Infinitesimal p-adic Manin-Mumford and applications to Hida theory

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We study analytic functions on the open unit p-adic poly-disk centered at the multiplicative identity and prove that such functions only vanish at finitely many n-tuples of roots of unity unless they vanish along a translate of the formal multiplicative group. (Note that a root of unity lies on the disk only if it has p-power order). For polynomial functions, this follows from the multiplicative Manin-Mumford Conjecture. Our results however allow for a much wider class of analytic functions; in particular we establish a rigidity result for formal tori. Moreover, we extend these results to Lubin-Tate formal groups beyond the formal multiplicative group. We then apply our methods to Hida theory where the rings of analytic functions in question parametrize families of automorphic forms by weight. In particular, this allows us to describe which families of cohomological automorphic forms for GL_2 over an imaginary quadratic field contain forms in infinitely many classical weights and addresses a question arising from the work of Calegari and Mazur. Finally, we discuss analogous results for functions vanishing near the torsion of the formal group of an abelian variety. In this case, Coleman's theory of p-adic abelian integrals allows us to obtain a p-adic infinitesimal refinement of the Manin-Mumford Conjecture.

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  • 02/13/2018
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