New Directions in Modularity

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The problem of understanding prime numbers goes back to antiquity. One natural problem which arose in the 19th century was to count the number of primes less than X for some variable quantity X. The resolution came in an unexpected way : this *discrete* counting problem turned out to be related to the properties of a *continuous* function first studied by Euler but now known as the Riemann zeta function. The breakthrough hinged on a crucial property of this function discovered by Riemann, namely, that it had a hidden symmetry which related the value of the function evaluated at the point "s" to the value evaluated at the point "1-s". This hidden symmetry turns out to be merely the tip of the iceberg; there are a wide class of similar functions which exhibit similar symmetries (at least conjecturally), and these functions arise in many areas of mathematics and physics. By proving that a certain class of such functions had this symmetry, Andrew Wiles in 1994 was able to prove Fermat's Last Theorem. The PI's work aims to prove that a certain class of functions exhibit the same symmetry as the Riemann zeta function. More specifically, there are a natural collections of such functions indexed by a natural number g. When g is zero, there is a single function which was the one considered by Euler and Riemann. When g is one, the class of functions are exactly the ones studied by Wiles. The PI plans to study the functions arising when g is two. An overarching theme of the PI's research (including collaborations with Emerton, Geraghty, and Venkatesh) has been the study of torsion classes in cohomology, both in the Betti cohomology of arithmetic groups and the coherent cohomology of Shimura varieties, and their conjectural relationship with the Langlands program. The PI's work suggests that understanding torsion is at the heart of understanding reciprocity beyond Shimura varieties. One of the main long term technical goals of the proposed project is to prove that L-functions attached to genus two curves satisfy the expected functional equations. For curves of genus one, this is the famous Taniyama-Shimura conjecture, now a theorem, which was first proved in many cases by Wiles. For curves of genus zero, this is a famous theorem of Riemann, namely, the functional equation of the Riemann zeta function. In order to approach this problem, the PI has (with David Geraghty) developed a generalization of the Taylor-Wiles method which may conceivably apply in this case. Much of his effort will be devoted to trying to overcome the numerous technical obstacles which are required to carry out this argument, including local-global compatibility for Galois representations, and vanishing of cohomology groups outside certain ranges. More generally, the PI intends to further develop this general approach to modularity questions in other contexts, including Galois representations conjecturally associated to automorphic forms for GL(n) over the rationals. He also plans to study applications of these results to the Bloch-Kato conjecture and establishing links between K-theory, completed cohomology, and the Langlands program.