Terms Of 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.

Matching branches of a nonperturbative conformal block at its singularity divisor

Conformal block is a function of many variables, usually represented as a formal series, with coefficients which are certain matrix elements in the chiral (e.g. Virasoro) algebra. Non-perturbative conformal block is a multi-valued function, defined globally over the space of dimensions, with many branches and, perhaps, additional free parameters, not seen at the perturbative level. We discuss additional complications of non-perturbative description, caused by the fact that all the best studied examples of conformal blocks lie at the singularity locus in the moduli space (at divisors of the coefficients or, simply, at zeroes of the Kac determinant). A typical example is the Ashkin-Teller point, where at least two naive non-perturbative expressions are provided by elliptic Dotsenko-Fateev integral and by the celebrated Zamolodchikov formula in terms of theta-constants, and they are different. The situation is somewhat similar at the Ising and other minimal model points.

Periods of second kind differentials of $(n,s)$-curves

For elliptic curves, expressions for the periods of elliptic integrals of the second kind in terms of theta-constants, have been known since the middle of the 19th century. In this paper we consider the problem of generalizing these results to curves of higher genera, in particular to a special class of algebraic curves, the so-called $(n,s)$-curves. It is shown that the representations required can be obtained by the comparison of two equivalent expressions for the projective connection, one due to Fay-Wirtinger and the other from Klein-Weierstrass. As a principle example, we consider the case of the genus two hyperelliptic curve, and a number of new Thomae and Rosenhain-type formulae are obtained. We anticipate that our analysis for the genus two curve can be extended to higher genera hyperelliptic curves, as well as to other classes of $(n,s)$ non-hyperelliptic curves.

On algebraic relations for Ramanujan’s functions

Let P,Q, and R denote the Ramanujan Eisenstein series. We compute algebraic relations in terms of P(qi) (i=1,2,3,4), Q(qi) (i=1,2,3), and R(qi) (i=1,2,3). For complex algebraic numbers q with 0<|q|<1 we prove the algebraic independence over ℚ of any three-element subset of {P(q),P(q2),P(q3),P(q4)} and of any two-element subset of {Q(q),Q(q2),Q(q3)} and {R(q),R(q2),R(q3)}, respectively. For all the results we use some expressions of \(P(q^{i_{1}}), Q(q^{i_{2}}) \), and \(R(q^{i_{3}}) \) in terms of theta constants. Computer-assisted computations of functional determinants and resultants are essential parts of our proofs.

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.

Theta constants associated to cubic threefolds

For a cubic surface X, by considering the intermediate Jacobian J(Y) of the triple covering Y of the 3-dimensional projective space branching along X, Allcock, Carlson and Toledo constructed a period map per from the family of marked cubic surfaces to the four dimensional complex ball embedded in the Siegel upper half space of degree 5. We give an expression of the inverse of per in terms of theta constants by constructing an isomorphism between J(Y) and a Prym variety of a cyclic 6-ple covering of the projective line branching at seven points.

Two-loop superstrings IV: The cosmological constant and modular forms

The slice-independent gauge-fixed superstring chiral measure in genus 2 derived in the earlier papers of this series for each spin structure is evaluated explicitly in terms of theta-constants. The slice-independence allows an arbitrary choice of superghost insertion points q 1, q 2 in the explicit evaluation, and the most effective one turns out to be the split gauge defined by S δ ( q 1, q 2)=0. This results in expressions involving bilinear theta-constants M . The final formula in terms of only theta-constants follows from new identities between M and theta-constants which may be interesting in their own right. The action of the modular group Sp(4, Z ) is worked out explicitly for the contribution of each spin structure to the superstring chiral measure. It is found that there is a unique choice of relative phases which insures the modular invariance of the full chiral superstring measure, and hence a unique way of implementing the GSO projection for even spin structure. The resulting cosmological constant vanishes, not by a Riemann identity, but rather by the genus 2 identity expressing any modular form of weight 8 as the square of a modular form of weight 4. The degeneration limits for the contribution of each spin structure are determined, and the divergences, before the GSO projection, are found to be the ones expected on physical grounds.

Mirror symmetry of elliptic curves and Ising model

We study the differential equations governing mirror symmetry of elliptic curves, and obtain a characterization of the ODEs which give rise to the integral q-expansion of mirror maps. Through theta function representation of the defining equation, we express the mirror correspondence in terms of theta constants. By investigating the elliptic curves in X 9-family, the identification of the Landau-Ginzburg potential with the spectral curve of Ising model is obtained. Through the Jacobi elliptic function parametrization of Boltzmann weights in the statistical model, an exact Jacobi form-like formula of mirror map is described.

Quasi-periodic solutions of the coupled nonlinear Schrödinger equations

We consider travelling periodic and quasi-periodic wave solutions of a set of coupled nonlinear Schrödinger equations. In fibre optics these equations can be used to model single mode fibres under the action of cross-phase modulation, with weak birefringence. The problem is reduced to the ‘1:2:1’ integrable case of the two-particle quartic potential. A general approach for finding elliptic solutions is given. New solutions which are associated with two-gap Treibich-Verdier potentials are found. General quasi-periodic solutions are given in terms of two dimensional theta functions with explicit expressions for frequencies in terms of theta constants. The reduction of quasi-periodic solutions to elliptic functions is discussed.

Expression of parameters of solutions of algebraically integrable nonlinear equations in terms of theta constants

Expression of parameters of solutions of algebraically integrable nonlinear equations in terms of theta constants