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The Birch--Swinnerton-Dyer exact formula for quadratic twists of elliptic curves

Shuai Zhai

Abstract

In the present paper, we obtain a general lower bound for the $2$-adic valuation of the algebraic part of the central value of the complex $L$-series for the quadratic twists of any elliptic curve over $\mathbb{Q}$, showing that when the $2$-part of the product of Tamagawa factors grows, the $2$-part of the algebraic central $L$-value grows as well, in accordance with the Birch--Swinnerton-Dyer exact formula. This generalises a result of Coates--Kim--Liang--Zhao to all elliptic curves defined over $\mathbb{Q}$. We also prove the existence of an explicit infinite family of quadratic twists with analytic rank $0$ for a large family of elliptic curves.

The Birch--Swinnerton-Dyer exact formula for quadratic twists of elliptic curves

Abstract

In the present paper, we obtain a general lower bound for the -adic valuation of the algebraic part of the central value of the complex -series for the quadratic twists of any elliptic curve over , showing that when the -part of the product of Tamagawa factors grows, the -part of the algebraic central -value grows as well, in accordance with the Birch--Swinnerton-Dyer exact formula. This generalises a result of Coates--Kim--Liang--Zhao to all elliptic curves defined over . We also prove the existence of an explicit infinite family of quadratic twists with analytic rank for a large family of elliptic curves.

Paper Structure

This paper contains 8 sections, 18 theorems, 94 equations.

Table of Contents

  1. Introduction
  2. The infinitude of $\mathcal{S}$
  3. Results from modular symbols
  4. Modular symbols
  5. Period lattice
  6. Integrality at 2
  7. Lower Bound
  8. Non-vanishing result

Key Result

Theorem 1.1

Let $E$ be any optimal elliptic curve over ${\mathbb {Q}}$ having conductor $C$. Then, for all square free integers $M \equiv1 \ \mathrm{mod}\ 4$ with $(M, C) = 1$ such that $L(E^{(M)}, 1) \neq 0$, we have where $\nu_E$ is the Manin constant of $E$.

Theorems & Definitions (35)

  • Theorem 1.1
  • Corollary 1.2
  • Conjecture 1.3
  • Theorem 1.4
  • proof
  • Corollary 1.5
  • Remark 1.6
  • Remark 1.7
  • Lemma 2.1
  • proof
  • ...and 25 more