Alladi Ramakrishnan Hall

On the conjecture of Harris and Venkatesh

Loic Merel

IMJ-PRG, Université Paris Cité

Over the last decade, Venkatesh has formulated vast conjectures. The first significant instance has been considered by Harris and Venkatesh. It asserts the proportionality of two collections of numbers (a_q) and (b_q) where q runs through the prime numbers, and a_q, b_q belong to the finite field F_q with q elements. But a_q and b_q arise from different worlds : modular forms on one side and representations of the Galois group on the other side. We will explain the conjecture other prerequisites than basic algebraic number theory. We will review the progress accomplished on the conjecture. (The new part of this talk is joint work with Emmanuel Lecouturier.)