arXiv subjects
Publications and source records attributed to Richard Griffon.
We prove inequalities relating the Taguchi heights, respectively the graded heights, of four Drinfeld modules arranged in a ``parallelogram of isogenies''. This inequality is the analogue for Drinfeld modules of the parallelogram inequality of R\'emond (2022) for abelian varieties over number fields and of Griffon--Le Fourn--Pazuki (2025) for abelian varieties over function fields.
We give optimal estimates on the variation of the differential and modular heights within an isogeny class of abelian varieties defined over the function field of a curve (in any characteristic). We also prove a parallelogram inequality for abelian varieties in this context, and deduce corollaries of these results.
Let $p$ be a prime, let $r$ and $q$ be powers of $p$, and let $a$ and $b$ be relatively prime integers not divisible by $p$. Let $C/\mathbb F_{r}(t)$ be the superelliptic curve with affine equation $y^b+x^a=t^q-t$. Let $J$ be the Jacobian of $C$. By work of Pries--Ulmer, $J$ satisfies the Birch and Swinnerton-Dyer conjecture (BSD). Generalizing work of Griffon--Ulmer, we compute the $L$-function of $J$ in terms of certain Gauss sums. In addition, we estimate several arithmetic invariants of $J$ appearing in BSD, including the rank of the Mordell--Weil group $J(\mathbb F_{r}(t))$, the Faltings height of $J$, and the Tamagawa numbers of $J$ in terms of the parameters $a,b,q$. For any $p$ and $r$, we show that for certain $a$ and $b$ depending only on $p$ and $r$, these Jacobians provide new examples of families of simple abelian varieties of fixed dimension and with unbounded analytic and algebraic rank as $q$ varies through powers of $p$. Under a different set of criteria on $a$ and $b$, we prove that the order of the Tate--Shafarevich group of $J$ grows quasilinearly in $q$ as $q \to \infty.$
We extend the Brauer-Siegel theorem to new families of number fields, both in the classical setting of asymptotically bad families and in the more general framework due to Tsfasman and Vl\u{a}du\c{t} of asymptotically exact families. We introduce a notion of Galois complexity for extensions of number fields, and show that the generalized Brauer-Siegel theorem, as conjectured by Tsfasman and Vl\u{a}du\c{t}, holds for families in which the complexity does not grow too fast. This allows to unify and extend all previously known results. The crucial step in our work is the proof of a new version -- stated in terms of our Galois complexity -- of a fundamental principle due to Stark descending exceptional zeroes of zeta functions down to quadratic number fields. Among the hitherto unknown cases we are able to treat are the families of number fields contained in the solvable Galois closure of a given number field.
We prove two theorems concerning isogenies of elliptic curves over function fields. The first one describes the variation of the height of the $j$-invariant in an isogeny class. The second one is an "isogeny estimate", providing an explicit bound on the degree of a minimal isogeny between two isogenous elliptic curves. We also give several corollaries of these two results.
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 $\text{Reg}(E)$, 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 Néron model of $E$.
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 $\mathrm{III}(E)$ is finite and satisfies $|\mathrm{III}(E)| = H(E)^{1+o(1)}$ 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 $\mathbb{F}_q$ be a finite field of odd characteristic and $K= \mathbb{F}_q(t)$. For any integer $d\geq 2$ coprime to $q$, consider the elliptic curve $E_d$ over $K$ defined by $y^2=x(x^2+t^{2d} x-4t^{2d})$. We show that the rank of the Mordell--Weil group $E_d(K)$ is unbounded as $d$ varies. The curve $E_d$ satisfies the BSD conjecture, so that its rank equals the order of vanishing of its $L$-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 $d$.
In this paper, we study a certain Artin--Schreier family of elliptic curves over the function field $\mathbb{F}_q(t)$. 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, 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 $d$ is an integer coprime to $q$, 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 prove an analogue of the Brauer-Siegel theorem for Fermat surfaces over a finite field. Namely, letting $F_d$ be the Fermat surface of degree $d$ over $\mathbb{F}_q$ and $p_g(F_d)$ be its geometric genus, we consider the product of the order of the Brauer group $\mathrm{Br}(F_d)$ of $F_d$ times the absolute value of a Gram determinant of the Néron-Severi group of $F_d$ with respect to the intersection form (the regulator $\mathrm{Reg}(F_d)$ of $F_d$). We show that this product grows like $q^{p_g(F_d)}$ when $d$ tends to infinity: $$ \log\left( |\mathrm{Br}(F_d)|\cdot \mathrm{Reg}(F_d)\right) \sim \log q^{p_g(F_d)}.$$