Tamagawa numbers and torsion of elliptic curves over function fields

Abstract

We study the divisibilities of Tamagawa numbers $c(E)$ of elliptic curves $E$ over global function fields $K$ in terms of their torsion subgroups $E(K{)}_{\mathrm{tors}}$. In particular, for a non-isotrivial elliptic curve $E/k(t)$, where $k$ is a finite field of characteristic greater than $3$, we prove that $|E(k(t){)}_{\mathrm{tors}}{|}^2$ divides $c(E)$, except possibly in four exceptional torsion families. More specifically, we give a complete characterisation of divisibilities for $c(E)$ in each torsion family, and provide explicit examples to prove that they are best possible. Over a general global function field, we also prove that a rational point of prime order $5\le N\le 101$ on $E/K$ forces $N^2$ to divide $c(E)$, which motivates our result. Finally, we formulate a conjecture on the leading coefficient of the $L$-function of $E/K$, motivated by the integrality of Birch--Swinnerton-Dyer quotients.

arXiv comment42 pages