Examples of surfaces with real multiplication
نویسندگان
چکیده
منابع مشابه
Real Multiplication Abelian Surfaces over Q
Recall: New eigen-cuspforms f of weight 2 on Γ0(N) with Q(f) totally real and [Q(f) : Q] = d ←→ dimension d factors of J0(N) ←→ abelian varieties A/Q of dimension d with real multiplication (=RM) by an order in K = Q(f) up to isogeny. d = 1: elliptic curves, which we understand quite well: moduli (the j-line, ∼= P), explicit formulas (Weierstrass etc.), isogenies, etc. d > 1: we understand thes...
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The endomorphism algebra of a K3 type Hodge structure is a totally real field or a CM field. In this paper we give a low brow introduction to the case of a totally real field. We give existence results for the Hodge structures, for their polarizations and for certain K3 surfaces. We consider the Kuga Satake variety of these Hodge structures and we discuss some examples. Finally we indicate vari...
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Let K be a field of characteristic different from 2, and let f(x) be a sextic polynomial irreducible over K with no repeated roots, whose Galois group is A5. If the Jacobian of the hyperelliptic curve y = f(x) admits real multiplication over the ground field from an order of a real quadratic number field, then either its endomorphism algebra is this quadratic field or the Jacobian is supersingu...
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Let h2 = Sp(4,R)/U(2) be the Siegel upperhalf space of rank 2. The quotient Sp(4,Z)\h2 has three remarkable properties: (a) it is the moduli space of principally polarized abelian surfaces, (b) it has the structure of a quasi-projective complex algebraic variety which is defined over the rational numbers Q, and (c) it has a natural compactification (the BailyBorel Satake compactification) which...
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An abelian surface A over a field K has potential quaternionic multiplication if the ring End K̄ (A) of geometric endomorphisms of A is an order in an indefinite rational division quaternion algebra. In this brief note, we study the possible structures of the ring of endomorphisms of these surfaces and we provide explicit examples of Jacobians of curves of genus two which show that our result is...
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ژورنال
عنوان ژورنال: LMS Journal of Computation and Mathematics
سال: 2014
ISSN: 1461-1570
DOI: 10.1112/s1461157014000199