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Let X / C be a K3 surface with complex multiplication by the ring of integers of a CM field E. We show that X can always be defined over an Abelian extension K / E explicitly determined by the discriminant form of the lattice NS (X). We then construct a model of X over K via Galois-descent and we study some of its basic properties, in particular we determine its Galois representation explicitly. Finally, we apply our results to give upper and lower bounds for a minimal field of definition for X in terms of the class number of E and the discriminant of NS (X). [ABSTRACT FROM AUTHOR]

We present a novel randomized algorithm to factor polynomials over a finite field F q of odd characteristic using rank 2 Drinfeld modules with complex multiplication. The main idea is to compute a lift of the Hasse invariant (modulo the polynomial f ∈ F q [ x ] to be factored) with respect to a random Drinfeld module ϕ with complex multiplication. Factors of f supported on prime ideals with supersingular reduction at ϕ have vanishing Hasse invariant and can be separated from the rest. Incorporating a Drinfeld module analogue of Deligne's congruence, we devise an algorithm to compute the Hasse invariant lift, which turns out to be the crux of our algorithm. The resulting expected runtime of n 3 / 2 + ε (log ⁡ q) 1 + o (1) + n 1 + ε (log ⁡ q) 2 + o (1) to factor polynomials of degree n over F q matches the fastest previously known algorithm, the Kedlaya-Umans implementation of the Kaltofen-Shoup algorithm. [ABSTRACT FROM AUTHOR]

Let K be an imaginary quadratic field of discriminant d K , and let n be a nontrivial integral ideal of K in which N is the smallest positive integer. Let Q N (d K) be the set of primitive positive definite binary quadratic forms of discriminant d K whose leading coefficients are relatively prime to N. We adopt an equivalence relation ∼ n on Q N (d K) so that the set of equivalence classes Q N (d K) / ∼ n can be regarded as a group isomorphic to the ray class group of K modulo n. We further establish an explicit isomorphism of Q N (d K) / ∼ n onto Gal (K n / K) in terms of Fricke invariants, where K n denotes the ray class field of K modulo n. This would be a certain extension of the classical composition theory of binary quadratic forms, originated and developed by Gauss and Dirichlet. [ABSTRACT FROM AUTHOR]

An isogeny graph is a graph whose vertices are principally polarizable abelian varieties and whose edges are isogenies between these varieties. In his thesis, Kohel describes the structure of isogeny graphs for elliptic curves and shows that one may compute the endomorphism ring of an elliptic curve defined over a finite field by using a depth-first search (DFS) algorithm in the graph. In dimension 2, the structure of isogeny graphs is less understood and existing algorithms for computing endomorphism rings are very expensive. In this article, we show that, under certain conditions, the problem of determining the endomorphism ring can also be solved in genus 2 with a DFS-based algorithm. We consider the case of genus-2 Jacobians with complex multiplication, with the assumptions that the real multiplication subring has class number one and is locally maximal at ℓ , for ℓ a fixed prime. We describe the isogeny graphs in that case, by considering cyclic isogenies of degree ℓ , under the assumption that there is an ideal l of norm ℓ in K 0 which is generated by a totally positive algebraic integer. The resulting algorithm is implemented over finite fields, and examples are provided. To the best of our knowledge, this is the first DFS-based algorithm in genus 2. [ABSTRACT FROM AUTHOR]

We study congruences involving truncated hypergeometric series of the form F 2 3 ( 1 / 2 , 1 / 2 , 1 / 2 1 , 1 ; λ ) ( m p s − 1 ) / 2 : = ∑ k = 0 ( m p s − 1 ) / 2 ( ( 1 / 2 ) k / k ! ) 3 λ k where p is a prime and m , s are positive integers. These truncated hypergeometric series are related to the arithmetic of a family of K3 surfaces. For special values of λ , with s = 1 , our congruences are stronger than those predicted by the theory of formal groups, because of the presence of elliptic curves with complex multiplications. They generalize a conjecture made by Stienstra and Beukers for the λ = 1 case and confirm some other supercongruence conjectures at special values of λ . [ABSTRACT FROM AUTHOR]

Abstract: We give a classification of all principally polarized abelian surfaces that admit an -isogeny to themselves, and show how to compute all the abelian surfaces that occur. We make the classification explicit in the simplest case . As part of our classification, we also show how to find all principally polarized abelian surfaces with multiplication by a given imaginary quadratic order. [Copyright &y& Elsevier]

Abstract: We give a new, somewhat elementary method for proving parity results about Iwasawa-theoretic Selmer groups and apply our method to certain Galois representations which are not self-dual. The main result is essentially that Iwasawa''s λ-invariants for these representations over dihedral -extensions are even. Our approach is a specialization argument and does not make use of Nekovář''s deformation-theoretic Cassels pairing, though Nekovář''s theory implies our results. Examples of the representations we consider arise naturally in the study of CM abelian varieties defined over the totally real subfield of the reflex field of the CM type. We also discuss connections with “large Selmer rank” in the sense of Mazur–Rubin and give several examples in the context of abelian varieties and modular forms. [Copyright &y& Elsevier]

Abstract: Using geometric tools introduced by P. Cohen, H. Shiga, J. Wolfart and G. Wüstholz, we show in Theorem 1 that when a certain Gauss hypergeometric function takes an algebraic value at an algebraic point, then another Gauss hypergeometric function takes a transcendental value at a related algebraic point. Using Appell hypergeometric functions, which generalize to two variables the Gauss functions, we study values at algebraic points of a new transcendental function defined in terms of these two functions. By Theorem 2, these values correspond to abelian varieties in the same isogeny class. Using a result of Edixhoven–Yafaev [B. Edixhoven, A. Yafaev, Subvarieties of Shimura varieties, Ann. of Math. 157 (2003) 621–645], this last result is in turn related to the distribution of the moduli of such abelian varieties in certain Shimura varieties. [Copyright &y& Elsevier]

In this paper, we deduce general multiplication formulas for hypercomplex monogenic and polymonogenic generalizations of Eisenstein series that are related to translation groups. In particular, a criterion for paravector multiplication of arbitrary finite-dimensional lattices in terms of being integral is developed. Under these number theoretical conditions it is then possible to transfer the concept of the complex multiplication of the ℘-function to the framework of its hypercomplex higher dimensional analogues within the Clifford analysis setting. We derive explicit formulas for the hypercomplex division values of the hypercomplex monogenic and polymonogenic Eisenstein series for lattices with hypercomplex multiplication.We also provide applications to the function theory. In particular, it will be shown that a non-constant polymonogenic function that satisfies one of the specific integer multiplication formulas of the Clifford-analytic Eisenstein series must have singularities. Furthermore, it coincides with one of those polymonogenic Eisenstein series whenever all the singularities are distributed in the form of a lattice and have all the same order and principal parts. [Copyright &y& Elsevier]

In this paper, we study the image of l-adic representations coming from Tate module of an abelian variety defined over a number field. We treat abelian varieties with complex and real multiplications. We verify the Mumford–Tate conjecture for a new class of abelian varieties with real multiplication. [Copyright &y& Elsevier]

We study K3 surfaces with complex multiplication following the classical work of Shimura on CM abelian varieties. After we translate the problem in terms of the arithmetic of the CM field and its idèles, we proceed to study some abelian extensions that arise naturally in this context. We then make use of our computations to determine the fields of moduli of K3 surfaces with CM and to classify their Brauer groups. More specifically, we provide an algorithm that given a number field K and a CM number field E , returns a finite list of groups which contains Br (X ‾) G K for any K3 surface X / K that has CM by the ring of integers of E. We run our algorithm when E is a quadratic imaginary field (a condition that translates into X having maximal Picard rank) generalizing similar computations already appearing in the literature. [ABSTRACT FROM AUTHOR]