On the congruences of Eisenstein series with polynomial indexes

Authors：Hu, Su; Kim, Min-Soo; Sha, Min^*

Source：RAMANUJAN JOURNAL, 2023, 62(2): 413-430.

DOI：10.1007/s11139-023-00731-3

• Eisenstein series
• Congruences
• Serre's p-adic family of Eisenstein series
• p-adic analysis
• Bernoulli numbers

In this paper, based on Serre's p-adic family of Eisenstein series, we prove a general family of congruences for Eisenstein series Gk in the formS(n) (i=1) gi(p)G fi (p) = g0(p)(mod pN),where f(1)(t), ... , f(n)(t) E Z[t] are non-constant integer polynomials with positive leading coefficients and g0(t), ... , gn(t) E Q(t) are rational functions. This generalizes the classical von Staudt-Clausen and Kummer congruences of Eisenstein series, and also some new Congruences.