First Grant for Ambrus Pal

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The proposed approach to tackle the main objective, making progress on the Artin-Tate conjecture, is to combine methods using the Langlands correspondence for function fields with tools of p-adic analytic nature, in particular crystalline cohomology. Our strategy will likely not just show the conjecture in its original form in special cases, but will lead to refined versions as well, therefore deepening our understanding in the general case. In particular we intend to show a generalized form of the Gross-Zagier formula for function fields, and derive as an application that every genus one curve defined over a function field has a solvable point. Moreover we will work towards proving that a weak form of the Tate conjecture implies a refined version of the Artin-Tate conjecture involving p-adic regulators. We also intend to study the divisibility properties of p-adic L-functions via the Langlands program, and as a closely related problem, we'll start to develop a p-adic version of the latter both as a tool for our particular problem, both for laying the foundations for future research.