General SectionQuadratic twists of X0(14)
Introduction
Let E be an elliptic curve defined over , and let be the complex L-series of E. For each square free non-zero integer , we write for the twist of E by the quadratic extension , and for its complex L-series. It has been proved in [2] that there are infinitely many d such that is non-vanishing at , and infinitely many d such that has a simple zero at . Thanks to the results of Gross–Zagier and Kolyvagin, we know that for those d such that has a zero at of order 0 or 1, the rank of is equal to the order of this zero at and the Tate–Shafarevich group of is finite. It is therefore natural to ask whether has a zero at of order 0 or 1 as d varies.
For the remainder of this section, we assume that . Let denote the conductor of E, and let denote the fundamental real period of the Néron differential on E. We define which is a non-zero rational number. Let denote the number of connected components of , and for each prime q dividing , let , where is the subgroup of points in with non-singular reduction modulo q. The conjectural Birch and Swinnerton-Dyer formula is then given by 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 . 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 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 curve We know that , the discriminant of A is , the j-invariant of A is and its Néron differential has the fundamental real period . Moreover, we have It is also known that the Tate–Shafarevich group of A is trivial and that , and . 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 . 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 be an integer. Let be a square free integer, where each is a prime which is inert both in and in , and the sign is chosen so that . Then , is finite, the Tate–Shafarevich group of is finite of odd cardinality, and the 2-part of Birch and Swinnerton-Dyer formula is valid for .
We remark that, using modular symbols, Cai–Li–Zhai [5] also proved this theorem in the special case when , where each is a prime which is inert in . 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 equations A new feature of this paper is that our elliptic curve does not have complex multiplication, and that there is no -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 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 . By changing of variables, A is given by the equation Let M be a square free non-zero integer ≠1 with and . Let be the twist of M by the quadratic extension . The equation for 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 has root number −1. Among those quadratic twists , we will find the family of the twists which have a simple zero at .
Let be a prime and let be the imaginary quadratic field. A sensitive supersingular prime for an elliptic curve E is defined to be a good supersingular prime for E where
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 by the tensor product over of W with , and by . Let B be a definite quaternion algebra over , and let U be an open subgroup of . Let X denote the finite set , and write 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 . We keep the notation of the previous section, but we take . Recall that B is the quaternion algebra over ramified exactly at ∞ and 2, that is the global order as given in (4.4), and that . We fix an embedding of K into B such that , which induces a homomorphism of into .
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 have been computed in the previous section. To complete the proof of Theorem 1.1, we should compute the Tamagawa factors for . In this section, we assume that M is an arbitrary square free non-zero odd integer with . Let denote the Tamagawa factor for at a finite prime p, and let be the number of connected components of . The has bad additive reduction at all primes dividing M.
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