Reference : A Conjecture of Coleman on the Eisenstein Family
 Document type : E-prints/Working papers : Already available on another site Discipline(s) : Physical, chemical, mathematical & earth Sciences : Mathematics To cite this reference: http://hdl.handle.net/10993/51649
 Title : A Conjecture of Coleman on the Eisenstein Family Language : English Author, co-author : Advocaat, Bryan [University of Luxembourg > Faculty of Science, Technology and Medicine (FSTM) > Department of Mathematics (DMATH) >] Kiming, Ian [] Wiese, Gabor [University of Luxembourg > Faculty of Science, Technology and Medicine (FSTM) > Department of Mathematics (DMATH) >] Publication date : 2022 Peer reviewed : No Abstract : [en] We prove for primes $p\ge 5$ a conjecture of Coleman on the analytic continuation of the family of modular functions $\frac{\Es_\k}{V(\Es_\k)}$ derived from the family of Eisenstein series $\Es_\k$. The precise, quantitative formulation of the conjecture involved a certain on $p$ depending constant. 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. Permalink : http://hdl.handle.net/10993/51649 source URL : https://arxiv.org/abs/2207.03822 FnR project : FnR ; FNR12246620 > Hugo Parlier > GPS > Geometry, Probability And Their Synergies > 01/01/2019 > 30/06/2025 > 2017

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