Eisenstein series, p-adic modular functions, and overconvergence
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Let p be a prime ≥ 5. We establish explicit rates of overconvergence for some members of the “Eisenstein family”, notably for the p-adic modular function V(E(1,0)∗)/E(1,0)∗ (V the p-adic Frobenius operator) that plays a pivotal role in Coleman’s theory of p-adic families of modular forms. The proof goes via an in-depth analysis of rates of overconvergence of p-adic modular functions of form V(Ek) / Ek where Ek is the classical Eisenstein series of level 1 and weight k divisible by p- 1. Under certain conditions, we extend the latter result to a vast generalization of a theorem of Coleman–Wan regarding the rate of overconvergence of V(Ep-1) / Ep-1. We also comment on previous results in the literature. These include applications of our results for the primes 5 and 7.
Original language | English |
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Article number | 65 |
Journal | Research in Number Theory |
Volume | 7 |
Issue number | 4 |
Number of pages | 33 |
ISSN | 2363-9555 |
DOIs | |
Publication status | Published - 2021 |
Bibliographical note
Publisher Copyright:
© 2021, The Author(s), under exclusive licence to Springer Nature Switzerland AG.
- Colman-Mazur eigencurve, Eisenstein series, Overconvergent modular forms
Research areas
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