Ottawa, June 7 - 11, 2021
More recently, the theory of diagonal cycles, arising from the work and collective effort of Bertolini, Darmon, Rotger, Seveso, and Venerucci, has proven to be a fertile environment for proving new instances of the Birch and Swinnerton-Dyer conjecture for elliptic curves over the rationals.
The aim of this talk is to discuss joint work in progress with Daniel Barrera, Santiago Molina, and Victor Rotger on the generalisation of the theory of diagonal cycles to quaternionic Shimura curves over totally real number fields F and its application to extending Kato’s result for twists of elliptic curves E/F by Hecke characters of F of finite order.
In this setting, the normalized L-values for
$sym^2(f)$ can be expressed in terms of the Petersson inner product of $f$
with a nearly holomorphic function. The Petersson inner product is modified and related to an abstractly
defined algebraic pairing due to Hida, and the two pairing are related up
to a "canonical period". As a result, it is shown that the p-adic L-function for the symmetric-square exhibit congruences, and this has consequences for analytic Iwasawa invariants.
We give an overview of plus/minus theory and its generalisation to automorphic representations of GL(2n), along with some neat applications.