Tamagawa numbers and torsion of elliptic curves over function fields

Source 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 $E(K)_{\operatorname{tors}}$. 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 $|E(k(t))_{\operatorname{tors}}|^2$ 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.