Divisors of Fourier coefficients in p-adic families of modular forms

Abstract

We prove that the divisor function d(n(κ)) for the absolute norm of Hecke eigenvalues a_p(κ) is unbounded in p-adic families of modular forms, yet its distribution is highly constrained by p-adic geometry. We establish a striking contrast with the archimedean setting, showing that Hecke eigenvalues with many divisors are exponentially sparse in the family. This rigidity, driven by the geometric properties of the eigencurve, concentrates large divisor counts exclusively near points where a_p(κ) has high p-adic valuation.