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For a modular elliptic curve A = X 0 (4 9) and its quadratic twists A (M) , we give equivalent conditions such that the 2 -Selmer group S 2 (A (M) / ℚ) is minimal, namely, it is of order 2. One of these conditions is described by the L -value L (A (M) / ℚ , 1). The other conditions are described by quadratic and biquadratic residue symbol, so explicit and computable (and one can compute the density of M). Also we prove the full Birch–Swinnerton-Dyer conjecture when the equivalent conditions are satisfied. This generalizes a result by J. Coates, Y. Li, Y. Tian and S. Zhai. [ABSTRACT FROM AUTHOR]

We analyze complex multiplication for Jacobians of curves of genus 3, as well as the resulting Shimura class groups and their subgroups corresponding to Galois conjugation over the reflex field. We combine our results with numerical methods to find CM fields K for which there exist both hyperelliptic and non-hyperelliptic curves whose Jacobian has complex multiplication by \mathbb {Z}_K. More precisely, we find all sextic CM fields K in the LMFDB for which (heuristically) Jacobians of both types with CM by \mathbb {Z}_K exist. There turn out to be 14 such fields among the 547,156 sextic CM fields that the LMFDB contains. We determine invariants of the corresponding curves, and in the simplest case we also give an explicit defining equation. [ABSTRACT FROM AUTHOR]

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]

A complex algebraic curve is said to have many automorphisms if it cannot be deformed nontrivially together with its automorphism group. Oort asked whether the jacobian of any such curve has complex multiplication. We answer this question completely when the curve is hyperelliptic. For this, we first classify all hyperelliptic curves with many automorphisms, finding 3 infinite sequences and 15 separate isomorphism classes. Of these, all members of the infinite sequences and 5 of the isolated curves have a jacobian with complex multiplication, but the remaining 10 curves do not. The methods used include a criterion of Wolfart, a representation theoretic criterion of streit and the non-integrality of the j-invariants of certain quotient curves of genus 1. For the hardest cases, we use an ad hoc developed computational criterion depending on the splitting fields of the characteristic polynomials of Frobenius acting on the Tate module of the Jacobian. [ABSTRACT FROM AUTHOR]

Let C be a smooth, projective, genus g ≥ 2 curve, defined over C. Then C has many automorphisms if its corresponding moduli point p ∈ Mg has a neighborhood U in the complex topology, such that all curves corresponding to points in U \ p have strictly fewer automorphisms than C. We compute completely the list of superelliptic curves C for which the superelliptic automorphism is normal in the automorphism group Aut (C) and C has many automorphisms. For each of these curves, we determine whether its Jacobian has complex multiplication. As a consequence, we prove the converse of Streit's complex multiplication criterion for these curves. [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 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 E / F be an elliptic curve defined over a number field F. Suppose that E has complex multiplication over F ¯ , i.e. End F ¯ (E) ⊗ ℚ is an imaginary quadratic field. With the aid of CM theory, we find elliptic curves whose quadratic twists have a constant root number. [ABSTRACT FROM AUTHOR]

We show that all triples (x 1 , x 2 , x 3) of singular moduli satisfying x 1 x 2 x 3 ∈ ℚ × are "trivial". That is, either x 1 , x 2 , x 3 ∈ ℚ ; some x i ∈ ℚ and the remaining x j , x k are distinct, of degree 2 , and conjugate over ℚ ; or x 1 , x 2 , x 3 are pairwise distinct, of degree 3 , and conjugate over ℚ. This theorem is best possible and is the natural three-dimensional analogue of a result of Bilu, Luca and Pizarro-Madariaga in two dimensions. It establishes an explicit version of the André–Oort conjecture for the family of subvarieties V α ⊂ ℂ 3 defined by an equation x 1 x 2 x 3 = α ∈ ℚ. [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]

Let Ω be a non-zero holomorphic 2-form on a K3 surface S. Suppose that S is projective algebraic and is defined over . Let be the -vector space generated by the numbers given by all the periods ∫γ Ω, γ ∈ H2(S, ℤ). We show that, if , then S has complex multiplication, meaning that the Mumford-Tate group of the rational Hodge structure on H2(S, ℚ) is abelian. This result was announced in [P. Tretkoff, Transcendence and CM on Borcea-Voisin towers of Calabi-Yau manifolds, J. Number Theory 152 (2015) 118-155], without a detailed proof. The converse is already well known. [ABSTRACT FROM AUTHOR]

This article clarifies the process behind the operation known as complex multiplication and demonstrates that it is less abstract and more closely linked to conventional serial procedures than was previously thought. The operation results from one of many transpositional schemes that Boulez regularly incorporated in his precompositional tables. An understanding of this scheme and the way it derives from the partial ordering in register of the multiplication factors reveals isomorphic relationships that are crucial to the structural properties of the music. Through an analysis of Chapitre I of Boulez's Structures II, this article brings to light the nature of the harmonic relationships that shape the musical language, shows how these relate to the precompositional tables, and describes how they are essential to his sophisticated concept of form. [ABSTRACT FROM AUTHOR]

Let (α, β, γ) be three non-zero coprime integers. Using Baker's method and classical technique of reduction, we describe an algorithm which computes every point (x1, x2) on the ℂ-line αX + βY + γ = 0 such that x1 and x2 are both singular moduli. We illustrate our algorithm by writing down some examples, in particular we prove that there are non-complex multiplication points on a line such that |α|, |β| and |γ| are less than 10. [ABSTRACT FROM AUTHOR]

This article clarifies the process behind the operation known as complex multiplication and demonstrates that it is less abstract and more closely linked to conventional serial procedures than was previously thought. The operation results from one of many transpositional schemes that Boulez regularly incorporated in his precompositional tables. An understanding of this scheme and the way it derives from the partial ordering in register of the multiplication factors reveals isomorphic relationships that are crucial to the structural properties of the music. Through an analysis of Chapitre I of Boulez's Structures II, this article brings to light the nature of the harmonic relationships that shape the musical language, shows how these relate to the precompositional tables, and describes how they are essential to his sophisticated concept of form. [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 this paper, we give a similarity classification for the multiplication operator M on the Sobolev disk algebra $$R(\mathbb{D})$$ with g analytic on the closure of the unit disk $$\mathbb{D}$$. [ABSTRACT FROM AUTHOR]

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]

Fix m greater than one and let E be an elliptic curve over Q with complex multiplication. We formulate conjectures on the density of primes p (congruent to one modulo m) for which the pth Fourier coefficient of E is an mth power modulo p; often these densities differ from the naive expectation of 1/m. We also prove our conjectures for m dividing the number of roots of unity lying in the CM field of E; the most involved case is m = 4 and complex multiplication by Q(i). [ABSTRACT FROM AUTHOR]

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]

Neither the full context surrounding Pierre Boulez's multiplication operation nor its repercussions in terms of the structure of the music have been well understood. Boulez's multiplication operations are tied to specific transpositional schemes. These transpositional schemes are contextual and depend on the intervallic configuration of the factors of the multiplication. Thus, they create important isomorphic relationships that cut across different levels of musical structure. This article presents examples from two pieces by Boulez: Structures II and Troisieme Sonate. The examples demonstrate that, because of the trajectories that Boulez chooses to move through his precompositional tables and the specific ways that the multiplication products are realized on the musical surface, the isomorphic relationships that are essential to the tables are also key determinants of structure in this music. [ABSTRACT FROM AUTHOR]

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]