Alberta Number Theory Days – L-functions

Alberta Number Theory Days fell at the very start of the newly formed PIMS CRG in Number Theory, and was thus an excellent opportunity to kickstart interactions between three of the participating institutions, the universities of Alberta, Calgary and Lethbridge. A broad range of topics in number theory were featured, but almost all talks were in areas motivated by the understanding of L-functions. Indeed, two key frameworks in which L-functions are viewed were addressed during the weekend, each forming a core of talks. The first framework is the analytic study of L-functions with the view to deriving properties of prime numbers. This included a reformulation of the generalized Riemann hypothesis by Brandon Fodden in terms of a decidable property of the natural numbers, thus giving a new insight into this most important of conjectures. In a related direction, Amir Akbary (discussing joint work with Brandon Fodden) gave improved bounds on the power moments of L-functions in the Selberg class, with applications to principal automorphic L-functions and Artin L-functions. Kaneenika Sinha’s presentation also concerned bounds, but this time bounds on the analytic rank of Jacobians of modular curves, an area of study with links to the famous conjecture of Birch and Swinnerton-Dyer. The second framework is the overarchingLanglands programme, one of whose principal goals is to equate L-functions of Galois representations, important objects associated with the absolute Galois group ofQ, with L-functions of automorphic representations. The talks of Clifton Cunningham, Vinayak Vatsal and Jeremy Sylvestre can be viewed under the umbrella of the Langlands programme, and a particular highlight was Cunningham’s criterion for when a certain L-packet arising from an elliptic curve was infinite, a criterion given in terms of the set of supersingular primes of the elliptic curve. It was also a delight to observe Dustin Moody’s talk on a novel way to count isogenies between elliptic curves over finite fields via isogeny volcanoes. The closing talk of the conference, by Matthew Greenberg, demonstrated the impressive contribution that Dembélé, Greenberg and Voight have made towards the verification of a conjecture of Gross, with only one case remaining. The context for the talks will be described more fully in Section 2.