On the congruences of Eisenstein series with polynomial indexes

2018/05/23 by Su Hu, Min-Soo Kim, Hu, Su +3
Mathematics · #Advanced Algebra and Geometry #Advanced Mathematical Identities #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1805.09225

openalex publication_date 2018/05/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 ∑_i=1ⁿ g_i(p)G_fi(p)≡ g₀(p)\mod p^N, where f₁(t),…,f_n(t)∈ℤ[t] are non-constant integer polynomials with positive leading coefficients and g₀(t),…,g_n(t)∈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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