Level stripping for degree 2 Siegel modular forms

Abstract

Abstract. In this paper, we consider stripping primes from the level of degree 2 cuspidal Siegel eigenforms. Specifically, given an eigenform of level N r under certain restrictions, where N is a prime, we construct an eigenform of level N , which is congruent in eigenvalues to our original form. To obtain our results, we use constructions of Eisenstein series and theta functions to adapt ideas from a level stripping result on elliptic modular forms.

Level stripping for vector-valued Siegel modular forms of genus 2

“…Bearing this in mind, the main result of this paper is a level stripping result for Siegel modular forms analogous to Theorem 1. Such results have been previously been obtained by Taylor in [27] under an ordinarity condition, by Brown and the author in [7] for Siegel modular forms which are lifted from elliptic modular forms, and by the author in [14] for scalar valued Siegel modular forms.…”

“…In particular, the level-stripping result of this paper and the subsequent application to Galois representations can be viewed as a direct generalization of the results in [14] to the vector-valued setting. The techniques used to prove the main results in this paper are identical to the techniques employed in [14], but the primary obstacle lies in the fact that the arithmetic of vector-valued Siegel modular forms can be quite a bit more delicate. Furthermore, it is important to remark that this paper seeks to correct a mistake which was overlooked in [14].…”

“…In this section, we introduce a certain operator on the space of Siegel modular forms which is analogous to the U N operator in [20] and then give the relevant properties which will be important for our purposes. Furthermore, we will provide a correction to the proof of the main result in [14].…”

“…We remark here on a mistake in the proof of Theorem 8 in [14]. In this proof, the author makes use of of the property given in Lemma 11.…”

“…The techniques used to prove the main results in this paper are identical to the techniques employed in [14], but the primary obstacle lies in the fact that the arithmetic of vector-valued Siegel modular forms can be quite a bit more delicate. Furthermore, it is important to remark that this paper seeks to correct a mistake which was overlooked in [14]. For more details see the end of Section 3.3.…”