Eisenstein series, p-adic modular functions, and overconvergence
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Eisenstein series, p-adic modular functions, and overconvergence. / Kiming, Ian; Rustom, Nadim.
I: Research in Number Theory, Bind 7, Nr. 4, 65, 2021.Publikation: Bidrag til tidsskrift › Tidsskriftartikel › Forskning › fagfællebedømt
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TY - JOUR
T1 - Eisenstein series, p-adic modular functions, and overconvergence
AU - Kiming, Ian
AU - Rustom, Nadim
N1 - Publisher Copyright: © 2021, The Author(s), under exclusive licence to Springer Nature Switzerland AG.
PY - 2021
Y1 - 2021
N2 - 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.
AB - 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.
KW - Colman-Mazur eigencurve
KW - Eisenstein series
KW - Overconvergent modular forms
U2 - 10.1007/s40993-021-00292-8
DO - 10.1007/s40993-021-00292-8
M3 - Journal article
AN - SCOPUS:85116403242
VL - 7
JO - Research in Number Theory
JF - Research in Number Theory
SN - 2363-9555
IS - 4
M1 - 65
ER -
ID: 284172945