La semaine de l’IMB

Lying between arithmetic and analytic aspects of modular forms and Galois representations, p-adic L-functions have been crucial in many of the incredible progress towards the Birch and Swinnerton-Dyer of the last decade, particularly in the work of Skinner, Loeffler, Zerbes, et al. Two main difficulties arise in their construction: an algebraic geometric input for controlling the rate of growth of p-adic L-functions, which characterizes them uniquely, and an automorphic input for relating them to classical L-functions. I will talk about recent work with Mladen Dimitrov on p-adic L-functions attached to general parahoric representations of GL(2n) of symplectic type. Previous constructions, based on Friedberg-Jacquet integral formulas and the work of Ash-Ginzburg, produce the 0 function. We construct an "improved" Shalika model, inspired by the construction of improved p-adic L-functions, and obtain the expected p-adic L-functions for all parahoric representations of symplectic type. Different constructions of p-adic L-functions are better suited for different applications, and we are using this construction for the study of trivial zeros.