On the Tate-Shafarevich group of certain elliptic curves

Abstract

Let K ⊂ C be an imaginary quadratic field with prime discriminant −p < −3, ring of integers OK = O and class group ClK of (odd) order hK = h. The jinvariant j(O) generates a field F/Q of degree h such that H = FK is the Hilbert class field of K. Suppose A/F is a Q-curve with j-invariant j(O); thus, A is an elliptic curve which, over H, is isogenous to each of its Galois conjugates, and has complex multiplication by O. We let B = ResF/QA be the h-dimensional abelian variety over Q obtained from A by restriction of scalars. Any two such A (and any two such B) are quadratic twists of one another: letting A(p) denote the canonical curve of discriminant ideal (−p3), with restriction B(p), we have A = A(p) (and B = B(p)) for some quadratic discriminant D. We refer the reader to Gross [Gr1] for general facts about CM Q-curves. Much progress has been made recently on proving the conjecture of Birch and Swinnerton-Dyer for these curves. For instance, if L(1, A/F ) = 0, and h = 1, Rubin [Ru2] has proved that the Birch-Swinnerton–Dyer conjecture for A/F holds up to a power of 2. Note that the ring R+ of Q-endomorphisms of B (or B(p)) is an order in T = R+ ⊗ Q, a totally real field of degree h over Q. The Tate-Shafarevich group ∐∐ B/Q is a finite module over R+; our main goal in this paper is to gain insight into the structure of this module via L-series in some special cases. Namely, suppose p ≡ 3 mod 8 and A = A(p)−3. Also, assume that R+ is integrally closed. Let ψ be a Hecke character of K such that ψ ◦ NH/K is the Hecke character attached to A/H. This choice of ψ gives rise to an embedding of T in R (see section 2). We define an algebraic integer s = 0 in FT as a sum of certain modified elliptic units first introduced by Gross [Gr3] and show that there is a (unique) integral ideal f of R+ whose lift to FT is generated by s. Our starting point is a formula (Theorem 2) expressing L(1, ψ) as a period times s, showing, in particular, that this central critical value does not vanish. Writing L(s, A/F ) as a product of Hecke L-series, calculating the local factors in the Birch-Swinnerton–Dyer conjecture, and applying our formula together with results of Coates, Wiles, Arthaud, and Rubin [CW], [Ar], [Ru1], we obtain Main Theorem(Theorem 5) With the above assumptions and notation, A(F ) = B(Q) is finite. If the Birch-Swinnerton–Dyer conjecture holds for A/F (or for