## STNB2020(34th edition)

### Restrictions on endomorphism algebras of Jacobians

Pip Goodman

#### Abstract

Zarhin has extensively studied restrictions placed on the endomorphism algebras of Jacobians of hyperelliptic curves $C : y^2 = f(x)$ when the Galois group ${\rm Gal}(f)$ is insoluble and large' relative to $g$ the genus of $C$. But what happens when ${\rm Gal}(f)$ is not large' or insoluble? We will see that for many values of $g$, much can be said if ${\rm Gal}(f)$ merely contains an element of `large' prime order.