mathbb{F}_q$ (i.e. the absolute value of a Gram determinant of the Néron-Severi group of $F_d$ with respect to the intersection form).
I did my PhD thesis under the supervision of Marc Hindry at Université Paris Diderot. My defence took place in July 2016. Here is the introduction (in English and in French) and the full text (in French).
Last updated : November 2019