Theta Constants Research Articles

Classical theta constants vs. lattice theta series, and super string partition functions

Recently, various possible expressions for the vacuum-to-vacuum superstring amplitudes has been proposed at genus g = 3, 4, 5. To compare the different proposals, here we will present a careful analysis of the comparison between the two main technical tools adopted to realize the proposals: the classical theta constants and the lattice theta series. We compute the relevant Fourier coefficients in order to relate the two spaces. We will prove the equivalence up to genus 4. In genus five we will show that the solutions are equivalent modulo the Schottky form and coincide if we impose the vanishing of the cosmological constant.

ARITHMETIC-GEOMETRIC MEANS FOR HYPERELLIPTIC CURVES AND CALABI–YAU VARIETIES

In this paper, we define a generalized arithmetic-geometric mean μg among 2g terms motivated by 2τ-formulas of theta constants. By using Thomae's formula, we give two expressions of μg when initial terms satisfy some conditions. One is given in terms of period integrals of a hyperelliptic curve C of genus g. The other is by a period integral of a certain Calabi–Yau g-fold given as a double cover of the g-dimensional projective space Pg.

BKM Lie superalgebras from dyon spectra in Z N CHL orbifolds for composite N

We show that the generating function of electrically charged 1/2-BPS states in N=4 supersymmetric Z_N-CHL orbifolds of the heterotic string on T^6 are given by multiplicative eta-products. The eta-products are determined by the cycle shape of the corresponding symplectic involution in the dual type II picture. This enables us to complete the construction of the genus-two Siegel modular forms due to David, Jatkar and Sen [arXiv:hep-th/0609109] for Z_N orbifolds when N is non-prime. We study the Z_4 CHL orbifold in detail and show that the associated Siegel modular forms, \Phi_3(Z) and \widetilde{\Phi}_3(Z), are given by the square of the product of three even genus-two theta constants. Extending work by us[arXiv:0807.4451] as well as Cheng and Dabholkar[arXiv:0809.4258], we show that their `square roots' appear as the denominator formulae of two distinct Borcherds-Kac-Moody (BKM) Lie superalgebras. The BKM Lie superalgebra associated with the generating function of 1/4-BPS states, i.e., \widetilde{\Phi}_3(Z) has a parabolic root system with a light-like Weyl vector and the walls of its fundamental Weyl chamber are mapped to the walls of marginal stability of the 1/4-BPS states.

THETA SERIES OVER THE HURWITZ QUATERNIONS

There are six theta constants over the Hurwitz quaternions on the quaternion half-space of degree 2. The paper describes the behavior of these theta constants under the transpose mapping, which can be derived from the Fourier expansions. The results are applied to the theta series of the first and second kind.

A variant of Jacobi type formula for Picard curves

The classical Jacobi formula for the elliptic integrals (Gesammelte Werke I, p. 235) shows a relation between Jacobi theta constants and periods of ellptic curves E ( λ ) : w 2 = z ( z - 1 ) ( z - λ ) . In other words, this formula says that the modular form ϑ 00 4 ( τ ) with respect to the principal congruence subgroup Γ ( 2 ) of PSL ( 2 , Z ) has an expression by the Gauss hypergeometric function F ( 1 / 2 , 1 / 2 , 1 ; 1 - λ ) via the inverse of the period map for the family of elliptic curves E ( λ ) (see Theorem 1.1). In this article we show a variant of this formula for the family of Picard curves C ( λ 1 , λ 2 ) : w 3 = z ( z - 1 ) ( z - λ 1 ) ( z - λ 2 ) , those are of genus three with two complex parameters. Our result is a two dimensional analogy of this context. The inverse of the period map for C ( λ 1 , λ 2 ) is established in [S] and our modular form ϑ 0 3 ( u , v ) (for the definition, see (2.7)) is defined on a two dimensional complex ball D = { 2 Re v + | u | 2 < 0 } , that can be realized as a Shimura variety in the Siegel upper half space of degree 3 by a modular embedding. Our main theorem says that our theta constant is expressed in terms of the Appell hypergeometric function F 1 ( 1 / 3 , 1 / 3 , 1 / 3 , 1 ; 1 - λ 1 , 1 - λ 2 ) .

Quaternionic theta constants

We investigate the six quaternionic theta constants introduced by Freitag and Hermann. More precisely we investigate their restrictions to the Hermitian resp. Siegel half-space of degree 2. It turns out that these theta constants generate the graded ring of symmetric Hermitian modular forms for the principal congruence subgroup of level 1 + i over the Gaussian number field resp. of Siegel modular forms for the principal congruence subgroup of level 2 and even weight. As an application we obtain a simple construction of Igusa’s Siegel modular form of degree 2 and weight 30 with respect to the non-trivial character.

Lattice theta constants vs Riemann theta constants and NSR superstring measures

We discuss relations between two different representations of hypothetical holomorphic NSR measures, based on two different ways of constructing the semi-modular forms of weight 8. One of these ways is to build forms from the ordinary Riemann theta constants and another — from the lattice theta constants. We discuss unexpectedly elegant relations between lattice theta constants, corresponding to 16-dimensional self-dual lattices, and Riemann theta constants and present explicit formulae expressing the former ones through the latter. Starting from genus 5 the modular-form approach to construction of NSR measures is clearly sick and it seems to fail completely already at genus 6.

The abelianization of a symmetric mapping class group

AbstractLet Σg,r be a compact oriented surface of genus g with r boundary components. We determine the abelianization of the symmetric mapping class group (g,r)(p2) of a double unbranched cover p2: Σ2g − 1,2r → Σg,r using the Riemann constant, Schottky theta constant, and the theta multiplier. We also give lower bounds on the order of the abelianizations of the level d mapping class group.

On the Chow-forms of elliptic normal curves of degree 4

In this note, we shall aim, using explicit theta constants,to describe the Chow-form of an elliptic normal curve of degree 4 embedded in $P^{3}$ by using theta functions, at any point of the upper half plane. In this case, the embedded elliptic curve is a complete intersection which is defined by two quadratic forms. In order to calculate its Chow-form, we use the elimination theory. Our main result is Th. 1' (§2). In our case, Chow-forms are divisors of the Grassmann variety of lines in $P^{3}$. From the theorem, we see that the Chow-forms of elliptic normal curves of degree 4 lie on a linear subspace $M$ (cf. 2.3) of dimension 23. §2 is the main part of this note.In §3, using the theory of elliptic modular forms of level 4, we consider the geometric meaning of Th. 1'. Then our theorem shows that the Chow point is given by modular forms of weight 2 (of weight 1 in the traditional sense) and of level 4. The compactification of the moduli space of level 4 is isomorphic to a plane conic. Thus, we see that the Chow points of the projective elliptic normal curves of degree 4 determined by the points of the upper half plane form a rational curve of degree 4 in $M$ which is essentially the image of the 2-uple embedding of the above plane conic (Th. 2, §3. cf. Comments after the proof of Th. 2). Furthermore, we see that the correspondence which assigns the Chow point to a point of the upper half plane can be extended to the inequivalent cusps of the principal congruence subgroup of level 4 (Cor. 1, §3). Finally, we give a remark about the structure of the corresponding component of the Chow variety parametrizing l-cycles of degree 4 in $P^{3}$ (Cor. 2, §3).

A p-Adic Quasi-Quadratic Time Point Counting Algorithm

In this article, we give an algorithm for the computation of the number of rational points on the Jacobian variety of a generic ordinary hyperelliptic curve defined over a finite field of cardinality q with time complexity O(n 2+o(1) ) and space complexity O(n 2 ), where n = log(q). In the latter complexity estimate the genus and the characteristic are assumed as fixed. Our algorithm forms a generalization of both, the AGM algorithm of J.-F. Mestre and the canonical lifting method of T. Satoh. We canonically lift a certain arithmetic invariant of the Jacobian of the hyperelliptic curve in terms of theta constants. The theta null values are computed with respect to a semi-canonical theta structure of level 2 ν p where ν>0 is an integer and . The results of this paper suggest a global positive answer to the question whether there exists a quasi-quadratic time algorithm for the computation of the number of rational points on a generic ordinary abelian variety defined over a finite field.

On arithmetic properties of values of theta-constants

This article describes results about the transcendence and algebraic independence of values of theta constants (Nullthetawerte) and direct methods for proving these results. Values of other functions related to theta constants are discussed. We also present some conjectures and open questions.

Automorphic forms on the $5$-dimensional complex ball with respect to the Picard modular group over $\mathbb Z[i

We represent the $105$ automorphic forms on the $5$-dimensional complex ball $\mathbb B^5$ constructed by Matsumoto-Terasoma as the products of four linear combinations of the pull backs of theta constants under an embedding of $\mathbb B^5$ into the Siegel upper half space of degree $6$. They were used to describe the inverse of the period map for the family of the $4$-fold coverings of the complex projective line branching at eight points.

Isogeny formulas for the Picard modular form and a three terms arithmetic geometric mean

In this paper we study the Picard modular forms and show a new three terms arithmetic geometric mean (AGM) system. This AGM system is expressed via the Appell hypergeometric function of two variables. The Picard modular forms are expressed via the theta constants, and they give the modular function for the family of Picard curves. Our theta constants are “Neben type” modular forms of weight 1 defined on the complex 2-dimensional hyperball with respect to an index finite subgroup of the Picard modular group. We define a simultaneous 3-isogeny for the family of Jacobian varieties of Picard curves. Our main theorem shows the explicit relations between two systems of theta constants which are corresponding to isogenous Jacobian varieties. This relation induces a new three terms AGM which is a generalization of Borweins' cubic AGM.

Proof of R. Salvati Manni and J. Top's conjectures on Siegel modular forms and Abelian surfaces

This paper gives complete proofs of R. Salvati Manni and J. Top's conjectures which are contained in the paper, "Cusp forms of weight 2 for the group Γ(4, 8), Amer. J. Math . 115 (1993), 455-486." One of the conjectures claims that the Hasse-Weil zeta function corresponding to the Jacobian variety defined over [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="01i" /] of the hyper-elliptic curve y 2 = x 5 - x equals the Andrianov L -function of Siegel cusp form Θ of degree 2 and weight 2 which is four products of the Igusa theta constants. Regarding the Θ as a function obtained by the Yoshida lifting from a pair of elliptic modular forms, we prove the conjecture.

On invariant subspaces and eigenfunctions of regular Hecke operators on spaces of multiple theta constants

Invariant subspaces and eigenfunctions of regular Hecke operators acting on spaces spanned by products of even number of Igusa theta constants with rational characteristics are constructed. For some of the eigenfunctions of genuses g = 1 and g = 2 , corresponding zeta functions of Hecke and Andrianov are explicitly calculated.

Asyzygies, modular forms, and the superstring measure II

Precise factorization constraints are formulated for the three-loop superstring chiral measure, in the separating degeneration limit. Several natural ansatzes in terms of polynomials in theta constants for the density of the measure are examined. None of these ansatzes turns out to satisfy the dual criteria of modular covariance of weight 6, and of tending to the desired degeneration limit. However, an ansatz is found which does satisfies these criteria for the square of the density of the measure, raising the possibility that it is not the density of the measure, but its square which is a polynomial in theta constants. A key notion is that of totally asyzygous sextets of spin structures. It is argued that the ansatz produces a vanishing cosmological constant.

Theta constants associated to coverings of P1 branching at eight points

In this paper, we construct automorphic forms on the five-dimensional complex ball which give the inverse of the period map for cyclic 4-ple coverings of the complex projective line branching at eight points. We use theta constants associated to the Prym varieties of these coverings.

AUTOMORPHIC FORMS ON THE COMPLEX AND REAL BALLS DERIVED FROM THETA CONSTANTS

Through the modular embedding of the complex n-dimensional ball BnC into the Siegel upper half-space Sn+1 of degree n + 1 with respect to the Eisenstein integers Z[ω], we pull back the theta constants on Sn+1. We find a condition on the characteristics of the theta constants so that the pullbacks are non-zero automorphic forms on BnC with respect to the congruence subgroup Γ(1−ω). These automorphic forms are real valued on the real ball naturally embedded in the complex ball.

Action of Hecke operators on products of Igusa theta constants with rational characteristics

We derive explicit formulas for the images of products of Igusa theta constants with rational characteristics under the action of regular Hecke operators. In particular, we prove that when the class number of the sum of 2k squares is one, images of products of 2k Igusa theta constants with rational characteristics under the action of regular Hecke operators are, in general, linear combinations of similar products with explicitly given coefficients.

On the Picard modular surface for ℚ(√–2) and related Theta constants

The family of abelian varieties over C which have the endomorphism √–d of type (n, 1) is parameterized on the complex ball Bn. Let G be a modular group for such a family equipped with a certain level structure. The quotient is called the Picard modular variety of level G. There are many studies on Picard modular varieties (for example, see [2]). But concrete models are known only for the cases d = 1, 3 and 163 (see [3], [6], [7] and [9]). In this paper, we study the Picard modular surface for d = 2. In Section 1, we give an explicit modular embedding Φ : Bn → Hn+1 for d ≡ 1, 2 mod 4. This is inspired by the modular embedding given in [7]. In Section 2, we study several congruence subgroups of the Picard modular group G(ℤ) for ℚ(√–2 ) and relations among them. Furthermore we show some special properties of the theta constants restricted on Φ(Bn) with d = 2. In Section 3, we state the main theorem. Namely, (1) We give a birational morphism from \documentclass{article}\usepackage{amssymb}\pagestyle{empty}\begin{document}$ \overline {{\bf B}_2/ G_2(\mathbb Z)} $\end{document} onto a hypersurface in the weighted projective space P(1, 2, 3, 4). (2) We describe the image as an explicit hypersurface of degree 6 defined over ℤ[√2 ].