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Congruences and ramified primes in fields of coefficients of newforms

Nuno Freitas, Filip Gawron

Abstract

We investigate the splitting behavior of $\ell$ in the coefficient field of a newform $f$ of level $N$, under the assumption that $f$ is congruent modulo a prime above $\ell$ to another newform $g$ whose level divides $N/p^2$ for some prime $p\mid N$. In particular, we show that the maximal real subfield of the $\ell$-th cyclotomic field, $\mathbb{Q}(ζ_\ell + ζ_\ell^{-1})$, is contained in the coefficient field of $f$. We conclude by presenting explicit examples that illustrate these results.

Congruences and ramified primes in fields of coefficients of newforms

Abstract

We investigate the splitting behavior of in the coefficient field of a newform of level , under the assumption that is congruent modulo a prime above to another newform whose level divides for some prime . In particular, we show that the maximal real subfield of the -th cyclotomic field, , is contained in the coefficient field of . We conclude by presenting explicit examples that illustrate these results.

Paper Structure

This paper contains 4 sections, 9 theorems, 20 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Auxiliary results
  3. Proof of Theorem \ref{['maintheorem']}
  4. Examples of congruences and ramified primes

Key Result

Theorem 1.2

Let $f\in S_k(\Gamma_0(N),\varepsilon)$ be a newform of weight $k\geq 2$ and field of coefficients $\mathbb{Q}_f$. Let $p$ be a prime such that $p^2 \mid N$. Let $\ell \neq p$ be an odd prime, $\lambda$ a prime of $\mathbb{Q}_f$ dividing $\ell$, and $N'$ the prime-to-$\ell$ part of the Artin conduct

Theorems & Definitions (22)

  • Theorem 1.2
  • Remark 1.3
  • Proposition 2.1
  • Lemma 2.2
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • ...and 12 more