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The 2-adic valuations of the algebraic central $L$-values for quadratic twists of weight 2 newforms

Taiga Adachi, Keiichiro Nomoto, Ryota Shii

TL;DR

This work provides sharp lower bounds for the $2$-adic valuations of algebraic central $L$-values $L(f, \chi_M, 1)$ for quadratic twists of a weight $2$ newform $f$ with rational coefficients, linking these valuations to the $2$-primary Tate–Shafarevich group when the analytic rank is zero. Building on Zhao’s method, the authors employ a refined modular-symbol framework that separates the $+/-$ modular symbols, enabling a decomposition of the twisted $L$-values into algebraic components and enabling 2-adic control even when $m \equiv 3 \bmod 4$. They formulate four main theorems across cases determined by the rectangularity of the period lattice and the parity parameter $n$, giving explicit lower bounds of the form $v_2\left( L(f, \chi_{M}, 1)/\Omega_f^{\mathrm{sgn}(\chi_M)} \right) \ge \mathfrak{w}_m\,r(m) + \min\{ \cdots \}$ and proving the existence of infinitely many $M$ for which equality holds. The paper also provides numerical examples at levels $N=34$ and $N=37$ illustrating the bounds and sharpness, highlighting cases where the conjectured equality is realized and the associated arithmetic finiteness results for the twists.

Abstract

Let $f$ be a normalized newform of weight 2 on $Γ_0(N)$ whose coefficients lie in $\mathbb{Q}$ and let $χ_M$ be a primitive quadratic Dirichlet character with conductor $M$. In this paper, under mild assumptions on $M$, we give a sharp lower bound of the 2-adic valuation of the algebraic central $L$-value $L(f, χ_M, 1)$ and evaluate the 2-adic valuation for an infinite number of $M$.

The 2-adic valuations of the algebraic central $L$-values for quadratic twists of weight 2 newforms

TL;DR

This work provides sharp lower bounds for the -adic valuations of algebraic central -values for quadratic twists of a weight newform with rational coefficients, linking these valuations to the -primary Tate–Shafarevich group when the analytic rank is zero. Building on Zhao’s method, the authors employ a refined modular-symbol framework that separates the modular symbols, enabling a decomposition of the twisted -values into algebraic components and enabling 2-adic control even when . They formulate four main theorems across cases determined by the rectangularity of the period lattice and the parity parameter , giving explicit lower bounds of the form and proving the existence of infinitely many for which equality holds. The paper also provides numerical examples at levels and illustrating the bounds and sharpness, highlighting cases where the conjectured equality is realized and the associated arithmetic finiteness results for the twists.

Abstract

Let be a normalized newform of weight 2 on whose coefficients lie in and let be a primitive quadratic Dirichlet character with conductor . In this paper, under mild assumptions on , we give a sharp lower bound of the 2-adic valuation of the algebraic central -value and evaluate the 2-adic valuation for an infinite number of .
Paper Structure (7 sections, 15 theorems, 76 equations, 3 tables)

This paper contains 7 sections, 15 theorems, 76 equations, 3 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Twisted L-function and Modular Symbols
  4. The 2-adic valuations of the algebraic parts
  5. Numerical Example
  6. Example of Theorem \ref{['thm:MainThm1']}
  7. Example of Theorem \ref{['thm:MainThm3']}

Key Result

Theorem 1.1

Assume that the Manin constant for $E$ is equal to 1. (See § sec:Preliminaries.) Write $M=4^nm=4^nq_1\cdots q_r$, where $n\in\{0, 1\}$ and $q_1, \dots, q_r$ are distinct odd primes. We set $\frakv_m\coloneqq\min_{1\leq i\leq r}\lbrace*\rbrace{v_2(a_{q_i}-2)}$. We put $\frakw_m$ as $0$ if $\frakv_m=0 where $\mathop{\mathrm{ sgn}}\nolimits(\chi_M)$ is the sign of $\chi_M(-1)$. Moreover, there exist

Theorems & Definitions (27)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 2.1: cf. manin1
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Proposition 3.3
  • proof
  • Lemma 3.4
  • ...and 17 more