On Cm Abelian Varieties over Imaginary Quadratic Fields

نویسندگان

  • Tonghai Yang
  • TONGHAI YANG
چکیده

In this paper, we associate canonically to every imaginary quadratic field K = Q(√−D) one or two isogenous classes of CM (complex multiplication) abelian varieties over K, depending on whether D is odd or even (D 6= 4). These abelian varieties are characterized as of smallest dimension and smallest conductor, and such that the abelian varieties themselves descend to Q. When D is odd or divisible by 8, they are the scalar restriction of ‘canonical’ elliptic curves first studied by Gross and Rohrlich. We prove that these abelian varieties have the striking property that the vanishing order of their L-function at the center is dictated by the root number of the associated Hecke character. We also prove that the smallest dimension of a CM abelian variety over K is exactly the ideal class number of K and classify when a CM abelian variety over K has the smallest dimension. 0. Introduction. Let K = Q( √−D) be an imaginary quadratic field with fundamental discriminant −D and ideal class number h. According to Rohrlich ([Ro2]), a Hecke character χ of K of conductor f is called canonical if it satisfies the following three conditions: (0.1) χ(ā) = χ(a) for all ideals a of K prime to f; (0.2) χ(αOK) = ±α for every principal ideal αOK of K prime to f; (0.3) The conductor f is divisible only by primes ramified in K/Q. The canonical Hecke characters have the following amazing properties. (A) The associated L-function L(s, χ) behaves extremely well at the symmetric center. Indeed, one has ([MR] for the value and [MY] for the derivative) (0.4) ords=1L(s, χ) = 1−W (χ) 2 . 1991 Mathematics Subject Classification. 11G05 11M20 14H52.

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تاریخ انتشار 2003