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Gaussian periods have been studied for centuries in the realms of number theory, field theory, cryptography, and elsewhere. However, it was only within the last decade or so that they began to be studied from a visual perspective. By plotting Gaussian periods in the complex plane, various interesting and insightful patterns emerge, leading to various conjectures and theorems about their properties.In this paper, we offer a description of Gaussian periods, along with examples of the structure that can occur when plotting them in the complex plane. In addition to this, we offer two ways in which this study can be generalized to other situations — one relating to supercharacter theory, the other relating to class field theory — along with discussions and visual examples of each. [ABSTRACT FROM AUTHOR]

We provide a lower bound for the number of components of the algebraic monodromy group in the situation of a K 3 surface over a number field k. In the CM case, our bound is sharp. As an application, we describe, in the case of CM, the jump character [3, Definition 2.4.6] entirely in terms of the endomorphism field and the geometric Picard rank. [ABSTRACT FROM AUTHOR]

We show that for every finite set of prime numbers $S$ , there are at most finitely many singular moduli that are $S$ -units. The key new ingredient is that for every prime number $p$ , singular moduli are $p$ -adically disperse. We prove analogous results for the Weber modular functions, the $\lambda$ -invariants and the McKay–Thompson series associated with the elements of the monster group. Finally, we also obtain that a modular function that specializes to infinitely many algebraic units at quadratic imaginary numbers must be a weak modular unit. [ABSTRACT FROM AUTHOR]

Let K be an imaginary quadratic field, and fix an odd prime p that does not divide the class number of K , and is excluded from a finite set of primes depending on K. In this paper we prove that Iwasawa's λ -invariant for the cyclotomic Z p -extension of K is greater than 1 if and only if the number of points on a certain reduced elliptic curve is divisible by p 2. [ABSTRACT FROM AUTHOR]

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]

In this paper, we present an alternative method to compute all imaginary octic non-CM fields with class number one, listed in Yamamura’s paper (Acta Arith 86:133–47, 1998). We consider such octic fields as an abelian extension over an appropriate quadratic subfield, whereas Yamaura’s method uses a known structure for quartic fields. In fact, we demonstrate that all such imaginary non-CM octic fields with class number one are subfields of the ray class fields of some imaginary quadratic fields with an appropriate conductor. Using the Pari software, as described in Sect. 3, we determine all imaginary octic non-CM fields with class number one. (The codes are included as an appendix.) [ABSTRACT FROM AUTHOR]

We define Jacobi forms with complex multiplication. Analogous to modular forms with complex multiplication, they are constructed from Hecke characters of the associated imaginary quadratic field. From this construction we obtain a Jacobi form which specializes to η (τ) 26 which we present to highlight an open question of Dyson and Serre. We give other examples and applications of Jacobi forms with complex multiplication including constructing theta blocks associated to elliptic curves with complex multiplication and new families of congruences and cranks for certain partition functions. [ABSTRACT FROM AUTHOR]

Given any cube-free integer λ > 0 , we study the 3 -adic valuation of the algebraic part of the central L -value of the elliptic curve X 3 + Y 3 = λ Z 3. We give a lower bound in terms of the number of distinct prime factors of λ , which, in the case 3 divides λ , also depends on the power of 3 in λ. This extends an earlier result of the author in which it was assumed that 3 is coprime to λ. We also study the 3 -part of the Tate–Shafarevich group for these curves and show that the lower bound is as expected from the conjecture of Birch and Swinnerton-Dyer, taking into account also the growth of the Tate–Shafarevich group. [ABSTRACT FROM AUTHOR]

Two abelian varieties A_1, A_2 over a number field K are strongly iso-Kummerian if the torsion fields K(A_1[d]) and K(A_2[d]) coincide for all d \geq 1. For all g \geq 4 we construct geometrically simple, strongly iso-Kummerian g-dimensional abelian varieties over number fields that are not geometrically isogenous. We also discuss related examples and put significant constraints on any further iso-Kummerian pair. [ABSTRACT FROM AUTHOR]

Based upon a new general vector-valued vector product, generalized complex numbers with respect to certain positively homogeneous functionals including norms and antinorms are introduced and a vector-valued Euler type formula for them is derived using a vector valued exponential function. Furthermore, generalized Cauchy–Riemann differential equations for generalized complex differentiable functions are derived. For random versions of the considered new type of generalized complex numbers, moments are introduced and uniform distributions on discs with respect to functionals of the considered type are analyzed. Moreover, generalized uniform distributions on corresponding circles are studied and a connection with generalized circle numbers, which are natural relatives of π , is established. Finally, random generalized complex numbers are considered which are star-shaped distributed. [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]

The Hilbert class polynomial has as roots the j-invariants of elliptic curves whose endomorphism ring is a given imaginary quadratic order. It can be used to compute elliptic curves over finite fields with a prescribed number of points. Since its coefficients are typically rather large, there has been continued interest in finding alternative modular functions whose corresponding class polynomials are smaller. Best known are Weber's functions, which reduce the size by a factor of 72 for a positive density subset of imaginary quadratic discriminants. On the other hand, Bröker and Stevenhagen showed that no modular function will ever do better than a factor of 100.83. We introduce a generalization of class polynomials, with reduction factors that are not limited by the Bröker–Stevenhagen bound. We provide examples matching Weber's reduction factor. For an infinite family of discriminants, their reduction factors surpass those of all previously known modular functions by a factor at least 2. [ABSTRACT FROM AUTHOR]

Grothendieck attache à une variété abélienne sur un corps de nombres des groupes de monodromie finie en chaque place de mauvaise réduction. On montre l'existence de variétés abéliennes avec monodromie finie de p -partie maximale pour p impair ainsi qu'une nouvelle borne sur la 2-partie de ces groupes dans le cas d'une variété abélienne avec multiplication complexe (CM). On montre que cette borne est atteinte. La construction se fait par torsion de variétés abéliennes CM ce qui fait intervenir la résolution de problèmes de Grunwald. In this paper we study the wild part of the finite monodromy groups of abelian varieties over number fields. We solve Grunwald problems for groups of the form Z / p Z ≀ S n over number fields to build CM abelian varieties with maximal wild finite monodromy in the odd prime case. For the even prime case we prove a new bound on the 2-part of the order of the finite monodromy group for CM abelian varieties and build varieties that reach it. [ABSTRACT FROM AUTHOR]

A family F of elliptic curves defined over number fields is said to be typically bounded in torsion if the torsion subgroups E (F) [ tors ] of those elliptic curves E / F ∈ F can be made uniformly bounded after removing from F those whose number field degrees lie in a subset of Z + with arbitrarily small upper density. For every number field F , we prove unconditionally that the family E F of elliptic curves defined over number fields and with F -rational j -invariant is typically bounded in torsion. For any integer d ∈ Z + , we also strengthen a result on typically bounding torsion for the family E d of elliptic curves defined over number fields and with degree d j -invariant. [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]

CM-type projective varieties X of complex dimension n are characterized by their CM-type rational Hodge structures on the cohomology groups. One may impose such a condition in a weakest form when the canonical bundle of X is trivial; the rational Hodge structure on the level- n subspace of H n (X ; Q) is required to be of CM-type. This brief note addresses the question whether this weak condition implies that the Hodge structure on the entire H ⁎ (X ; Q) is of CM-type. We study in particular abelian varieties when the dimension of the level- n subspace is two or four, and K3 × T 2. It turns out that the answer is affirmative. Moreover, such an abelian variety is always isogenous to a product of CM-type elliptic curves or abelian surfaces. This extends a result of Shioda and Mitani in 1974. [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]

For each algebraic number α ∈ ℚ ̄ , a result of Habegger [P. Habegger, Singular moduli that are algebraic units, Algebra Number Theory 9(7) (2015) 1515–1524] shows that there are only finitely many singular moduli j such that j − α is an algebraic unit. His result uses Duke's Equidistribution Theorem and is thus not effective. In this paper, we give an effective proof of Habegger's result assuming that α is not a singular modulus itself. We give an explicit bound, which depends only on α , on the discriminant Δ associated with a singular modulus j such that j − α is a unit. This implies explicit bounds on the number of these singular moduli. [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]

In this paper, we show that an action on the set of elliptic curves with j = 1728 preserves a certain kind of symmetry on the local root number of Hecke characters attached to such elliptic curves. As a consequence, we give results on the distribution of the root numbers and their average of the aforementioned Hecke characters. [ABSTRACT FROM AUTHOR]

A power series is called lacunary if "almost all" of its coefficients are zero. Integer partitions have motivated the classification of lacunary specializations of Han's extension of the Nekrasov–Okounkov formula. More precisely, we consider the modular forms F a , b , c (z) : = η (24 a z) a η (24 a c z) b - a η (24 z) , defined in terms of the Dedekind η -function, for integers a , c ≥ 1 , where b ≥ 1 is odd throughout. Serre (Publications Mathématiques de l'IHÉS 123–201:2959–2968, 1981) determined the lacunarity of the series when a = c = 1 . Later, Clader et al. (Am Math Soc 137(9):2959–2968, 2009) extended this result by allowing a to be general and completely classified the F a , b , 1 (z) which are lacunary. Here, we consider all c and show that for a ∈ { 1 , 2 , 3 } , there are infinite families of lacunary series. However, for a ≥ 4 , we show that there are finitely many triples (a, b, c) such that F a , b , c (z) is lacunary. In particular, if a ≥ 4 , b ≥ 7 , and c ≥ 2 , then F a , b , c (z) is not lacunary. Underlying this result is the proof the t-core partition conjecture proved by Granville and Ono (Trans Am Math Soc 348(1):331–347, 1996). [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]

In [M. Asakura, N. Otsubo and T. Terasoma, An algebro-geometric study of special values of hypergeometric functions 3 F 2 , to appear in Nagoya Math. J.; https://doi.org/10.1017/nmj.2018.36 ], we proved that the value of 3 F 2 a , b , q a + b , q ; 1 of the generalized hypergeometric function is a ℚ ¯ -linear combination of log of algebraic numbers if rational numbers a , b , q satisfy a certain condition. In this paper, we present a method to obtain an explicit description of it. [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 generalize the definition of CM cycles beyond the 'small' and 'big' CM ones studied by various authors, such as in [BY09] and [BKY12] and give a uniform formula for the CM values of Green functions associated to these special cycles in general using the idea of regularized theta lifts. Finally, as an application to Siegel 3-fold X 2 (2) , we can compute special values of theta functions and Rosenhain λ -invariants at a CM cycle, which is useful for genus two curve cryptography as in [CDSLY14]. [ABSTRACT FROM AUTHOR]

In this paper we construct the Shimura canonical model for the quaternion algebra B of discriminant 6 over Q in the sense of the original paper [SmrB] (see also [Shg2]). The "unit group" of B becomes to be a quadrangle group Γ = □ (0 ; 2 , 2 , 3 , 3) acting on the complex upper half space H . For our purpose we use the defining equation of the Shimura curve V = H / □ (0 ; 2 , 2 , 3 , 3) given by Y. Ihara which found in the work of A. Kurihara (1979). We are requested to find a modular function ψ defined on H with respect to Γ which realizes V satisfying several arithmetic conditions stated later. Basically our ψ is the modular function for the family of abelian surfaces with quaternion multiplication by B , so called false elliptic curves, with some arithmetic conditions. Our main tool is the hypergeometric modular function given by M. Petkova and H. Shiga (2011). This is the full paper of the survey report [Shg2] (2018). [ABSTRACT FROM AUTHOR]

A CM-order is a reduced order equipped with an involution that mimics complex conjugation. The Witt--Picard group of such an order is a certain group of ideal classes that is closely related to the "minus part" of the class group. We present a deterministic polynomial-time algorithm for the following problem, which may be viewed as a special case of the principal ideal testing problem: given a CM-order, decide whether two given elements of its Witt--Picard group are equal. In order to prevent coefficient blow-up, the algorithm operates with lattices rather than with ideals. An important ingredient is a technique introduced by Gentry and Szydlo in a cryptographic context. Our application of it to lattices over CM-orders hinges upon a novel existence theorem for auxiliary ideals, which we deduce from a result of Konyagin and Pomerance in elementary number theory. [ABSTRACT FROM AUTHOR]

Abstract Let L be an extended ring class field of an imaginary quadratic field K other than Q (− 1) and Q (− 3). We show that there is a form class group induced from a congruence subgroup which describes the Galois group of L over K in a concrete way. We also construct a primitive generator of L over K as a real algebraic integer which can be applied to certain quadratic Diophantine equations. [ABSTRACT FROM AUTHOR]

We treat two different equations involving powers of singular moduli. On the one hand, we show that, with two possible (explicitly specified) exceptions, two distinct singular moduli j (τ) , j (τ ′) such that the numbers 1 , j (τ) m and j (τ ′) n are linearly dependent over ℚ for some positive integers m , n , must be of degree at most 2. This partially generalizes a result of Allombert, Bilu and Pizarro-Madariaga, who studied CM-points belonging to straight lines in ℂ 2 defined over ℚ. On the other hand, we show that, with obvious exceptions, the product of any two powers of singular moduli cannot be a non-zero rational number. This generalizes a result of Bilu, Luca and Pizarro-Madariaga, who studied CM-points belonging to a hyperbola x y = A , where A ∈ ℚ. [ABSTRACT FROM AUTHOR]

Abstract Oort has conjectured that there do not exist Shimura varieties of dimension >0 contained generically in the Torelli locus of genus- g curves when g is sufficiently large. In this paper we prove the Oort conjecture for Shimura curves of Mumford type and Shimura curves parameterizing principally polarized g -dimensional abelian varieties isogenous to g -fold self-products of elliptic curves for g > 11. As a consequence, we obtain a finiteness result regarding smooth genus- g curves with completely decomposable Jacobians, which is related to a question of Ekedahl and Serre. [ABSTRACT FROM AUTHOR]

We show that the field Q ( x , y ) , generated by two singular moduli x and y , is generated by their sum x + y , unless x and y are conjugate over Q , in which case x + y generates a subfield of degree at most 2. We obtain a similar result for the product of two singular moduli. [ABSTRACT FROM AUTHOR]

Many cryptosystems are based on the difficulty of the discrete logarithm problem in finite groups. In this case elliptic and hyperelliptic cryptosystems are more noticed because they provide good security with smaller size keys. Since these systems were used for cryptography, it has been an important issue to transform a random value in finite field into a random point on an elliptic or hyperelliptic curve in a deterministic and efficient method. In this paper we propose a deterministic encoding to hyperelliptic curves over finite field. For cryptographic desires it is important to have an injective encoding. In finite fields with characteristic three we obtain an injective encoding for genus two hyperelliptic curves. [ABSTRACT FROM AUTHOR]

For an elliptic curve E over a finite field Fq, where q is a prime power, we propose new algorithms for testing the supersingularity of E. Our algorithms are based on the polynomial identity testing problem for the p-th division polynomial of E. In particular, an efficient algorithm using points of high order on E is given. [ABSTRACT FROM AUTHOR]

In this paper, we first prove that given pairwise distinct algebraic numbers α1, . . . , αn, the numbers α1 + t, . . . , αn + t are multiplicatively independent for all sufficiently large integers t. Then, for a pair (a, b) of distinct integers, we study how many pairs (a + t, b + t) are multiplicatively dependent when t runs through the set integers Z. Assuming the ABC conjecture we show that there exists a constant C1 such that for any pair (a, b) ∈ Z2, a ≠ b, there are at most C1 values of t ∈ Z such that (a + t, b + t) are multiplicatively dependent. For a pair (a, b) ∈ Z² with difference b - a = 30 we show that there are 13 values of t ∈ Z for which the pair (a+t, b+ t) is multiplicatively dependent. We further conjecture that 13 is the largest number of such translations for any such pair (a, b) and prove this for all pairs (a, b) with difference at most 1010. [ABSTRACT FROM AUTHOR]

Abstract: In 1987, Dan Gordon defined an elliptic curve analogue to Carmichael numbers known as elliptic Carmichael numbers. In this paper, we prove that there are infinitely many elliptic Carmichael numbers. In doing so, we resolve in the affirmative the question of whether there exist infinitely many Lucas–Carmichael numbers (that is, squarefree, composite integers n such that for every prime p that divides n, p + 1 | n + 1) . [ABSTRACT FROM AUTHOR]

In this paper we present two classes of scalar multiplication hardware architectures that compute a constant-time variable-base point multiplication over the Galbraith–Lin–Scott (GLS) family of binary elliptic curves. Our first architecture is speed-optimized and utilizes the available hardware resources to achieve the fastest possible elliptic curve point multiplication operation. The second architecture, on the other hand, targets a more effective resource utilization by optimizing time-area product. Our hardware designs are especially tailored for the quadratic extension field F 2 2 n , with n = 127 , which allows us to attain a security level close to 128 bits. We explore extensively the usage of digit-based and Karatsuba multipliers for performing the quadratic field arithmetic associated to GLS elliptic curves and report the area and time performance obtained by these two types of multipliers. Targeting a Xilinx Kintex-7 FPGA device, we report on real hardware implementations of our designs, the fastest of which achieves a delay of just 7.97 µs for computing one scalar multiplication on a 1-core design. [ABSTRACT FROM AUTHOR]

A fast Montgomery multiplier design utilizing the DSP resources in modern FPGAs is presented. In the proposed design, the operand size is the multiples of 528 bits and the digit size is 48 bits. The design has 48  ×  48 bit digit multipliers built from the DSP slices performing 24  ×  16 bit multiplications and a carry select accumulator built from the DSP slices performing 48 bit additions. The proposed Montgomery multiplier works iteratively. In each iteration, a digit of an operand is multiplied by the digits of the other, the result is accumulated, and reduced by Montgomery method. An iteration takes not one but eight cycles to keep the digit multiplier count low and save some hardware resources. The proposed design is implemented for Virtex-7 FPGAs. The performance results are comparable with the best results in the literature. Substantial savings in FPGA logic resources are obtained. [ABSTRACT FROM AUTHOR]

In this note, we prove a finiteness result for fibers that are canonical lifts in a given elliptic fibration. The question was motivated by the authors' construction of an arithmetic Euler top, and it highlights an interesting discrepancy between the arithmetic and the classical case: in the former, it is impossible to extend the flows to a compactification of the phase space, viewed as an elliptic fibration over the space of action variables. [ABSTRACT FROM AUTHOR]

Shimura and Taniyama proved that if A is a potentially CM abelian variety over a number field F with CM by a field K linearly disjoint from F, then there is an algebraic Hecke character λA of FK such that L(A/F, s) = L(λA, s). We consider a certain converse to their result. Namely, let A be a potentially CM abelian variety appearing as a factor of the Jacobian of a curve of the form ye = γxf + δ. Fix positive integers a and n such that n/2 < a ≤ n. Under mild conditions on e, f, γ, δ, we construct a Chow motive M, defined over F = Q(γ, δ), such that L(M/F, s) and L(λaAλ-n-aA, s) have the same Euler factors outside finitely many primes. [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]

For an elliptic curve E / Q , Hasse's theorem asserts that # E ( F p ) = p + 1 − a p , where | a p | ≤ 2 p . Assuming that E has complex multiplication, we establish asymptotics for primes p for which a p is in subintervals of the Hasse interval [ − 2 p , 2 p ] of measure o ( p ) . In particular, given a function f = o ( 1 ) satisfying some mild conditions, we provide counting functions for primes p where | a p | ∈ ( 2 p ( 1 − f ( p ) ) , 2 p ) , and for primes where a p ∈ ( 2 p ( c − f ( p ) ) , 2 c p ) , where c ∈ ( 0 , 1 ) is a constant. [ABSTRACT FROM AUTHOR]

The goal of this article is two-fold: First, to prove a (two-variable) main conjecture for a CM field F without assuming the p-ordinary hypothesis of Katz, making use of what we call the Rubin-Stark L-restricted Kolyvagin systems, which is constructed out of the conjectural Rubin-Stark Euler system of rank g. (We are also able to obtain weaker unconditional results in this direction.) The second objective is to prove the Park-Shahabi plus/minus main conjecture for a CM elliptic curve E defined over a general totally real field again using (a twist of the) Rubin-Stark Kolyvagin system. This latter result has consequences towards the Birch and Swinnerton-Dyer conjecture for E. [ABSTRACT FROM AUTHOR]

Suppose is an elliptic curve over of conductor with complex multiplication (CM) by , and is the corresponding cuspidal Hecke eigenform in . Then th Fourier coefficient of is nonzero in the short interval for all and for some . As a consequence, we produce infinitely many cuspidal CM eigenforms level and weight for which holds, for all . [ABSTRACT FROM AUTHOR]

We unconditionally determine $I_{\mathbb Q}(d)$, the set of possible prime degrees of cyclic K-isogenies of elliptic curves with ${\mathbb Q}$-rational j-invariants and without complex multiplication over number fields K of degree ≤ d, for d ≤ 7, and give an upper bound for $I_{\mathbb Q}(d)$ for d > 7. Assuming Serre's uniformity conjecture, we determine $I_{\mathbb Q}(d)$ exactly for all positive integers d. [ABSTRACT FROM AUTHOR]