Arithmetic of higher coherent cohomology of Shimura varieties

Abstract

In this thesis, we study various arithmetic properties of the higher (namely, Hi withi > 0) coherent cohomology of Shimura varieties and their applications on automorphic forms. Coherent cohomology, being only loosely connected to the theory of motives, carries interesting information that cannot be seen in other cohomology theories. On the other hand, its indirect definition makes a coherent cohomology class a highly interesting object which requires careful interpretation. The current thesis hopes to suggest that understanding various aspects of the mystery of coherent cohomology often yields fruitful and interesting study of arithmetic of automorphic forms. This thesis is naturally divided into three parts. In the first part, we interpret harmonic Maass forms as classes in the local cohomology of modular curves. This reinterpretation suggests natural generalizations of harmonic Maass forms to other Shimura varieties, which are engineered to avoid the Koecher’s principle. In the second part, we suggest a conjecture on the relationship between coherent cohomology of Shimura varieties of different cohomological degrees. The conjecture is motivated by a philosophy of Venkatesh, which suggests that each such part of cohomology is a graded module over the exterior algebra of the motivic cohomology of the corresponding adjoint motive. The case of low-weight automorphic forms on Shimura varieties is in some sense a degenerate case where more structures get involved, such as archimedean L-packets. In the final part, we consider the theme of p-adic variation of higher coherent cohomology, which is recently developed under the name of higher Hida theory. We carry out this strategy for the base-change L-function of U(2, 1), following the approach of Loeffler–Pilloni–Skinner–Zerbes for the spin L-function of GSp4. As such strategy relies heavily on the cohomological period integrals, we suggest a more streamlined approach based on the generalized Whittaker models