This project will investigate the impact of automorphic forms on geometry and number theory. The first part of the project, joint with K. Paranjape, will associate Calabai-Yau manifolds over the rationals of dimension m to certain basic holomorphic cusp forms of weight m+1 with rational coefficients. The goal of the second part, joint with J. Cogdell, is to refine the omnipotent converse theorem for GL(n) so as to require less on the associated L-functions, thereby increasing the potential applicability in more interesting situations. The third part, joint with N. Dunfield, will use CM automorphic forms to construct an explicit tower of compact hyperbolic 3-manifolds M(n) of arithmetic type such that the numbered of fibred faces in the Thurston unit ball goes to infinity, in fact exponentially, as n becomes large. The fourth project, joint with D. Prasad, will use global methods to settle a conjecture on when self dual irreducible representations of the multiplicative group of a division algebra D over a local field leave invariant a symmetric, or alternating, bilinear form, and this will have potential applications to the root number.
The main thrust of the project is to comprehend some of the manifestations of symmetry in Mathematics and beyond. One could say that everything interesting in the natural world has some aspect of symmetry, though not always visibly so. Often mathematicians and physicists build generating functions out of discrete collections of numbers arising from observations or calculations, and it is a pressing problem to know if these functions admit hidden symmetries, like the invariance under inversion of a hidden variable. When such symmetries arise, they are often describing the tones of automorphic functions, which are continuous, pulchritudinous entities such as the waveforms on a disk. Their discrete tones are linked, experimentally and theoretically, to exciting quantities such as lengths of geodesics, primes, and congruence solutions of polynomial equations. Exploiting them is a worthy endeavor, and there are many gold mines yet to be discovered.
