Deformations of Saito-Kurokawa type Galois representations

This proposal sets out to prove the modularity of abelian surfaces and of elliptic curves over imaginary quadratic fields, the next major challenges in the Langlands programme linking algebraic geometry and automorphic forms. This series of conjectures made by the mathematician Robert Langlands in the 1960s and 70s predicts precise links between three seemingly unrelated classes of objects. These come from representation theory (in the form of modular forms), number theory (Galois representations) and algebraic geometry (e.g. elliptic curves or abelian surfaces). This project will lead to a much better understanding of the arithmetic of abelian surfaces and of Siegel modular forms, which are of interest not only to number theorists and geometers, but also physicists and cryptographers.

Establishing links in the Langlands programme enables number theorists to understand more deeply the properties of the objects involved and allows them to prove theorems, such as the famous example of the proof of Fermat's last theorem by Wiles and Taylor in 1994. The key ingredient in Wiles' proof was to establish that there is a modular form whose associated Galois representation agrees with that of an elliptic curve. This proposal will study the modularity of abelian surfaces, one dimension up from the case of elliptic curves. A precise conjecture for this case was recently formulated by Brumer and Kramer predicting that abelian surfaces should correspond to paramodular Siegel modular forms of weight 2. We propose to prove the first general result for this "paramodular conjecture" without assuming residual modularity.

For this we will study cases where the abelian surface has a rational torsion point of a prime order p. This means that the corresponding p-adic Galois representation becomes reducible modulo p. When this residual representation has three irreducible constituents, Serre's conjecture (a theorem of Khare-Wintenberger) tells us that its semi-simplification is isomorphic to the Galois representation associated to the Siegel modular form obtained by lifting an elliptic modular form via the Saito-Kurokawa lift. We call such residual representations "of SK-type".

The approach pioneered by Wiles for proving the modularity of a Galois representation is to consider deformations of its residual representation, i.e. p-adic Galois representations reducing to this representation modulo p, and to show that they all arise from modular forms. The residually reducible situation, however, poses major challenges for the study of deformations. In joint work with Krzysztof Klosin the PI developed a new approach to the modularity of residually reducible Galois representations with two residual pieces, showing that modularity often follows from congruences between modular forms and instances of the Bloch-Kato conjectures.

By generalizing our method we are going to prove so-called R=T theorems for p-adic Galois representations that residually are of SK type, establishing the modularity of all their deformations. In addition to developing new tools in the deformation theory of residually reducible Galois representations this requires studying the p-adic properties of Saito-Kurokawa lifts. In particular, we will construct congruences between Saito-Kurokawa lifts and other Siegel modular forms. To access the non-cohomological weight 2 case, for which classical techniques do not apply, we will prove such congruences for p-adic families. This will allow us to prove the paramodular conjecture for abelian surfaces with rational p-torsion. In addition, we will study the Bianchi modularity of elliptic curves over imaginary quadratic fields, another famous case that has resisted efforts so far, by proving the paramodularity of the abelian surface given by their base change to Q.

As with many projects in pure mathematics, the main benefits of this proposal will be academic in the short and medium term. The most direct impact is through the training of the Research Assistant in specialized research methods and the culture and literature of the area. This helps to maintain mathematical capability both in research and for teaching future users of mathematics.

Longer term, as yet unforeseen economic impact might be possible, e.g. in the area of cyber security in the EPSRC digital economy theme via the use of hyperelliptic curves in cryptography. To enable such economic impact we will strive for the widest and clearest possible dissemination of our research output.

PhD students in number theory will also benefit from this work by learning skills and knowledge related to the research subject and exposure to a wider research community. In addition they will attain the sort of analytical, methodical skills necessary for future careers also outside academia, thus benefiting their employability and the UK economy.