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Non-vanishing of central $L$-values of the Gross family of elliptic curve

Yukako Kezuka, Yong-Xiong Li

TL;DR

This work proves non-vanishing of central L-values for quadratic twists of Gross’s CM elliptic curve over Hilbert class fields, establishing exact 2-adic valuations of the corresponding algebraic L-values and deducing finiteness of Mordell–Weil and Tate–Shafarevich groups. It combines a 2-descent on the Gross-related abelian variety B = Res_{H/K}(A), infinite descent in Iwasawa theory at the special prime above 2, and a generalized Zhao induction for abelian varieties to control L-values of twists A^{(R)}/H. A P-part Birch–Swinnerton-Dyer refinement is proven for the restriction of scalars, and a density result for non-vanishing central L-values is derived. The results extend prior work to q ≡ 7 mod 8, including the case q ≡ 15 mod 16, and contribute to the broader understanding of non-vanishing and BSD phenomena in CM-elliptic curve families.

Abstract

We prove non-vanishing theorems for the central values of $L$-series of quadratic twists of the Gross elliptic curve with complex multiplication by the imaginary quadratic field $\mathbb{Q}(\sqrt{-q})$, where $q$ is any prime congruent to $7$ modulo $8$. This completes the non-vanishing theorems proven by Coates and the second author in which the primes $q$ were taken to be congruent to $7$ modulo $16$. From this, we obtain the finiteness of the Mordell-Weil group and the Tate-Shafarevich group for these curves. For a prime $\mathfrak{P}$ lying above the prime $2$, we also prove a converse theorem in the rank $0$ case and the $\mathfrak{P}$-part of the Birch-Swinnerton-Dyer conjecture for the higher-dimensional abelian varieties obtained by restriction of scalars.

Non-vanishing of central $L$-values of the Gross family of elliptic curve

TL;DR

This work proves non-vanishing of central L-values for quadratic twists of Gross’s CM elliptic curve over Hilbert class fields, establishing exact 2-adic valuations of the corresponding algebraic L-values and deducing finiteness of Mordell–Weil and Tate–Shafarevich groups. It combines a 2-descent on the Gross-related abelian variety B = Res_{H/K}(A), infinite descent in Iwasawa theory at the special prime above 2, and a generalized Zhao induction for abelian varieties to control L-values of twists A^{(R)}/H. A P-part Birch–Swinnerton-Dyer refinement is proven for the restriction of scalars, and a density result for non-vanishing central L-values is derived. The results extend prior work to q ≡ 7 mod 8, including the case q ≡ 15 mod 16, and contribute to the broader understanding of non-vanishing and BSD phenomena in CM-elliptic curve families.

Abstract

We prove non-vanishing theorems for the central values of -series of quadratic twists of the Gross elliptic curve with complex multiplication by the imaginary quadratic field , where is any prime congruent to modulo . This completes the non-vanishing theorems proven by Coates and the second author in which the primes were taken to be congruent to modulo . From this, we obtain the finiteness of the Mordell-Weil group and the Tate-Shafarevich group for these curves. For a prime lying above the prime , we also prove a converse theorem in the rank case and the -part of the Birch-Swinnerton-Dyer conjecture for the higher-dimensional abelian varieties obtained by restriction of scalars.
Paper Structure (6 sections, 28 theorems, 87 equations)

This paper contains 6 sections, 28 theorems, 87 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The first $2$-descent on $B$ over $K$
  4. Descent on $B$ over the fields $K$ and $F = K(B_{{\mathfrak{P}}^2})$
  5. Infinite descent over the field $F_\infty=K(B_{{\mathfrak{P}}^\infty})$ and a main conjecture for $F_\infty/K$
  6. Generalised Zhao's method and $2$-adic valuation of central $L$-values

Key Result

Theorem 1.1

Let $q$ be a prime congruent to $7$ modulo $8$ and $K = \mathbb{Q}(\sqrt{-q})$. Let $H$ be the Hilbert class field of $K$ and $A$ the Gross curve defined over $H$ with complex period $\Omega_\infty$. For $R = r_1 \cdots r_k \in {\mathfrak{R}}$, denote by $A^{(R)}$ the twist of $A$ by $H(\sqrt{R})/H$ and for any prime $\mathcal{P}$ of $TH$ lying above ${\mathfrak{P}}$, we have In particular, $L(A^

Theorems & Definitions (39)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Proposition 2.1
  • Proposition 3.1
  • proof
  • Lemma 3.2
  • proof
  • Proposition 3.3
  • proof
  • ...and 29 more