On CM abelian varieties over imaginary quadratic fields
نویسندگان
چکیده
منابع مشابه
On Cm Abelian Varieties over Imaginary Quadratic Fields
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 divisibl...
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A. Weil proved that the geometric Frobenius π = Fa of an abelian variety over a finite field with q = pa elements has absolute value √ q for every embedding. T. Honda and J. Tate showed that A 7→ πA gives a bijection between the set of isogeny classes of simple abelian varieties over Fq and the set of conjugacy classes of q-Weil numbers. Higher-dimensional varieties over finite fields, Summer s...
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An analogy between abelian Anderson T-motives of rank r and dimension n , and abelian varieties over C with multiplication by an imaginary quadratic field K, of dimension r and of signature (n, r − n), permits us to get two elementary results in the theory of abelian varieties. Firstly, we can associate to this abelian variety a (roughly speaking) K-vector space of dimension r in C. Secondly, i...
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Let A be a modular abelian variety of GL2-type over a totally real field F of class number one. Under some mild assumptions, we show that the Mordell-Weil rank of A grows polynomially over Hilbert class fields of CM extensions of F .
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We first develop a criterion to determine normal bases (Theorem 2.4), and by making use of necessary lemmas which were refined from [3] we further prove that singular values of certain Siegel functions form normal bases of ray class fields over all imaginary quadratic fields other than Q( √−1) and Q( √−3) (Theorem 4.5 and Remark 4.6). This result would be an answer for the Lang-Schertz conjectu...
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ژورنال
عنوان ژورنال: Mathematische Annalen
سال: 2004
ISSN: 0025-5831,1432-1807
DOI: 10.1007/s00208-004-0511-8