Abstract Recent works [2] , [5] , [13] , [14] studied how Hecke operators map between subspaces of modular forms of the type A r , s = { η ( δ z ) r F ( δ z ) : F ( z ) ∈ M s } for suitable δ. For primes p ≥ 5 , we give results on how U p maps between subspaces of these spaces with p-integral coefficients modulo powers of p. As a corollary, we prove congruences between Shimura images of a natural family of half-integral weight eigenforms modulo p. Specifically, we classify all congruences of the type Shim r ( η ( 24 z ) r E s ( 24 z ) ) ≡ Shim r ′ ( η ( 24 z ) r ′ E s ′ ( 24 z ) ) ( mod p ) for 1 ≤ r , r ′ ≤ 23 with gcd ⁡ ( r r ′ , 6 ) = 1 , 5 ≤ p ≤ 23 , and s , s ′ ∈ { 0 , 4 , 6 , 8 , 10 , 14 } . Congruences of this type were used in [7] , [8] to extend Ramanujan's congruences for p ( n ) modulo powers of 5, 7, and 11.

中文翻译：

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- Shimura 图像的运算符和同余

摘要 最近的工作 [2] , [5] , [13] , [14] 研究了 Hecke 算子如何在 A r , s = { η ( δ z ) r F ( δ z ) : F 类型的模形式的子空间之间映射( z ) ∈ M s } 为合适的δ。对于素数 p ≥ 5 ，我们给出了 U p 如何在这些空间的子空间之间映射的结果，这些空间的 p 积分系数是 p 的模幂。作为推论，我们证明了模 p 的半整数权重特征形式自然族的 Shimura 图像之间的一致性。具体来说，我们对 Shim r ( η ( 24 z ) r E s ( 24 z ) ) ≡ Shim r ′ ( η ( 24 z ) r ′ E s ′ ( 24 z ) ) ( mod p ) 类型的所有同余进行分类1 ≤ r , r ′ ≤ 23 且 gcd ⁡ ( rr ′ , 6 ) = 1 , 5 ≤ p ≤ 23 ，并且 s , s ′ ∈ { 0 , 4 , 6 , 8 , 10 , 14 } 。这种类型的同余在 [7] 、 [8] 中用于扩展 Ramanujan 对 5、7 和 11 的 p ( n ) 模幂的同余。