Thursday 01 Mar 2018Number Theory Seminar: CONGRUENCES BETWEEN DERIVATIVES OF TWISTED HASSE-WEIL L-FUNCTIONS

Daniel Macias Castillo - Universidad Autonoma de Madrid

H103 14:30-16:30


The results discussed in this talk include joint work with David Burns, Christian Wuthrich, Werner Bley and StŽephane ViguiŽe. Let A be an abelian variety defined over a number field k. Then for any finite Galois extension F of k with group G the equivariant Tamagawa number conjecture (‘ETNC’) for the pair h 1 (A/F )(1), Z[G]  was formulated by Burns and Flach as a natural refinement of the seminal conjecture of Bloch and Kato. Under certain not-too-stringent conditions we give a reformulation involving the finite support cohomology of Bloch and Kato of the p-part of this conjecture (for a given prime number p). We next describe several (conjectural) consequences of this reinterpretation. These concern elements which interpolate the values at s = 1 of higher derivatives of the Hasse-Weil L-functions of twists of A by irreducible complex characters of G, suitably normalised by a product of explicit equivariant regulators and periods. They include integrality properties, statements concerning the Galois structure of Tate-Shafarevich and Selmer groups and ‘refined conjectures of the Birch and Swinnerton-Dyer type’ in the spirit of, but finer and more general than, the ones formulated by Mazur and Tate. In another direction, we briefly discuss how our approach leads to the first verifications of the p-part of the ETNC for elliptic curves A defined over Q for certain families within the technically most demanding case in which A has strictly positive rank over F and the Galois group G is both non-abelian and of order divisible by p


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