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Non-optimal levels of some reducible mod $p$ modular representations

Shaunak V. Deo

Abstract

Let $p \geq 5$ be a prime, $N$ be an integer not divisible by $p$, $\barρ_0$ be a reducible, odd and semi-simple representation of $G_{\mathbb{Q},Np}$ of dimension $2$ and $\{\ell_1,\cdots,\ell_r\}$ be a set of primes not dividing $Np$. After assuming that a certain Selmer group has dimension at most $1$, we find sufficient conditions for the existence of a cuspidal eigenform $f$ of level $N\prod_{i=1}^{r}\ell_i$ and appropriate weight lifting $\barρ_0$ such that $f$ is new at every $\ell_i$. Moreover, suppose $p \mid \ell_{i_0}+1$ for some $1 \leq i_0 \leq r$. Then, after assuming that a certain Selmer group vanishes, we find sufficient conditions for the existence of a cuspidal eigenform of level $N\ell_{i_0}^2 \prod_{j \neq i_0} \ell_j$ and appropriate weight which is new at every $\ell_i$ and which lifts $\barρ_0$. As a consequence, we prove a conjecture of Billerey--Menares in many cases.

Non-optimal levels of some reducible mod $p$ modular representations

Abstract

Let be a prime, be an integer not divisible by , be a reducible, odd and semi-simple representation of of dimension and be a set of primes not dividing . After assuming that a certain Selmer group has dimension at most , we find sufficient conditions for the existence of a cuspidal eigenform of level and appropriate weight lifting such that is new at every . Moreover, suppose for some . Then, after assuming that a certain Selmer group vanishes, we find sufficient conditions for the existence of a cuspidal eigenform of level and appropriate weight which is new at every and which lifts . As a consequence, we prove a conjecture of Billerey--Menares in many cases.
Paper Structure (17 sections, 21 theorems, 49 equations)

This paper contains 17 sections, 21 theorems, 49 equations.

Table of Contents

  1. Introduction
  2. History
  3. Set-up and Main results
  4. Cyclicity of $H^1_{\{p\}}(G_{\mathbb{Q},Np},\bar{\chi}^{-1})$
  5. Sketch of the proofs of the main results
  6. Organization of the paper
  7. Notations and Conventions
  8. Acknowledgments
  9. Structure of deformation rings
  10. Deformation rings and ordinary deformations
  11. Relationship between $R(M)^{\text{pd}}_{\bar{\rho}_0}$ and $R(M)_{\bar{\rho}_c}$
  12. Increasing the ramification
  13. Relationship between $R(M)_{\bar{\rho}_c}$ and $R(N)_{\bar{\rho}_c}$
  14. Proof of Theorem \ref{['thmb']}
  15. Proof of Theorem \ref{['thmc']}
  16. ...and 2 more sections

Key Result

Theorem 1.2

Suppose we are in the Set-up setup as above and $\dim(H^1_{\{p\}}(G_{\mathbb{Q},Np},\bar{\chi}^{-1})) = 1$. Let $k>2$ be an integer such that $k \equiv k_0 \pmod{p-1}$ and $\ell_1,\cdots,\ell_r$ be primes such that $\ell_i \nmid Np$, $\bar{\chi}|_{G_{\mathbb{Q}_{\ell_i}}} = \omega_p|_{G_{\mathbb{Q}_

Theorems & Definitions (49)

  • Theorem 1.2
  • Theorem 1.3
  • Remark 1.4
  • Remark 1.5
  • Remark 1.6
  • Corollary 1.7
  • Corollary 1.8
  • Remark 1.9
  • Remark 1.10
  • Remark 1.11
  • ...and 39 more