Abstract
The purpose of this thesis is to study interactions between the endomorphisms of an abelian variety and its residual representations. We examine two principal questions in this direction. First, what constraints do the images of the residual representations impose on the endomorphism ring? Second, which Galois groups can be realised via Galois representations attached to abelian varieties with a non-trivial endomorphism ring?Thanks to work of Zarhin, it is known that if the image of the mod 2 representation attached to a hyperelliptic jacobian is particularly 'large', then the endomorphism ring will be trivial, that is, isomorphic to the integers. There are many results due to Zarhin et al. in this direction. But all require the image to be a large insoluble group acting absolutely irreducibly on the underlying module. It is shown here that many restrictions on the endomorphism ring persist even if the Galois group is merely cyclic of 'large' prime order.
The second question has been studied in detail when the endomorphism ring is trivial.
This began with Serre's famous Open Image Theorem. He showed that for any elliptic curve, there exists some constant $C$ such that for any prime $ell > C$, the mod $ell$ representation is surjective. In the same paper, Serre gives explicit examples of elliptic curves where he works out for which primes $ell$ the associated representations are not surjective. Recently there have been similar examples given for higher dimensional abelian varieties, also with trivial endomorphism ring. In the final chapter explicit examples of abelian varieties with non-trivial endomorphism ring having large images are given. The group structure of the associated images is highly influenced by the endomorphism algebra, and we give a description of the new Galois groups obtained in this manner.
| Date of Award | 28 Sept 2021 |
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| Original language | English |
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| Supervisor | Tim Dokchitser (Supervisor) |