Modular Equations Research Articles

Modular anomaly equations and S-duality in N = 2 $$ \mathcal{N}=2 $$ conformal SQCD

We use localization techniques to study the non-perturbative properties of an $$ \mathcal{N}=2 $$ superconformal gauge theory with gauge group SU(3) and six fundamental flavours. The instanton corrections to the prepotential, the dual periods and the period matrix are calculated in a locus of special vacua possessing a ℤ 3 symmetry. In a semiclassical expansion, we show that these observables are constrained by S-duality via a modular anomaly equation which takes the form of a recursion relation. The solutions of the recursion relation are quasi-modular functions of Γ1 (3), which is a subgroup of the S-duality group and is also a congruence subgroup of SL(2, ℤ).

Class invariants from a new kind of Weber-like modular equation

A technique is described for explicitly evaluating quotients of the Dedekind eta function at quadratic integers. These evaluations do not make use of complex approximations but are found by an entirely `algebraic' method. They are obtained by means of specialising certain modular equations related to Weber's modular equations of `irrational type'. The technique works for certain eta quotients evaluated at points in an imaginary quadratic field with discriminant d1 (mod 8).

Certain New Schläfli Type Mixed Modular Equations

Schlafli (J. Reine Angew. Math. 72, 360–369, 1870) has established modular equations involving kk′ and λλ′ for degrees 3, 5, 7, 9, 11, 13, 17, and 19. On pages 86 and 88 of his first notebook, Ramanujan recorded 11 Schlafli-type modular equations for composite degrees. In this paper, we establish several new Schlafli type mixed modular equations for composite degrees by elementary algebraic manipulations which are analogous to those recorded by Ramanujan.

Modular-type functions attached to mirror quintic Calabi–Yau varieties

In this article we study a differential algebra of modular-type functions attached to the periods of a one-parameter family of Calabi–Yau varieties which is mirror dual to the universal family of quintic threefolds. Such an algebra is generated by seven functions satisfying functional and differential equations in parallel to the modular functional equations of classical Eisenstein series and the Ramanujan differential equation. Our result is the first example of automorphic-type functions attached to varieties whose period domain is not Hermitian symmetric. It is a reformulation and realization of a problem of Griffiths from the seventies on the existence of automorphic functions for the moduli of polarized Hodge structures.

The polynomial approximate common divisor problem and its application to the fully homomorphic encryption

We propose and examine the approximate polynomial common divisor problem, which can be viewed as a polynomial analogue to the approximate integer common divisor problem. Since our problem is rather new, we perform extensive cryptanalysis, applying various known attacks against the structurally similar problems. Moreover, we propose a small root finding algorithm for multivariate modular equation system, and apply it to the proposed problem. Those analyses confirm that the proposed problem is difficult with appropriate parameters.Additionally, we construct a simple somewhat homomorphic encryption scheme, which can efficiently accommodate large message spaces. When the evaluation of a low degree polynomial of very large integers is required, our scheme is more efficient than the recent RLWE-based scheme, YASHE, by Bos et al. (2013). In particular, multiplication is ten times faster when evaluating degree-10 polynomial of 1638-bit integers. We convert this scheme to a leveled fully homomorphic encryption scheme by applying Brakerski’s scale invariant technique, and the resulting scheme has features similar to the variant of van Dijk et al.’s scheme by Coron et al. (2014). Our scheme, however, does not use the subset sum, which makes its design much simpler.

Some new identities for colored partition

In this paper, we discover several new identities on colored partitions and provide proofs for them. Many colored partition identities presented in the paper do not belong to the general and unified combinatorial framework provided by Sandon and Zanello. Most of our proofs depend upon new modular equations given by employing the method of reciprocation.

Identities for Partitions with Distinct Colors

We establish several identities for partitions with distinct colors that arise from Ramanujan’s formulas for multipliers in the theory of modular equations. In the course of our investigations, we prove three conjectures made by C. Sandon and F. Zanello. However, combinatorial proofs of these three conjectures and most of the theorems in this paper remain to be given.

Ramanujan’s cubic transformation and generalized modular equation

We study the quotient of hypergeometric functions $\mu _a^* (r) = \frac{\pi } {{2\sin (\pi a)}}\frac{{F(a,1 - a;1;1 - r^3 )}} {{F(a,1 - a;1;r^3 )}},r \in (0,1) $ in the theory of Ramanujan’s generalized modular equation for a ∈ (0, 1/2], find an infinite product formula for µ1/3*(r) by use of the properties of µ a * and Ramanujan’s cubic transformation. Besides, a new cubic transformation formula of hypergeometric function is given, which complements the Ramanujan’s cubic transformation.

On Ramanujan's modular equation of degree 7

In this paper, we give an alternative proof of a Ramanujan's modular equation of degree 7 by employing certain theta function identities.

ON EVALUATIONS OF THE MODULAR j-INVARIANT BY MODULAR EQUATIONS OF DEGREE 2

We derive modular equations of degree 2 to establish explicit relations for the parameterizations for the theta functions <TEX>${\varphi}$</TEX> and <TEX>${\psi}$</TEX>. We then find specific values of the parameterizations to evaluate some new values of the modular j-invariant in terms of <TEX>$J_n$</TEX>.

On Some Ramanujan Identities for the Ratios of Eta-Functions

We give direct proofs of some of Ramanujan’s P-Q modular equations based on simply proved elementary identities from Chapter 16 of his Second Notebook.

Hypergeometric series, modular linear differential equations and vector-valued modular forms

We survey the theory of vector-valued modular forms and their connections with modular differential equations and Fuchsian equations over the three-punctured sphere. We present a number of numerical examples showing how the theory in dimensions 2 and 3 leads naturally to close connections between modular forms and hypergeometric series.

A reduced BPS index of E-strings

We study the BPS spectrum of E-strings in a situation where the global E_8 symmetry is broken down to D_4 + D_4 by a certain twist. We find that the refined BPS index in this setup serves as a reduced BPS index of E-strings, which gives a novel trigonometric generalization of the Nekrasov partition function for four-dimensional N=2 supersymmetric SU(2) gauge theory with N_f=4 massless flavors. We determine the perturbative part of this index and also first few unrefined instanton corrections. By using these results together with the modular anomaly equation, the genus expansion of the free energy can be computed efficiently up to very high order. We also determine the unrefined elliptic genus of four E-strings under the twist.

Modular anomaly equations in N $$ \mathcal{N} $$ =2* theories and their large-N limit

We propose a modular anomaly equation for the prepotential of the N=2* super Yang-Mills theory on R^4 with gauge group U(N) in the presence of an Omega-background. We then study the behaviour of the prepotential in a large-N limit, in which N goes to infinity with the gauge coupling constant kept fixed. In this regime instantons are not suppressed. We focus on two representative choices of gauge theory vacua, where the vacuum expectation values of the scalar fields are distributed either homogeneously or according to the Wigner semi-circle law. In both cases we derive an all-instanton exact formula for the prepotential. As an application, we show that the gauge theory partition function on S^4 at large N localises around a Wigner distribution for the vacuum expectation values leading to a very simple expression in which the instanton contribution becomes independent of the coupling constant.

PROOFS OF CONJECTURES OF SANDON AND ZANELLO ON COLORED PARTITION IDENTITIES

In a recent systematic study, C. Sandon and F. Zanello of- fered 30 conjectured identities for partitions. As a consequence of their study of partition identities arising from Ramanujan's formulas for mul- tipliers in the theory of modular equations, the present authors in an earlier paper proved three of these conjectures. In this paper, we provide proofs for the remaining 27 conjectures of Sandon and Zanello. Most of our proofs depend upon known modular equations and formulas of Ra- manujan for theta functions, while for the remainder of our proofs it was necessary to derive new modular equations and to employ the process of duplication to extend Ramanujan's catalogue of theta function formulas.

Cryptanalysis of two cryptosystems based on multiple intractability assumptions

Two public key cryptosystems based on the two intractable number-theoretic problems, integer factorisation and simultaneous Diophantine approximation, were proposed in 2005 and 2009, respectively. In this study, the authors break these two cryptosystems for the recommended minimum parameters by solving the corresponding modular linear equations with small unknowns. For the first scheme, the public modulus is factorised and the secret key is recovered with the Gauss algorithm. By using the LLL basis reduction algorithm for a seven-dimensional lattice, the public modulus in the second scheme is also factorised and the plaintext is recovered from a ciphertext. The author's attacks are efficient and verified by experiments which were done within 5s.

THE GENERAL CHINESE REMAINDER THEOREM

The Chinese remainder theorem deals with systems of modular equations. The classical variant requires the modules to be pairwise coprime. In this paper we discuss the general variant, which does not require this restriction on modules. We have selected and implemented several algorithms for the general Chinese remainder theorem. Moreover, we point out some interesting applications of this variant in secret sharing and threshold cryptography.

Series for 1/π and π with free parameters

In terms of the hypergeometric method, we establish the extension of a series for 1/π due to Ramanujan [Modular equations and approximations to π. Quart J Math (Oxford). 1914;45:350–372] with two free parameters and a beautiful series for π with one free parameter. Meanwhile, two series for 1/π with three free parameters are also derived in the same way.

Congruences for 9-regular partitions modulo 3

In view of the modular equation of fifth order, we give a simple proof of Keith’s conjecture which is some infinite families of congruences modulo 3 for the 9-regular partition function. Meanwhile, we derive some new congruences modulo 3 for the 9-regular partition function.

On modular solutions of fractional weights for the Kaneko–Zagier equation for $$\varGamma ^*_0(2) \,\hbox {and}\, \varGamma ^*_0(3)$$ Γ 0 ∗ ( 2 ) and Γ 0 ∗ ( 3 )

For the full modular group, Kaneko gave a modular form solution of fractional weights for a holomorphic modular differential equation of second order. In this paper, we give modular form solutions of fractional weights for a modular differential equation on the Fricke group of levels 2 and 3.