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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]

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]

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]

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]

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]

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]

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]

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]

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]

For a field of definition of an abelian variety and prime ideal of which is of a good reduction for , the structure of as abelian group is: where , , and for . We are interested in finding an asymptotic formula for the number of prime ideals with , has a good reduction at , . We succeed in proving this under the assumption of the Generalized Riemann Hypothesis (GRH). Unconditionally, we achieve a short range asymptotic for abelian varieties of CM type, and the full cyclicity theorem for elliptic curves over a number field containing the CM field. [ABSTRACT FROM AUTHOR]

It is shown that the quartic Fermat equation x4 + y4 = 1 has nontrivial integral solutions in the Hilbert class field Σ of any quadratic field whose discriminant satisfies -d ≡ 1 (mod 8). A corollary is that the quartic Fermat equation has no nontrivial solution in , for p (> 7) a prime congruent to 7 (mod 8), but does have a nontrivial solution in the odd degree extension Σ of K. These solutions arise from explicit formulas for the points of order 4 on elliptic curves in Tate normal form. The solutions are studied in detail and the results are applied to prove several properties of the Weber singular moduli introduced by Yui and Zagier. [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]

The main purpose of this paper is to give an intrinsic interpretation of the space Θ(D) generated by the binary theta series ϑf attached to the positive binary quadratic forms f whose discriminant has the form D(f) = D/t2, for some integer t. It turns out that , the space of modular forms of weight 1 and of level |D| which have complex multiplication (CM) by their Nebentypus character . As an application, we obtain a structure theorem of the space . The proof of this theorem rests on the results of [The space of binary theta series, Ann. Sci. Math. Québec 36 (2012) 501-534] together with a characterization of the newforms f which have CM by their Nebentypus character in terms of properties of the associated Deligne-Serre Galois representationρf. [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]

We present seven theorems on the structure of prime order torsion points on CM elliptic curves defined over number fields. The first three results refine bounds of Silverberg and Prasad-Yogananda by taking into account the class number of the CM order and the splitting of the prime in the CM field. In many cases we can show that our refined bounds are optimal or asymptotically optimal. We also derive asymptotic upper and lower bounds on the least degree of a CM-point on X1(N). Upon comparison to bounds for the least degree for which there exist infinitely many rational points on X1(N), we deduce that, for sufficiently large N, X1(N) will have a rational CM point of degree smaller than the degrees of at least all but finitely many non-CM points. [ABSTRACT FROM AUTHOR]

We study parameter estimation problem for diagonalizable stochastic partial differential equations driven by a multiplicative fractional noise with any Hurst parameter H ∈ (0, 1). Two classes of estimators are investigated: traditional maximum likelihood type estimators, and a new class called closed-form exact estimators. Finally the general results are applied to stochastic heat equation driven by a fractional Brownian motion. [ABSTRACT FROM AUTHOR]

We show that every periodic element of the free idempotent generated semigroup on an arbitrary biordered set belongs to a subgroup of the semigroup. [ABSTRACT FROM AUTHOR]

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]

A sequence $(a_n)_{n=0}^{\infty}$ over ℤ is an LP-sequence if for every prime p and integer n ≥ 0 we have $a_n \equiv \prod_{i=0}^{l-1} a_{n_{i}}$ (mod p), when $n= \sum_{i=0}^{l-1} n_ip^i$ is a base p expansion of n. In this paper, we study sequences $(a_n)_{n=0}^{\infty}$ such that both $(a_n)_{n=0}^{\infty}$, $(a_{dn})_{n=0}^{\infty}$ are LP-sequences for some d ≥ 2. One of those sequences is a counter-example to a conjecture of McIntosh [15]. [ABSTRACT FROM AUTHOR]

We prove that certain powers of the gap function for the newform associated to an elliptic curve without complex multiplication are "finite" on average. In particular we obtain quantitative results on the number of large values of the gap function. [ABSTRACT FROM AUTHOR]

An arithmetic framework to string compactification is described. The approach is exemplified by formulating a strategy that allows to construct geometric compactifications from exactly solvable theories at c = 3. It is shown that the conformal field theoretic characters can be derived from the geometry of space–time, and that the geometry is uniquely determined by the two-dimensional field theory on the worldsheet. The modular forms that appear in these constructions admit complex multiplication, and allow an interpretation as generalized McKay–Thompson series associated to the Mathieu and Conway groups. This leads to a string motivated notion of arithmetic moonshine. [ABSTRACT FROM AUTHOR]

Based on the formula for multiplying Schubert classes obtained in [17], we develop an algorithm computing the product of two arbitrary Schubert classes in a flag manifold G/H, where G is a compact connected Lie group and H ⊂ G is the centralizer of a one-parameter subgroup in G. Since all Schubert classes on G/H constitute a basis for the integral cohomology H*(G/H), the algorithm gives also a method to compute the integral cohomology ring H*(G/H) independent of the classical spectral sequence method of Leray and Borel [32, 33, 8, 9]. [ABSTRACT FROM AUTHOR]

Four operands Real Multiply Accumulation (4op-RMA) is a method to implement transversal circuits, direct form circuits and 2D type circuits with real coefficients by using a complex multiplier. Digital Signal Processors (DSPs) with 4op-RMA implement these circuits twice as fast as the conventional ones. Besides, we can simplify the programs on such DSPs. However 4op-RMA have not been applied to the lattice circuits yet because of their circuit structures. In this paper, we propose an implementation method for the lattice circuits by using 4op-RMA. The lattice part can be calculated by a 4op-RMA operation, which corresponds to a complex multiplication. [ABSTRACT FROM AUTHOR]