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# Mathematical Research Letters

## Volume 22 (2015)

### Number 3

### The structure of Siegel modular forms modulo $p$ and $U(p)$ congruences

Pages: 899 – 928

DOI: http://dx.doi.org/10.4310/MRL.2015.v22.n3.a14

#### Authors

#### Abstract

We determine the ring structure of Siegel modular forms of degree $g$ modulo a prime $p$, extending Nagaoka’s result in the case of degree $g = 2$. We characterize $U(p)$ congruences of Jacobi forms and Siegel modular forms, and surprisingly find different behaviors of Siegel modular forms of even and odd degrees.

#### Keywords

Siegel modular forms modulo $p$, theta cycles and $U(p)$ congruences, Jacobi forms modulo $p$

#### 2010 Mathematics Subject Classification

Primary 11F33, 11F46. Secondary 11F50.