Computing Class Polynomials for Abelian Surfaces
نویسندگان
چکیده
منابع مشابه
Computing Class Polynomials for Abelian Surfaces
We describe a quasi-linear algorithm for computing Igusa class polynomials of Jacobians of genus 2 curves via complex floating-point approximations of their roots. After providing an explicit treatment of the computations in quartic CM fields and their Galois closures, we pursue an approach due to Dupont for evaluating θconstants in quasi-linear time using Newton iterations on the Borchardt mea...
متن کاملComputing Hilbert Class Polynomials
We present and analyze two algorithms for computing the Hilbert class polynomial HD. The first is a p-adic lifting algorithm for inert primes p in the order of discriminant D < 0. The second is an improved Chinese remainder algorithm which uses the class group action on CM-curves over finite fields. Our run time analysis gives tighter bounds for the complexity of all known algorithms for comput...
متن کاملComputing Igusa class polynomials
We give an algorithm that computes the genus two class polynomials of a primitive quartic CM field K, and we give a runtime bound and a proof of correctness of this algorithm. This is the first proof of correctness and the first runtime bound of any algorithm that computes these polynomials. Our algorithm uses complex analysis and runs in time e O(∆), where ∆ is the discriminant of K.
متن کاملNon-Abelian Class Field Theory for Riemann Surfaces
Let T be a Tannakian category with a fiber functor ω : T → VerC, where VerC denotes the category of finite dimensional C-vector spaces. An object t ∈ T is called reducible if there exist non-zero objects x, y ∈ T such that t = x ⊕ y. An object is called irreducible if it is not reducible. If moreover every object x of T can be written uniquely as a sum of irreducible objects x = x1 ⊕ x2 ⊕ . . ....
متن کاملComputing Hilbert class polynomials with the Chinese remainder theorem
We present a space-efficient algorithm to compute the Hilbert class polynomial HD(X) modulo a positive integer P , based on an explicit form of the Chinese Remainder Theorem. Under the Generalized Riemann Hypothesis, the algorithm uses O(|D|1/2+ log P ) space and has an expected running time of O(|D|1+ ). We describe practical optimizations that allow us to handle larger discriminants than othe...
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ژورنال
عنوان ژورنال: Experimental Mathematics
سال: 2014
ISSN: 1058-6458,1944-950X
DOI: 10.1080/10586458.2013.878675