Pip Goodman

Restrictions on Endomorphism Algebras

Given a hyperelliptic curve C: y^2=f(x), Zarhin has studied the restrictions placed on the endomorphism algebra of its Jacobian under the assumption that the Galois group Gal(f) is 'large' and insoluble. These restrictions occur for almost purely group theoretic reasons. By establishing a partial converse to a result of Guralnick and Kedlaya, in recent work, we have shown that many of these restrictions persist when Gal(f) merely contains an element of 'large' prime order. In this talk, I will focus on the algebraic ideas arising in the proof.