A conjecture of Coleman on the Eisenstein family

Bryan W. Advocaat, Ian Kiming, Gabor Wiese*

*Corresponding author af dette arbejde

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Abstract

We prove for primes p≥5 a conjecture of Coleman on the analytic continuation of the family of modular functions Eκ∗V(Eκ∗) derived from the family of Eisenstein series Eκ∗. The precise, quantitative formulation of the conjecture involved a certain constant depending on p. We show by an example that the conjecture with the constant that Coleman conjectured cannot hold in general for all primes. On the other hand, the constant that we give is also shown not to be optimal in all cases. The conjecture is motivated by its connection to certain central statements in works by Buzzard and Kilford, and by Roe, concerning the “halo” conjecture for the primes 2 and 3, respectively. We show how our results generalize those statements and comment on possible future developments.

OriginalsprogEngelsk
TidsskriftRamanujan Journal
Vol/bind66
Udgave nummer1
Sider (fra-til)1-22
ISSN1382-4090
DOI
StatusUdgivet - 2025

Bibliografisk note

Funding Information:
This work was supported by the Luxembourg National Research Fund PRIDE17/1224660/GPS.

Publisher Copyright:
© The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature 2024.

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