EPSRC Reference: 
EP/R006563/1 
Title: 
Deformations of SaitoKurokawa type Galois representations 
Principal Investigator: 
Berger, Dr TT 
Other Investigators: 

Researcher CoInvestigators: 

Project Partners: 

Department: 
Mathematics and Statistics 
Organisation: 
University of Sheffield 
Scheme: 
Standard Research 
Starts: 
01 December 2017 
Ends: 
30 November 2020 
Value (£): 
331,917

EPSRC Research Topic Classifications: 

EPSRC Industrial Sector Classifications: 
No relevance to Underpinning Sectors 


Related Grants: 

Panel History: 

Summary on Grant Application Form 
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 padic Galois representation becomes reducible modulo p. When this residual representation has three irreducible constituents, Serre's conjecture (a theorem of KhareWintenberger) tells us that its semisimplification is isomorphic to the Galois representation associated to the Siegel modular form obtained by lifting an elliptic modular form via the SaitoKurokawa lift. We call such residual representations "of SKtype".
The approach pioneered by Wiles for proving the modularity of a Galois representation is to consider deformations of its residual representation, i.e. padic 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 BlochKato conjectures.
By generalizing our method we are going to prove socalled R=T theorems for padic 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 padic properties of SaitoKurokawa lifts. In particular, we will construct congruences between SaitoKurokawa lifts and other Siegel modular forms. To access the noncohomological weight 2 case, for which classical techniques do not apply, we will prove such congruences for padic families. This will allow us to prove the paramodular conjecture for abelian surfaces with rational ptorsion. 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.

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