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Iwasawa invariants of modular forms with $a_p=0$

Rylan Gajek-Leonard

Abstract

Fix a prime $p$ and a cuspidal newform $f$ of level coprime to $p$ with $a_p=0$. Attached to $f$ are signed $p$-adic $L$-functions $L_p^\pm(f)$ and Mazur-Tate elements $θ_n(f)$, both of which encode arithmetic data about $f$ along the cyclotomic $\mathbf{Z}_p$-extension of $\mathbf{Q}$. We compute the Iwasawa invariants of Mazur-Tate elements in terms of the corresponding invariants of the signed $p$-adic $L$-functions. As corollaries, we determine the $p$-adic valuation of critical values of the $L$-function of $f$, and describe a relation between the Iwasawa invariants of congruent modular forms of weights 2 and $p+1$. Our results provide an asymptotic method for computing the signed Iwasawa invariants attached to newforms of any weight $k\geq 2$ with $a_p=0$.

Iwasawa invariants of modular forms with $a_p=0$

Abstract

Fix a prime and a cuspidal newform of level coprime to with . Attached to are signed -adic -functions and Mazur-Tate elements , both of which encode arithmetic data about along the cyclotomic -extension of . We compute the Iwasawa invariants of Mazur-Tate elements in terms of the corresponding invariants of the signed -adic -functions. As corollaries, we determine the -adic valuation of critical values of the -function of , and describe a relation between the Iwasawa invariants of congruent modular forms of weights 2 and . Our results provide an asymptotic method for computing the signed Iwasawa invariants attached to newforms of any weight with .
Paper Structure (32 sections, 38 theorems, 161 equations, 2 figures, 4 tables)

This paper contains 32 sections, 38 theorems, 161 equations, 2 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Sketch of proof of Theorem \ref{['main1']}
  3. Outlook
  4. Acknowledgements
  5. Notation
  6. Preliminaries
  7. Iwasawa invariants
  8. Mazur-Tate Elements
  9. Modular symbols
  10. Cohomological symbols
  11. Mazur-Tate elements of modular forms
  12. $p$-adic $L$-functions
  13. Signed $p$-adic $L$-functions
  14. Lifts of Mazur-Tate elements
  15. Polynomial approximation
  16. ...and 17 more sections

Key Result

Theorem A

Let $f\in S_k(\Gamma_1(N))$ be a newform with $a_p(f)=0$. There are units $u_{n}\in \mathbf{Z}_p[\varepsilon(p)]^\times$ such that for all $n\geq 0$ we have where $\omega_n=(1+T)^{p^n}-1$ and $\log_{k,n}^{\pm}\in\mathbf{Z}_p[T]$ (see §section_lifts) are explicit products of cyclotomic polynomials.

Figures (2)

  • Figure 1: In gray are the regions $\mathcal{B}_n(F)$ in the cases $Q(F)=0$ and $Q(F)>0$, under the assumption $\mu=0$. In bold are the points $(i,v_i(F))$. The diagram illustrates Proposition \ref{['newton']}: if the Newton polygon of $\mathcal{T}_{\lambda-1}(F)$ is contained entirely in $\mathcal{B}_n(F)$ then each vertex of the Newton polygon is a lattice point within the gray region, implying that $\operatorname{ord}_pa_i\geq v_i(F)$ for all $i\in \{0,\dots, \lambda-1\}$ and therefore that $F$ is $p$-large at $n$.
  • Figure 2: The boundary of $\operatorname{NP}_n$ (dotted) and $\mathcal{B}_n$ (bold) when $Q_n=0$ and $Q_n>0$, illustrating the containment \ref{['sufnp']}.

Theorems & Definitions (94)

  • Theorem A: Theorem \ref{['lifts']}
  • Theorem B: Theorems \ref{['theorem_smallweights']} and \ref{['main']}
  • Example 1.1
  • Remark 1.2
  • Corollary C: Corollary \ref{['coro_p1']}
  • Corollary D
  • Theorem E: Theorem \ref{['mainpl']}
  • Definition 2.1
  • Lemma 2.2
  • proof
  • ...and 84 more