Inverse Galois Problem and torsion points on abelian varieties

Lead Research Organisation: University of Bristol
Department Name: Mathematics

Abstract

One of the biggest unsolved questions in number theory is the Inverse Galois Problem. It is to show that every finite group can occur as a Galois group over the rationals. Abelian groups are well-understood through class field theory, and it is known (though it is very hard) that soluble groups can be realised as well. So a lot of work has been dedicated to simple and almost simple groups, and they can sometimes be realised from Galois groups of torsion points on curves and abelian varieties. This requires being able to construct curves that have some specific behaviour - automorphisms, endomorphisms, reduction types etc. Pip Goodman's research topic is to work in this direction, using recent developments in the theory of curves, their models, and their Jacobians.

Publications

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Studentship Projects

Project Reference Relationship Related To Start End Student Name
EP/N509619/1 01/10/2016 30/09/2021
1940106 Studentship EP/N509619/1 18/09/2017 31/03/2021 Pip Goodman
 
Description The first key finding to report is a theoretical result which helps determine the endomorphism algebra of an abelian variety from its 2-torsion points. This result is both of theoretical and computational interest. In particular, it helps identify abelian varieties with special endomorphism algebras which are needed in the Inverse Galois Problem.

Next, I have discovered certain data attached to endomorphism algebras of abelian varieties which encodes part of their Galois representations. The Galois representations are the crucial objects to study when trying to prove results in Inverse Galois Theory. However, my supervisor and I believe this data to be important in the general study of abelian varieties, going much beyond the original goal of Inverse Galois Theory. That said, it looks like this data and other methods will show several infinite families of non-soluble groups occur as Galois groups over the rationals, hence resolving many cases of the Inverse Galois Problem, but I am yet to work out all the details, and will report on this at the end of the grant.
Exploitation Route The first finding mentioned above improves our understanding of how an abelian variety's endomorphism algebra is restricted by its Galois representation. We believe this to be of general theoretical interest, and have applications in future algorithms computing endomorphism algebras, and possibly other situations where one needs control over the endomorphism algebra, or the Galois representations, both being ubiquitous in modern number theory.

The data mentioned above should have several applications to the general theory of abelian varieties. In particular, it along with other methods developed in trying to resolve cases of the Inverse Galois Problem should be applicable to other abelian varieties with different endomorphism algebras and hence could be taken forward to resolve even more cases.
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