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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 P and Q be two points on an elliptic curve defined over a number field K. For α ∈ End (E) , define B α to be the O K -integral ideal generated by the denominator of x (α (P) + Q) . Let O be a subring of End (E) , that is a Dedekind domain. We will study the sequence { B α } α ∈ O . We will show that, for all but finitely many α ∈ O , the ideal B α has a primitive divisor when P is a non-torsion point and there exist two endomorphisms g ≠ 0 and f so that f (P) = g (Q) . This is a generalization of previous results on elliptic divisibility sequences. [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 abelian varieties and K3 surfaces with complex multiplication defined over number fields of fixed degree. We show that these varieties fall into finitely many isomorphism classes over an algebraic closure of the field of rational numbers. As an application we confirm finiteness conjectures of Shafarevich and Coleman in the CM case. In addition we prove the uniform boundedness of the Galois invariant subgroup of the geometric Brauer group for forms of a smooth projective variety satisfying the integral Mumford–Tate conjecture. When applied to K3 surfaces, this affirms a conjecture of Várilly-Alvarado in the CM case. [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]

The article discusses the indecomposability of p-adic Galois representations emerging from the Tate module of an abelian variety with real multiplication defined over a number field. The indecomposability for the nearly ordinary Galois representation of each weight of Hilbert cusp form is mentioned. The importance of the p-local indecomposability of the p-adic Tate module as a characterization of non-CM/CM abelian varieties is also addressed.

In 1987, Gordon gave an integer primality condition similar to the familiar test based on Fermat’s little theorem, but based instead on the arithmetic of elliptic curves with complex multiplication. We prove the existence of infinitely many composite numbers simultaneously passing all elliptic curve primality tests assuming a weak form of a standard conjecture on the bound on the least prime in (special) arithmetic progressions. Our results are somewhat more general than both the 1999 dissertation of the first author (written under the direction of the third author) and a 2010 paper on Carmichael numbers in a residue class written by Banks and the second author. [ABSTRACT FROM PUBLISHER]

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]