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L-values of elliptic curves twisted by cubic characters

David Kurniadi Angdinata

Abstract

Given a rational elliptic curve $ E $ of analytic rank zero, its L-function can be twisted by an even primitive Dirichlet character $ χ$ of order $ q $, and in many cases its associated central algebraic L-value $ \mathcal{L}(E, χ) $ is known to be integral. This paper derives some arithmetic consequences from a congruence between $ \mathcal{L}(E, 1) $ and $ \mathcal{L}(E, χ) $ arising from this integrality, with an emphasis on cubic characters $ χ$. These include $ q $-adic valuations of the denominator of $ \mathcal{L}(E, 1) $, determination of $ \mathcal{L}(E, χ) $ in terms of Birch--Swinnerton-Dyer invariants, and asymptotic densities of $ \mathcal{L}(E, χ) $ modulo $ q $ by varying $ χ$.

L-values of elliptic curves twisted by cubic characters

Abstract

Given a rational elliptic curve of analytic rank zero, its L-function can be twisted by an even primitive Dirichlet character of order , and in many cases its associated central algebraic L-value is known to be integral. This paper derives some arithmetic consequences from a congruence between and arising from this integrality, with an emphasis on cubic characters . These include -adic valuations of the denominator of , determination of in terms of Birch--Swinnerton-Dyer invariants, and asymptotic densities of modulo by varying .
Paper Structure (8 sections, 19 theorems, 56 equations, 2 tables)

This paper contains 8 sections, 19 theorems, 56 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Background and conventions
  3. Modular symbols
  4. Denominators of L-values
  5. Units of twisted L-values
  6. Residual densities of twisted L-values
  7. Twisted L-values of Kisilevsky--Nam
  8. Tables of Galois images

Key Result

Theorem 1.1

Let $E$ be an elliptic curve over $\mathbb{Q}$ of conductor $N$, and let $\chi$ be a cubic Dirichlet character of odd prime conductor $p \nmid N$ such that $3 \nmid \mathrm{c}_0\!\del{E}\mathrm{BSD}\!\del{E}\#E\!\del{\mathbb{F}_p}$. Assume that Stevens's conjecture holds for $E$, and that the Birch- for some sign $u = \pm1$, chosen such that

Theorems & Definitions (58)

  • Theorem 1.1: Corollary \ref{['cor:cubic']}
  • Theorem 1.2: Proposition \ref{['prop:cubic']}
  • Theorem 1.3: Theorem \ref{['thm:density']}
  • Theorem 1.4: Theorem \ref{['thm:valuation']}
  • Conjecture 2.1: Stevens
  • Conjecture 2.2: Birch--Swinnerton-Dyer
  • Remark 2.3
  • Remark 2.4
  • Lemma 3.1
  • proof
  • ...and 48 more