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arXiv · 2201.09278

On the Lang--Trotter conjecture for Siegel modular forms

Arvind Kumar·Moni Kumari·Ariel Weiss
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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.

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Arvind Kumar, Moni Kumari, Ariel Weiss. 2022-01-23. On the Lang--Trotter conjecture for Siegel modular forms. https://doi.org/10.1112/mtk.70108

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