On the Lang–Trotter conjecture for Siegel modular forms

Abstract Let be a genus‐two cuspidal Siegel eigenform. We prove an adelic open image theorem for the compatible system of Galois representations associated with , generalizing the results of Ribet and Momose for elliptic modular forms. Using this result, we investigate the distribution of the Hecke eigenvalues of , and obtain upper bounds for the sizes of the sets for fixed , in the spirit of the Lang–Trotter conjecture for elliptic curves.