On the Lang--Trotter conjecture for Siegel modular forms

Abstract

Let $f$ be a genus two cuspidal Siegel modular eigenform. We prove an adelic open image theorem for the compatible system of Galois representations associated to $f$, generalising the results of Ribet and Momose for elliptic modular forms. Using this result, we investigate the distribution of the Hecke eigenvalues $a_p$ of $f$, and obtain upper bounds for the sizes of the sets $\{p\le x:a_p=a\}$ for fixed $a\in \mathbf{C}$, in the spirit of the Lang--Trotter conjecture for elliptic curves.

29 pages. Accepted version, to appear in Mathematika. Comments welcome!