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arXiv · 1805.09225

On the congruences of Eisenstein series with polynomial indexes

Su Hu·Min-Soo Kim·Min Sha
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Abstract

In this paper, based on Serre's $p$-adic family of Eisenstein series, we prove a general family of congruences for Eisenstein series $G_k$ in the form $$ \sum_{i=1}^n g_i(p)G_{f_i(p)}\equiv g_0(p)\mod p^N, $$ where $f_1(t),\ldots,f_n(t)\in\mathbb{Z}[t]$ are non-constant integer polynomials with positive leading coefficients and $g_0(t),\ldots,g_n(t)\in\mathbb{Q}(t)$ are rational functions. This generalizes the classical von Staudt-Clausen's and Kummer's congruences of Eisenstein series, and also yields some new congruences.

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Su Hu, Min-Soo Kim, Min Sha. 2018-05-23. On the congruences of Eisenstein series with polynomial indexes. https://arxiv.org/abs/1805.09225

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