The Explicit Hypergeometric Modularity Method (EHMM)

The EHMM, was first introduced by Allen, Grove, Long, and Tu. My advisors suggested my dissertation topic based on this paper, and much of my work in [1] and [2] was written concurrently with the original two papers. The main theoretical contribution of those papers was the realization that the EHMM outputs a vector space of modular forms attached to a hypergeometric motive over a number field, rather than a single modular form. These vector spaces can be identified with the holomorphic piece of the analytic de Rham cohomology for the direct sume of the underlying hypergeometric motives, which enables cleaner geometric proofs and stronger results in some cases. The basic outline of the EHMM is in the pictures below. Recently, with Grove, Long, and Tu, we have written a part III of the EHMM, [3], which extends the ideas of the original papers to a larger class of examples. The focus of my work on this paper was using the commutative formal group law to identify a single modular form in our vector space attached to a hypergeometric motive that is defined over Q.

In [4], I explain how the geometric formulation used in my previous papers applies to irregular cases, i.e. where the Hodge numbers of the hypergeometric motive do not satisfy Hodge symmetry (this can occur when the coefficient field is not real). Such cases are are notoriously more difficult to understand. In [4], I use the complexified de Rham cohomology to relate the periods in these cases with two CM periods with distinct CM discriminant. The proofs for this paper are completed, and a preprint will be posted soon. Eventually, there will also be a part II, focused on the p-adic unit roots and their relation with modularity.