Modular varieties, arithmetic and geometry

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This project will investigate the arithmetic and geometry encoded in the modular varieties arising from the study of automorphic forms, as well as in certain Calabi-Yau manifolds. The first part of the project will investigate the possibility of proving the Tate conjecture for the entire class of quaternionic Shimura surfaces, which if successful, will lead to progress on this question for divisors on all Shimura varieties of classical type. The second project, joint with K. Paranjape, will aim to associate Calabi-Yau varieties with involution over the rationals of dimension m to certain basic holomorphic cusp forms of weight m+1 and rational coefficients. The emphasis here will be on the 3-dimensional case, with a view to understanding quadratic twists and functorial products. The PI will also continue his investigation into certain other topics, including the works with N. Dunfield on the circle fibrations of hyperbolic 3-manifolds of arithmetic type, and with P. Michel concerning the exact averages of L-values.Many problems one encounters become amenable to elucidation by the mathematical method when they exhibit some symmetry such as periodicity or invariance under mirror reflection. This is important in the cracking of codes and in the study of crystals and precious stones, for example. The main thrust of this project is to comprehend some of the manifestations of symmetry, especially when their presence is not evident. Mathematicians, Physicists, and others often start with discrete collections of numbers, possibly from experimentation, then form their generating functions, and ask if they encode hidden symmetries. When such harmonious arrangements arise in nature, they frequently describe the tones of automorphic functions, which are continuous entities like the waveforms on a disk. Their discrete frequencies are linked to exciting constructs like lengths of curves, prime numbers, and congruence solutions of polynomial equations. A particular focus of the project is to determine when the presence of Galois symmetries implies the existence of special curved subspaces of the ambient space, relating in turn to the poles of certain zeta functions. The ultimate aim is to understand the ubiquity and power of number formations better through geometry and analysis.