Newform Eisenstein Congruences of Local Origin

Dan Fretwell; Jenny Roberts

doi:10.48550/arXiv.2306.09842

Newform Eisenstein Congruences of Local Origin

Dan Fretwell^*, Jenny Roberts

^*Corresponding author for this work

Research output: Contribution to journal › Article (Academic Journal) › peer-review

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Abstract

We give a general conjecture concerning the existence of Eisenstein congruences between weight 𝑘≥3 newforms of square-free level NM and weight k new Eisenstein series of square-free level N. Our conjecture allows the forms to have arbitrary character 𝜒 of conductor N. The special cases 𝑀=1 and 𝑀=𝑝 prime are fully proved, with partial results given in general. We also consider the relation with the Bloch–Kato conjecture, and finish with computational examples demonstrating cases of our conjecture that have resisted proof.

Original language	English
Pages (from-to)	505–527
Number of pages	23
Journal	Ramanujan Journal
Volume	64
Issue number	2
Early online date	23 Mar 2024
DOIs	https://doi.org/10.48550/arXiv.2306.09842 https://doi.org/10.1007/s11139-024-00838-1
Publication status	Published - 1 Jun 2024

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Publisher Copyright:
© The Author(s) 2024.

Keywords

math.NT
11F33, 11F03, 11F30, 11F80

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10.48550/arXiv.2306.09842Licence: CC BY
10.1007/s11139-024-00838-1Licence: CC BY

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title = "Newform Eisenstein Congruences of Local Origin",

abstract = "We give a general conjecture concerning the existence of Eisenstein congruences between weight 𝑘≥3 newforms of square-free level NM and weight k new Eisenstein series of square-free level N. Our conjecture allows the forms to have arbitrary character 𝜒 of conductor N. The special cases 𝑀=1 and 𝑀=𝑝 prime are fully proved, with partial results given in general. We also consider the relation with the Bloch–Kato conjecture, and finish with computational examples demonstrating cases of our conjecture that have resisted proof.",

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T1 - Newform Eisenstein Congruences of Local Origin

AU - Fretwell, Dan

AU - Roberts, Jenny

PY - 2024/6/1

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N2 - We give a general conjecture concerning the existence of Eisenstein congruences between weight 𝑘≥3 newforms of square-free level NM and weight k new Eisenstein series of square-free level N. Our conjecture allows the forms to have arbitrary character 𝜒 of conductor N. The special cases 𝑀=1 and 𝑀=𝑝 prime are fully proved, with partial results given in general. We also consider the relation with the Bloch–Kato conjecture, and finish with computational examples demonstrating cases of our conjecture that have resisted proof.

AB - We give a general conjecture concerning the existence of Eisenstein congruences between weight 𝑘≥3 newforms of square-free level NM and weight k new Eisenstein series of square-free level N. Our conjecture allows the forms to have arbitrary character 𝜒 of conductor N. The special cases 𝑀=1 and 𝑀=𝑝 prime are fully proved, with partial results given in general. We also consider the relation with the Bloch–Kato conjecture, and finish with computational examples demonstrating cases of our conjecture that have resisted proof.

KW - math.NT

KW - 11F33, 11F03, 11F30, 11F80

U2 - 10.48550/arXiv.2306.09842

DO - 10.48550/arXiv.2306.09842

M3 - Article (Academic Journal)

SN - 1382-4090

VL - 64

SP - 505

EP - 527

JO - Ramanujan Journal

JF - Ramanujan Journal

IS - 2

ER -

Newform Eisenstein Congruences of Local Origin

Abstract

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