Richard Griffon

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20182025

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math.NT2025

A parallelogram height inequality for Drinfeld modules

Liam Baker, Richard Griffon, Fabien Pazuki

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 a…

math.NT2025

Variation of height in an isogeny class over a function field

Richard Griffon, Samuel Le Fourn, Fabien Pazuki

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 c…

math.NT2021

On the arithmetic of a family of superelliptic curves

Sarah Arpin, Richard Griffon, Libby Taylor +1

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 $C / F_{r} (t)$ be the superelliptic curve wit…

math.NT2020

Isogenies of elliptic curves over function fields

Richard Griffon, Fabien Pazuki

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. Th…

math.NT2019

Elliptic curves with large Tate-Shafarevich groups over $F_{q} (t)$

Richard Griffon, Guus de Wit

Let $F_{q}$ be a finite field of odd characteristic $p$ . We exhibit elliptic curves over the rational function field $K = F_{q} (t)$ whose Tate-Shafarevich groups are…

math.NT2019

On the arithmetic of a family of twisted constant elliptic curves

Richard Griffon, Douglas Ulmer

Let $F_{r}$ 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…