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Automorphic Forms and Moduli Spaces of Galois Representations

Final Report Summary - AF AND MSOGR (Automorphic Forms and Moduli Spaces of Galois Representations)

This main achievement of this project has been in establishing many new cases of a 70 year old conjecture of Hasse, on the meromorphic continuation of Hasse--Weil zeta functions. These are analogues of the Riemann zeta function, whose origin is geometric: they are associated to curves and to higher-dimensional geometric objects. In the special case of a single point, the corresponding Hasse--Weil zeta function is just the Riemann zeta function. It is expected that all Hasse--Weil zeta functions satisfy analogues of the properties of the Riemann zeta function (including the famous Riemann hypothesis), but very little is known about them in general. In particular, it is expected that all Hasse--Weil zeta functions admit a meromorphic continuation to the whole complex plane, but even for curves, this was only known for the projective line and for elliptic curves. The case of elliptic curves was a consequence of Andrew Wiles' proof 25 years ago of Fermat's Last Theorem, and subsequent improvements on those results.

Elliptic curves are curves of genus one; they look like doughnuts with a single hole. Curves of higher genus look like many doughnuts stuck together. Recently announced joint work of the PI with George Boxer, Frank Calegari and Vincent Pilloni proves that the Hasse--Weil zeta functions of genus 2 curves (doughnuts with two holes) have meromorphic continutations. This was previously only known for a handful of examples, as the result of extensive computer calculations.

The project has also lead to an improvement of our understanding of the p-adic Langlands correspondence, which is a huge conjectural framework in which many important aspects of modern number theory can be united.
In particular, the project has clarified the relationship of this correspondence to the Taylor--Wiles patching method (used in the proof of Fermat's Last Theorem mentioned above), and has introduced new geometric techniques to the subject, by constructing new moduli stacks of Galois representations, which promise to be important in future investigations.