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 Office : 3.003

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.

# Research

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 special values of their special value at s=1, Artin-Tate conjecture, Brauer-Siegel theorem and analogues.

### Articles and preprints

• Elliptic curves with large Ш over function fields, with Guus de Wit,
In preparation.

Let $\mathbb{F}_q$ be a finite field of odd characteristic. We exhibit a sequence $\{E_a\}_{a\geq 1}$ of elliptic curves over $K:=\mathbb{F}_q(t)$ such that, for all $a\geq 1$, the Tate-Shafarevich group $Ш(E_a)$ is finite and satisfies $$\forall \epsilon>0, \qquad H(E_a)^{1-\epsilon}\ll|Ш(E_a)|\ll H(E_a)^{1+\epsilon},$$ while $Ш(E_a)[p^\infty]$ is trivial. Here $H(E_a)$ denotes the exponential differential height of $E_a$. In contrast with other constructions, the curves $E_a$ here are non-pairwise $\overline{K}$-isomorphic and non-pairwise $K$-isogenous. We also provide further arithmetic information about the curves $E_a$, most notably a complete description of their rational points and an explicit expression for their $L$-function.

• On the arithmetic of a family of twisted constant elliptic curves, with Douglas Ulmer,
Preprint (submitted); 38 pages.
[ ArXiv:1903.03901 | Local version ] -

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 $E$.

• A new family of elliptic curves with unbounded rank,
Accepted for publication in Moscow Mathematical Journal; 22 pages.
[ ArXiv:1809.07755 | Local version ] -

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.

• Bounds on special values of L-functions of elliptic curves in an Artin-Schreier family,
European Journal of Mathematics, Vol. 5 (2), June 2019, pp. 476-517.
[ ArXiv:1801.08492 | Local version | Journal ] -

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.

• Explicit L-function and a Brauer-Siegel theorem for Hessian elliptic curves,
Journal de Théorie des Nombres de Bordeaux (to appear); 16 pages.
[ ArXiv:1709.02761 | Local version ] -

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 $d\to\infty$.

• Analogue of the Brauer-Siegel theorem for Legendre elliptic curves,
Journal of Number Theory, Vol. 193, December 2018, pp. 189-212.
[ ArXiv:1706.07728 | Local version | Journal ] -

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 $d\to\infty$.

• A Brauer-Siegel theorem for Fermat surfaces over finite fields,
Journal of the London Math. Soc, Vol. 97 (3), June 2018, pp. 523-549.
[ ArXiv:1612.08721 | Local version | Journal ] -

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).

Find my papers on arXiv, ORCID, MathSciNet.

### Thesis

I did my PhD thesis under the supervision of Marc Hindry at Université Paris Diderot, I defended in July 2016: here is the introduction (in English and in French) and the full text (in French).

### Others

• Analogues du théorème de Brauer-Siegel pour quelques familles de courbes elliptiques,
my PhD thesis, 245 pages (in French).
• Analogue of the Brauer-Siegel theorem for some families of elliptic curves over function fields,
a poster I presented at the conference Silvermania 2015.
• La conjecture d’uniformité de Serre dans le cas Cartan déployé (d’après Bilu, Parent, Rebolledo),
my Masters thesis (in French).

# Teaching & Supervision

### Courses taught

• Spring 2019: Teaching Assistant for ''Introduction to Arithmetic of Dynamical Systems''
Bachelor course at Universität Basel.
• Fall 2018: Teaching Assistant for ''Einführung in die Topologie''
Bachelor course at Universität Basel.
• Fall 2017: Lecturer for ''Curves over finite fields''
Bachelor/Master course at Universiteit Leiden.
• Fall 2017: Teaching assistant for ''Algebraic Number Theory''
Mastermath course at VU Amsterdam.
• Spring 2017: Lecturer for ''Curves over finite fields''
Bachelor/Master course at Universiteit Leiden.
• Fall 2016: Teaching assistant for ''Algebraic Number Theory''
Mastermath course at VU Amsterdam.
• 2ème Semestre 2015/16 : Chargé de TD pour le cours ''Algèbre et Analyse Approfondies II''
Licence 2 (groupe CPEI : classes préparatoires intégrées), Université Paris Diderot.
• 2ème Semestre 2014/15 : Chargé de TD pour le cours ''Algèbre et Analyse Fondamentales II''
Licence 2 (MASS), Université Paris Diderot.
• 2ème Semestre 2013/14 : Chargé de TD pour le cours ''Algèbre et Analyse Élementaires II''
Licence 1 (Info), Université Paris Diderot.
• 1er Semestre 2012/13 : Chargé de TD pour le cours ''Algèbre et Analyse Élementaires I''
Licence 1 (MASS), Université Paris Diderot.

### Students supervised

Last updated : March 2019