Integrality of \mathfrakp-Adic L-Functions at Eisenstein Primes

2023/12/06 by Matthew Verheul, Verheul, Matthew
Mathematics · #Advanced Algebra and Geometry #Analytic Number Theory Research #Advanced Mathematical Identities

paper · pdf · doi:10.48550/arxiv.2312.03281

Abstract

Let f be a normalized, ordinary newform of weight ≥ 2. For each prime \mathfrakp of F=ℚ(a_n)_{n∈ ℕ}, there is an associated \mathfrakp-adic L-function L_\mathfrakp(f)∈ Λ⊗ ℚ interpolating special values of the classical L-function. If f is not congruent modulo \mathfrakp to an Eisenstein series, one knows L_\mathfrakp(f)∈ Λ. In this paper, we show, under mild hypotheses on the ramification of f, that this integrality result holds when f is congruent to an Eisenstein series. Moreover, we also obtain a divisibility in the main conjecture for L_\mathfrakp(f). As an application, we show that the integrality result and the divisibility hold in particular when f is of weight 2.

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