Theta Constants Research Articles

Solution of Mumford’s second problem on theta functions

A complete solution of Mumford’s second problem about representation of theta derivatives with rational characteristics in terms of theta constants with rational characteristics is found. An explicit formula for computing such an expression for theta derivative with an arbitrary rational characteristic is derived, and illustrated with examples. This approach fails only on four characteristics: [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text]; however expressions for theta derivatives with these four characteristics are found by other authors. Expressions for theta derivatives appear to be homogeneous of degree 3 with respect to theta constants.

Certified Newton schemes for the evaluation of low-genus theta functions

Theta functions and theta constants in low genus, especially genus 1 and 2, can be evaluated at any given point in quasi-linear time in the required precision using Newton schemes based on Borchardt sequences. Our goal in this paper is to provide the necessary tools to implement these algorithms in a provably correct way. In particular, we obtain uniform and explicit convergence results in the case of theta constants in genus 1 and 2, and theta functions in genus 1: the associated Newton schemes will converge starting from approximations to N bits of precision for N = 60, 300, and 1600 respectively, for all suitably reduced arguments. We also describe a uniform quasi-linear time algorithm to evaluate genus 2 theta constants on the Siegel fundamental domain. Our main tool is a detailed study of Borchardt means as multivariate analytic functions.

Eigenvalues and Fourier coefficients of degree two Siegel eigenforms constructed from Igusa theta constants

Eigenvalues and Fourier coefficients of degree two Siegel eigenforms constructed from Igusa theta constants

Sign choices in the AGM for genus two theta constants

Existing algorithms to compute genus 2 theta constants in quasi-linear time use Borchardt sequences, an analogue of the arithmetic-geometric mean for four complex numbers. In this paper, we show that these Borchardt sequences are given by good choices of square roots only, as in the genus 1 case. This removes the sign indeterminacies in the algorithm without relying on numerical integration.

Spanning the isogeny class of a power of an elliptic curve

Let E E be an ordinary elliptic curve over a finite field and g g be a positive integer. Under some technical assumptions, we give an algorithm to span the isomorphism classes of principally polarized abelian varieties in the isogeny class of E g E^g . The varieties are first described as hermitian lattices over (not necessarily maximal) quadratic orders and then geometrically in terms of their algebraic theta null point. We also show how to algebraically compute Siegel modular forms of even weight given as polynomials in the theta constants by a careful choice of an affine lift of the theta null point. We then use these results to give an algebraic computation of Serre’s obstruction for principally polarized abelian threefolds isogenous to E 3 E^3 and of the Igusa modular form in dimension 4 4 . We illustrate our algorithms with examples of curves with many rational points over finite fields.

Mirror Symmetry for a Cusp Polynomial Landau–Ginzburg Orbifold

AbstractFor any triple of positive integers $A^{\prime} = (a_1^{\prime},a_2^{\prime},a_3^{\prime})$ and $c \in{{\mathbb{C}}}^*$, cusp polynomial ${ f_{A^\prime }} = x_1^{a_1^{\prime}}+x_2^{a_2^{\prime}}+x_3^{a_3^{\prime}}-c^{-1}x_1x_2x_3$ is known to be mirror to Geigle–Lenzing orbifold projective line ${{\mathbb{P}}}^1_{a_1^{\prime},a_2^{\prime},a_3^{\prime}}$. More precisely, with a suitable choice of a primitive form, the Frobenius manifold of a cusp polynomial ${ f_{A^\prime }}$ turns out to be isomorphic to the Frobenius manifold of the Gromov–Witten theory of ${{\mathbb{P}}}^1_{a_1^{\prime},a_2^{\prime},a_3^{\prime}}$. In this paper we extend this mirror phenomenon to the equivariant case. Namely, for any $G$—a symmetry group of a cusp polynomial ${ f_{A^\prime }}$, we introduce the Frobenius manifold of a pair$({ f_{A^\prime }},G)$ and show that it is isomorphic to the Frobenius manifold of the Gromov–Witten theory of Geigle–Lenzing weighted projective line ${{\mathbb{P}}}^1_{A,\Lambda }$, indexed by another set $A$ and $\Lambda $, distinct points on ${{\mathbb{C}}}\setminus \{0,1\}$. For some special values of $A^{\prime}$ with the special choice of $G$ it happens that ${{\mathbb{P}}}^1_{A^{\prime}} \cong{{\mathbb{P}}}^1_{A,\Lambda }$. Combining our mirror symmetry isomorphism for the pair $(A,\Lambda )$, together with the “usual” one for $A^{\prime}$, we get certain identities of the coefficients of the Frobenius potentials. We show that these identities are equivalent to the identities between the Jacobi theta constants and Dedekind eta–function.

An explicit solution to the weak Schottky problem

We give an explicit weak solution to the Schottky problem, in the spirit of Riemann and Schottky. For any genus $g$, we write down a collection of polynomials in genus $g$ theta constants, such that their common zero locus contains the locus of Jacobians of genus $g$ curves as an irreducible component. These polynomials arise by applying a specific Schottky-Jung proportionality to an explicit collection of quartic identities for theta constants in genus $g-1$, which are suitable linear combinations of Riemann's quartic relations.

Double cover of modular S4 for flavour model building

We develop the formalism of the finite modular group Γ4′≡S4′, a double cover of the modular permutation group Γ4≃S4, for theories of flavour. The integer weight k>0 of the level 4 modular forms indispensable for the formalism can be even or odd. We explicitly construct the lowest-weight (k=1) modular forms in terms of two Jacobi theta constants, denoted as ε(τ) and θ(τ), τ being the modulus. We show that these forms furnish a 3D representation of S4′ not present for S4. Having derived the S4′ multiplication rules and Clebsch-Gordan coefficients, we construct multiplets of modular forms of weights up to k=10. These are expressed as polynomials in ε and θ, bypassing the need to search for non-linear constraints. We further show that within S4′ there are two options to define the (generalised) CP transformation and we discuss the possible residual symmetries in theories based on modular and CP invariance. Finally, we provide two examples of application of our results, constructing phenomenologically viable lepton flavour models.

General derivative Thomae formula for singular half-periods

The paper develops the result of second Thomae theorem in hyperelliptic case. The main formula, called general Thomae formula, provides expressions for values at zero of the lowest non-vanishing derivatives of theta functions with singular characteristics of arbitrary multiplicity in terms of branch points and period matrix. We call these values derivative theta constants. First and second Thomae formulas follow as particular cases. Some further results are derived. Matrices of second derivative theta constants (Hessian matrices of zero-values of theta functions with characteristics of multiplicity two) have rank three in any genus. Similar result about the structure of order $3$ tensor of third derivative theta constants is obtained, and a conjecture regarding higher multiplicities is made. As a byproduct a generalization of Bolza formulas are deduced.

On Frobenius' Theta Formula

Mumford's well-known characterization of the hyperelliptic locus of the moduli space of ppavs in terms of vanishing and non-vanishing theta constants is based on Neumann's dynamical system. Poor's approach to the characterization uses the cross ratio. A key tool in both methods is Frobenius' theta formula, which follows from Riemann's theta formula. In a 2004 paper Grushevsky gives a different characterization in terms of cubic equations in second order theta functions. In this note we first show the connection between the methods by proving that Grushevsky's cubic equations are strictly related to Frobenius' theta formula and we then give a new proof of Mumford's characterization via Gunning's multisecant formula.

On Siegel invariants of certain CM-fields

We first construct Siegel invariants of some CM-fields in terms of special values of theta constants, which would be a generalization of Siegel–Ramachandra invariants of imaginary quadratic fields. We further describe Galois actions on these invariants and provide a numerical example to show that this invariant really generates the ray class field of a CM-field.

Graded rings of modular forms of rational weights

In a previous paper, for any odd integer $$N\ge 5$$ we constructed $$(N-1)/2$$ modular forms of $$\varGamma (N)$$ of rational weight $$(N-3)/2N$$ and proved that the graded rings of modular forms of weight $$\ell (N-3)/2N$$ ($$\ell \in {\mathbb {Z}}_{\ge 0}$$) are generated by our forms for $$N=5$$, 7, 9. The proof was given by a direct calculation of the structure of the ring. In this paper, we generalize the result to cases when $$N=11$$ and 13 by using Castelnuovo–Mumford criterion on normal generation and Fujita criterion on relations of sections of invertible sheaves. For this purpose, it is needed to handle modular forms of small weight where Riemann Roch theorem has cohomological obstruction. Both rings for $$N=11$$ and 13 are generated by 5 and 6 generators with 15 and 35 concrete fundamental relations, respectively. These relations also give equations of the corresponding modular varieties. We will show that the similar claim does not hold for $$N=15$$ and $$N=23$$. We also give remarks on relations of some theta constants and further problems.

Weber’s formula for the bitangents of a smooth plane quartic

In a section of his 1876 treatise Theorie der Abel'schen Functionen vom Geschlecht 3 Weber proved a formula that expresses the bitangents of a non-singular plane quartic in terms of Riemann theta constants (Thetanullwerte). The present note is devoted to a modern presentation of Weber's formula. In the end a connection with the universal bitangent matrix is also displayed.

Thomae formula for abelian covers of ℂℙ¹

Abelian covers of C P 1 \mathbb {CP}^{1} , with fixed Galois group A A , are classified, as a first step, by a discrete set of parameters. Any such cover X X , of genus g ≥ 1 g\geq 1 , say, carries a finite set of A A -invariant divisors of degree g − 1 g-1 on X X that produce nonzero theta constants on X X . We show how to define a quotient involving a power of the theta constant on X X that is associated with such a divisor Δ \Delta , some polynomial in the branching values, and a fixed determinant on X X that does not depend on Δ \Delta , such that the quotient is constant on the moduli space of A A -covers with the given discrete parameters. This generalizes the classical formula of Thomae, as well as all of its known extensions by various authors.

On the variety associated to the ring of theta constants in genus 3

Due to fundamental results of Igusa and Mumford the $N=2^{g-1}(2^g+1)$ theta constants of first kind $$ \sum_{n\;{\rm integral}}\exp\pi{\rm i}\big(Z[n+a/2]+2b’(n+a/2)\big),\quad a,b\ {\rm integral}. $$ define for each genus $g$ an injective holomorphic map of the Satake compactification $X_g(4,8)= \overline{\scr{H}_g/\Gamma_g[4,8]}$ into the projective space $P^{N-1}$. Moreover, this map is biholomorphic onto the image outside the Satake boundary. It is not biholomorphic on the whole in the cases $g\ge 6$. Igusa also proved that in the cases $g\le 2$ this map is biholomorphic onto the image. In this paper we extend this result to the case $g=3$. So we show that the theta map $$ X_3(4,8)\longrightarrow\Bbb{P}^{35} $$ is biholomorphic onto the image. This is equivalent to the statement that the image is a normal subvariety of $\Bbb{P}^{35}$.

Generators of Siegel modular function fields of higher genus and level

For positive integers g and N , let F N ( g ) be the field of meromorphic Siegel modular functions of genus g and level N whose Fourier coefficients belong to the N th cyclotomic field. We present explicit generators of F N ( g ) over F 1 ( g ) in terms of quotient of theta constants, when g ≥ 2 and N ≥ 3 .

The Fermat septic and the Klein quartic as moduli spaces of hypergeometric Jacobians

We study the Schwarz triangle function with the monodromy group $\Delta(7,7,7)$, and we construct its inverse by theta constants. As consequences, we give uniformizations of the Klein quartic curve and the Fermat septic curve as Shimura curves parametrizing Abelian $6$-folds with endomorphisms $\mathbb{Z}[\zeta_7]$.

Heat equation for theta functions and vector-valued modular forms

We give a method for constructing vector-valued modular forms from singular scalar-valued ones of a suitable type. As an application, we prove that two remarkable spaces of vector-valued modular forms, which seem to be unrelated at a first look since they are constructed in two very different ways, are the same. More precisely, let $$V_{grad}$$ be the vector space generated by vector-valued modular forms constructed with gradients of odd theta functions and let $$V_\Theta $$ be the one generated by vector-valued modular forms arising from second order theta constants with our construction. We will prove that $$V_{grad}=V_\Theta $$ .

A character of the Siegel modular group of level 2 from theta constants

Given a characteristic, we define a character of the Siegel modular group of level 2, the computations of their values are obtained. Using our theorems, some key theorems of Igusa [2] can be recovered.

Generation of Siegel modular function fields of even level

For positive integers g g and N N , let F N ( g ) \mathcal {F}_N^{(g)} be the field of meromorphic Siegel modular functions of genus g g and level N N whose Fourier coefficients belong to the N N th cyclotomic field. We construct explicit generators of F N ( g ) \mathcal {F}_N^{(g)} over F 1 ( g ) \mathcal {F}_1^{(g)} by making use of a quotient of theta constants, when g ≥ 2 g\geq 2 and N N is even.