Abstract
The paper uses Iwasawa theory at the prime $p=2$ to prove non-vanishing theorems for the value at $s=1$ of the complex $L$-series of certain quadratic twists of the Gross family of elliptic curves with complex multiplication by the field $K = \BQ(\sqrt{-q})$, where $q$ is any prime $\equiv 7 \mod 8$. Our results establish some broad generalizations of the non-vanishing theorem first proven by D. Rohrlich using complex analytic methods. Such non-vanishing theorems are important because it is known that they imply the finiteness of the Mordell-Weil group and the Tate-Shafarevich group of the corresponding elliptic curves over the Hilbert class field of $K$. It is essential for the proofs to study the Iwasawa theory of the higher dimensional abelian variety with complex multiplication which is obtained by taking the restriction of scalars to $K$ of the particular elliptic curve with complex multiplication introduced by Gross.
Highlights
Let K = Q(√−q) be an imaginary quadratic field, where q is any prime number with q ≡ 7 mod 8
Since q ≡ 7 mod 8, the prime 2 splits in K, say 2OK = pp∗, a fact which will underly all of our subsequent arguments with Iwasawa theory. √We fix one of these pr√imes p, and we assume on that we have chosen the sign of −q so that ordp((1 − −q)/2) > 0
Gross [21, Theorem 12.2.1] has proven that there exists a unique elliptic curve A defined over Q(j(OK )), with complex multiplication by OK, minimal discriminant (−q3), and which is a Q-curve in the sense that it is isogenous over H to all of its conjugates
Summary
We point out that, for any imaginary quadratic field K, equation (1.2) defines an elliptic curve with complex multiplication by the full ring of integers of this imaginary quadratic field, which is defined over an extension of degree at most 6 of the Hilbert class field of K, and which has good reduction outside the set of primes dividing 2 and 3. In further joint work in preparation with Kezuka and Tian [10], we hope to use Iwasawa theory to prove the exact Birch–Swinnerton-Dyer formula for the order of the Tate–Shafarevich group of all the elliptic curves with complex multiplication appearing in Theorems 1.3 and 1.5. We thank Zhibin Liang for making some numerical computations related to Theorem 8.2, and Jianing Li for informing us of his extremely ingenious elementary proof of Corollary 8.3
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