Congruences of automorphic forms and Galois representations

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This PhD concerns questions in algebraic number theory about connections between modular forms and Galois representations and applications of these to arithmetic problems. Establishing the precise links between automorphic forms and Galois representations is part of the famous Langlands programme. The particular focus of the project would be to work on the Bloch-Kato conjectures on special values of L-functions. These conjectures predict that the p-divisibility of a value of the analytic L-function associated to a Galois representation corresponds exactly to the p-divisibility of the size of an algebraic object associated to the representation, the so-called Selmer group.Results on the p-integrality of theta lifts and their norm make it possible to provide evidence for a higher rank case of the Bloch-Kato conjecture. The PhD project studies this conjecture for the Asai representation of the Galois representation associated to an automorphic representation (Bianchi modular form) p for GL(2) over an imaginary quadratic field. The goal is to prove that if a prime p divides the L-value for the Asai representation then there exists a congruence between a theta lift (a Siegel modular form associated to p) and other "stable" Siegel modular forms. The Galois representations associated to these forms can then be used to construct elements in the Selmer group. This strategy involving congruences of modular forms was pioneered by Ribet in his proof of the converse of Herbrand's theorem and has since been used by, amongst others, Wiles and Skinner and Urban for Eisenstein series in the proof of Iwasawa main conjectures. The project is significantly different to previous work on the Bloch-Kato conjectures as it provides evidence in a context where it is not known whether the representation is "motivic". Amongst other techniques it requires extending the pullback method of proving congruences to low weight Siegel modular forms.