Elsevier

Journal of Number Theory

Volume 224, July 2021, Pages 142-164
Journal of Number Theory

General Section
Quadratic twists of X0(14)

https://doi.org/10.1016/j.jnt.2021.01.011Get rights and content

Abstract

In the present paper, we prove the 2-part of Birch and Swinnerton-Dyer conjecture for an explicit infinite family of rank 0 quadratic twists of the modular elliptic curve X0(14), using an explicit form of the Waldspurger formula. We also give an explicit infinite family of rank 1 quadratic twists of X0(14) whose Tate–Shafarevich group is of odd cardinality.

Introduction

Let E be an elliptic curve defined over Q, and let L(E,s) be the complex L-series of E. For each square free non-zero integer d1, we write E(d) for the twist of E by the quadratic extension Q(d)/Q, and L(E(d),s) for its complex L-series. It has been proved in [2] that there are infinitely many d such that L(E(d),s) is non-vanishing at s=1, and infinitely many d such that L(E(d),s) has a simple zero at s=1. Thanks to the results of Gross–Zagier and Kolyvagin, we know that for those d such that L(E(d),s) has a zero at s=1 of order 0 or 1, the rank of E(d)(Q) is equal to the order of this zero at s=1 and the Tate–Shafarevich group Ш(E(d)) of E(d) is finite. It is therefore natural to ask whether L(E(d),s) has a zero at s=1 of order 0 or 1 as d varies.

For the remainder of this section, we assume that L(E,1)0. Let C(E) denote the conductor of E, and let Ω(E) denote the fundamental real period of the Néron differential on E. We defineL(alg)(E,1)=L(E,1)/Ω(E), which is a non-zero rational number. Let c(E) denote the number of connected components of E(R), and for each prime q dividing C(E), let cq(E)=[E(Qq):E0(Qq)], where E0(Qq) is the subgroup of points in E(Qq) with non-singular reduction modulo q. The conjectural Birch and Swinnerton-Dyer formula is then given byL(alg)(E,1)=c(E)q|C(E)cq(E)#(Ш(E))(#E(Q))2. For each prime p, the equality of the p-primary parts of the two sides of (1.1) is called the p-part of Birch and Swinnerton-Dyer formula. While E has complex multiplication, Rubin [14] proved that the p-part of Birch and Swinnerton-Dyer formula for E is valid for all primes p5. On the other hand, the results of Skinner–Urban [17] and Kato [11] on Iwasawa theory give the p-part of Birch and Swinnerton-Dyer formula for all elliptic curves E without complex multiplication and for all primes p but a finite number of primes p, including p=2,3 always. The aim of the present paper is to prove the remaining 2-part of Birch and Swinnerton-Dyer formula for an infinite family of quadratic twists of the elliptic curveA=X0(14):y2+xy+y=x3+4x6. We know that A(Q)=Z/6Z, the discriminant of A is 2673, the j-invariant of A is 265373433 and its Néron differential ω=dx/(2y+x) has the fundamental real period Ω(A)=1.98134. Moreover, we haveL(alg)(A,1)=1/6. It is also known that the Tate–Shafarevich group of A is trivial and that c2(A)=2, c7(A)=3 and c(A)=1. Therefore the full Birch and Swinnerton-Dyer conjecture is valid for A. However, the full Birch and Swinnerton-Dyer conjecture is still unknown for arbitrary quadratic twists of A that have non-vanishing L-values at s=1. The following main theorem gives a family of quadratic twists of A for which the 2-part of Birch and Swinnerton-Dyer conjecture is valid.

Theorem 1.1

Let r0 be an integer. Let M=ϵq1qr be a square free integer, where each qi is a prime which is inert both in Q(7) and in Q(2), and the sign ϵ=±1 is chosen so that M1mod4. Then L(A(M),1)0, A(M)(Q) is finite, the Tate–Shafarevich group of A(M) is finite of odd cardinality, and the 2-part of Birch and Swinnerton-Dyer formula is valid for A(M).

We remark that, using modular symbols, Cai–Li–Zhai [5] also proved this theorem in the special case when M=q1qr, where each qi is a prime 5mod8 which is inert in Q(7). In this paper, the main tool to prove Theorem 1.1 is an explicit form of the Waldspurger formula given in [6]. The key to establishing this formula is an explicit calculation on the test vector associated to A. For this, we will use the test vector theory in [10] and the explicit description in [1] of the Hecke action on the Shimura set associated to A.

This paper is motivated from the subsequent papers [18], [7] and [3]. They proved the 2-part of Birch and Swinnerton-Dyer formula for certain families of quadratic twists of elliptic curves with complex multiplication, which have minimal Weierstrass equationsE:y2=x3x,X0(49):y2+xy=x3x22x1andX0(36):y2=x3+1. A new feature of this paper is that our elliptic curve A=X0(14) does not have complex multiplication, and that there is no Q-isogenous quadratic twist of A. Thus the induction methods of [7], [4], [3], [18] cannot directly apply to A, in a sense that one can only get the 2-part of Birch and Swinnerton-Dyer formula for a product of two quadratic twists of A. To overcome this difficulty, we will use both the induction method via the Euler system of Gross points [3], and the induction method via the unramified periods [18], [7]. In the former induction method, we will use both the inert and the split Kolyvagin primes' Euler system properties (see also [4] for the case of Heegner points), and some 2-adic properties of the Fourier coefficients of the newform associated to A at these primes (see Lemma 3.1) play an essential role in our whole argument.

Overview of the paper. To compute the 2-Selmer groups, we introduce a classical 2-descent method in §2. In §3, as an application of Corollary 2.4, we give an infinite family of the rank 1 twists whose Tate–Shafarevich group is of odd cardinality. Most of the rest of the paper is devoted to the L-series. We establish an explicit form of the Waldspurger formula in §4, and use induction arguments to obtain the 2-adic valuation of the L-values at s=1 in §5. Finally, in §6, we compute Tamagawa factors and prove Theorem 1.1.

Acknowledgments

The authors would like to thank John Coates for suggesting this problem. The first author is supported by a KIAS Individual Grant MG070401 at Korea Institute for Advanced Study. The second author would like to thank Myungjun Yu to invite him to a workshop held at KIAS in November 2019 for providing us with an excellent opportunity to discuss some ideas developed here. The second author is supported by NSFC-11901332.

Section snippets

Classical 2-descents

In this section, we will compute the 2-Selmer groups for a certain family of quadratic twists of A. We will use a classical 2-descent method (see Chapter X of [15] for details).

For a classical 2-descent method, we first give another expression for A and A(M). By changing of variables, A is given by the equationA:y2=x3+13x2+128x. Let M be a square free non-zero integer ≠1 with M1mod4 and (M,7)=1. Let A(M) be the twist of M by the quadratic extension Q(M)/Q. The equation for A(M) is then given by

Rank 1 twists

In this section, we will briefly discuss an application of Corollary 2.4, using Theorem 2.5 in [7]. We remark that for those M in Corollary 2.4, the formula (2.2) shows that A(M) has root number −1. Among those quadratic twists A(M), we will find the family of the twists which have a simple zero at s=1.

Let 0>3 be a prime 3mod4 and letL=Q(0) be the imaginary quadratic field. A sensitive supersingular prime for an elliptic curve E is defined to be a good supersingular prime q1 for E where q11

Explicit form of the Waldspurger formula

In this section, we will establish the explicit Waldspurger formula for the quadratic twists of A in Theorem 1.1. We begin with a brief review of the test vector theory in [10] to get an appropriate test vector for the formula.

If W is any abelian group, we will denote Wˆ by the tensor product over Z of W with Zˆ=<Z, and W by WZ. Let B be a definite quaternion algebra over Q, and let U be an open subgroup of Bˆ×. Let X denote the finite set B×\Bˆ×/U, and write g1,,gn for a set of

Induction method via Euler system of Gross points

In this subsection, we will use an induction argument of the Euler system of Gross points to compute the 2-adic valuation of the product L(alg)(A(M),1)L(alg)(A(pM),1). We keep the notation of the previous section, but we take K=Q(p). Recall that B is the quaternion algebra over Q ramified exactly at ∞ and 2, that R is the global order as given in (4.4), and that U=Rˆ×. We fix an embedding of K into B such that KR=OK, which induces a homomorphism of Kˆ× into Bˆ×.

Definition 5.1

For a positive integer m

Proof of the main theorem

The 2-adic valuations (2.3) of the L-values for the quadratic twists A(M) have been computed in the previous section. To complete the proof of Theorem 1.1, we should compute the Tamagawa factors for A(M). In this section, we assume that M is an arbitrary square free non-zero odd integer with (M,7)=1. Let cp(A(M)) denote the Tamagawa factor for A(M) at a finite prime p, and let c(A(M)) be the number of connected components of A(M). The A(M) has bad additive reduction at all primes dividing M.

References (18)

  • M. Bertolini et al.

    Heegner points on Mumford–Tate curves

    Invent. Math.

    (1996)
  • D. Bump et al.

    Non-vanishing theorems for L-functions for modular forms and their derivatives

    Invent. Math.

    (1990)
  • L. Cai et al.

    Heegner points on modular curves

    Trans. Am. Math. Soc.

    (2018)
  • L. Cai et al.

    Special automorphisms on Shimura curves and non-triviality of Heegner points

    Sci. China Math.

    (2016)
  • L. Cai et al.

    On the 2-part of the Birch and Swinnerton-Dyer conjecture for quadratic twists of elliptic curves

    J. Lond. Math. Soc.

    (2020)
  • L. Cai et al.

    Explicit Gross–Zagier formula and Waldspurger formulae

    Algebra Number Theory

    (2014)
  • J. Coates et al.

    Quadratic twists of elliptic curves

    Proc. Lond. Math. Soc. (3)

    (2015)
  • J. Coates

    Lectures on the Birch-Swinnerton-Dyer conjecture

  • T. Dokchitser et al.

    On the Birch-Swinnerton-Dyer quotients modulo squares

    Ann. Math.

    (2010)
There are more references available in the full text version of this article.

Cited by (0)

View full text