Arithmetic on curves with complex multiplication by √ − 2

Abstract

This paper is a contribution to the verification of conjectures of Birch and Swinnerton-Dyer about elliptic curves (1). The evidence that they produce is largely derived from curves with complex multiplication by i. It is natural to consider other kinds of complex multiplications and here we shall make a start on the case when the ring of complex multiplications is isomorphic to the ring Z[σ], where σ2 = − 2.

The Diophantine equationy2= x(x2+ 21Dx+ 112D2)and the conjectures of Birch and Swinnerton-Dyer

“…Roughly speaking, these tell us that the group E Q of rational points on E is described to a great extent by the L -function of E. The evidence they produced in support of these conjectures was then largely derived from curves E, with End(Ei) = Z[V -1] (see Birch and Swinnerton-Dyer (1965)). Further evidence was obtained by Rajwade (1968Rajwade ( , 1969 for the curves E 2 with End(E 2 ) = Z [ V^2 ] and for E 3 with End(E 3 ) = Z [a>] (w = ( -1 + V^3)/2) and by Stephens (1968) for the curve X* + Y* = D. See also Damerell (1970Damerell ( , 1971. In all thse verifications, L E (1), the value of the L-function of E at 1, is calculated in finite form in terms of Weierstrass's p-function.…”

“…for the case m =2 (SeeRajwade (1968)). We have used the following results (analogous to the cases m =2,3):1.…”

“…We proceed as in paragraph 5 of Rajwade (1968). The definitions and results needed here are exactly the same as for the cases in =2,3.…”

Group order formulas for reductions of CM elliptic curves

“…For example, the next result, which appears in a paper of Rajwade [25], deals with all the elliptic curves with CM by Z[ √ −2]. Related work that uses the theory of cyclotomy includes papers by B. W. Brewer, A. L. Whiteman, and others.…”

“…In a series of papers beginning in the late 1960's and continuing into the 1980's, Rajwade and co-authors (see for example [25,26,27,28,29]) dealt with elliptic curves over Q with complex multiplication by the ring of integers in Q( √ −d) for some small values of d, including d = 1, 2, 3, 7, 11, 19, using cyclotomy and the theory of complex multiplication.…”