Jacobi Theta Functions Research Articles

On the energy of topological defect lattices

Since the logarithm function is the solution of Poisson's equation in two dimensions, it appears as the Coulomb interaction in two dimensions, the interaction between Abrikosov flux lines in a type II superconductor, or between line defects in elastic media, and so on. Lattices of lines interacting logarithmically are, therefore, a subject of intense research due to their manifold applications. The solution of the Poisson equation for such lattices is known in the form of an infinite sum since the late 1990's. In this article we present an alternative analytical solution, in closed form, in terms of the Jacobi theta function.

Extremal determinants of Laplace\u2013Beltrami operators for rectangular tori

In this work we study the determinant of the Laplace–Beltrami operator on rectangular tori of unit area. We will see that the square torus gives the extremal determinant within this class of tori. The result is established by studying properties of the Dedekind eta function for special arguments. Refined logarithmic convexity and concavity results of the classical Jacobi theta functions of one real variable are deeply involved.

The Elliptic Tail Kernel

AbstractWe introduce and study a new family of $q$-translation-invariant determinantal point processes on the two-sided $q$-lattice. We prove that these processes are limits of the $q$–$zw$ measures, which arise in the $q$-deformation of harmonic analysis on $U(\infty )$, and express their correlation kernels in terms of Jacobi theta functions. As an application, we show that the $q$–$zw$ measures are diffuse. Our results also hint at a link between the two-sided $q$-lattice and rows/columns of Young diagrams.

Romik's conjecture for the Jacobi theta function

Dan Romik recently considered the Taylor coefficients of the Jacobi theta function around the complex multiplication point i. He then conjectured that the Taylor coefficients d(n) either vanish or are periodic modulo any prime p; this was proved by the combined efforts of Scherer and Guerzhoy-Mertens-Rolen, with the latter trio considering arbitrary half integral weight modular forms. We refine previous work for p≡1(mod4) by displaying a concise algebraic relation between d(n+p−12) and d(n) related to the p-adic factorial, from which we can deduce periodicity with an effective period.

Summation of Gaussian shifts as Jacobi's third Theta function

A proper choice of parameters of the Jacobi modular identity (Jacobi Imaginary transformation) implies that the summation of Gaussian shifts on infinity periodic grids can be represented as the Jacobi's third Theta function. As such, connection between summation of Gaussian shifts and the solution to a Schrodinger equation is explicitly shown. A concise and controllable upper bound of the saturation error for approximating constant functions with summation of Gaussian shifts can be immediately obtained in terms of the underlying shape parameter of the Gaussian. This sheds light on how to choose a shape parameter and provides further understanding on using Gaussians with increasingly flatness.

Mock modular forms whose shadows are Eisenstein series of integral weight

The purpose of this article is to give a simple and explicit construction of mock modular forms whose shadows are Eisenstein series of arbitrary integral weight, level, and character. As application, we construct forms whose shadows are Hecke's Eisenstein series of weight one associated to imaginary quadratic fields, recovering some results by Kudla, Rapoport and Yang (1999), and Schofer (2009), and forms whose shadows equal $\Theta^{2k}(z)$ for $k\in \{1,2,3,4\}$, where $\Theta(z)$ denotes Jacobi's theta function.

Proofs for certain q-trigonometric identities of Gosper

Applying an addition formula of Liu (2007), we deduce certain Jacobi thetafunction identities. From these results we confirm several $q$-trigonometricidentities conjectured by Gosper (2001). Another conjectured identityon the constant $\\Pi_{q}$ is also settled.

Quantum Jacobi forms in number theory, topology, and mathematical physics

We establish three infinite families of quantum Jacobi forms, arising in the diverse areas of number theory, topology, and mathematical physics, and unified by partial Jacobi theta functions.

Generalised heat coefficients and associated spectral zeta functions on complex projective spaces Pn(ℂ)

ABSTRACTThe heat invariants or the Minakshisundaram-Pleijel heat coefficients describe the residues of the spectral zeta functions on any compact Riemannian manifold M. In this paper, we first give an explicit description of the heat coefficients via the Jacobi theta functions and their higher order derivatives, and then express these Minakshisundaram-Pleijel coefficients in terms of the residues of the associated zeta functions . Different interesting formulae are obtained for these zeta functions. A new class of heat coefficients, the Maclaurin heat coefficients () (i.e. the coefficients appearing in the Maclaurin expansion of the heat kernel ), are explicitly studied in terms of the classical and generalised Minakshisundaram-Pleijel coefficients and () respectively. Remarkable asymptotic expansions for the Maclaurin spectral functions are established. We also introduce and construct new zeta functions associated with these Maclaurin heat coefficients (generalised Minakshisundaram-Pleijel zeta functions), and it is interesting to see that these generalised zeta functions are explicitly understood in terms of the classical (Minakshisundaram-Pleijel) zeta functions.

Density of algebraic points on Noetherian varieties

Let $${\Omega \subset \mathbb{R}^{n}}$$ be a relatively compact domain. A finite collection of real valued functions on $${{\Omega}}$$ is called a Noetherian chain if the partial derivatives of each function are expressible as polynomials in the functions. A Noetherian function is a polynomial combination of elements of a Noetherian chain. We introduce Noetherian parameters (degrees, size of the coefficients) which measure the complexity of a Noetherian chain. Our main result is an explicit form of the Pila–Wilkie theorem for sets defined using Noetherian equalities and inequalities: for any $${\varepsilon > 0}$$ , the number of points of height H in the transcendental part of the set is at most C· $${{H}^ \varepsilon}$$ where C can be explicitly estimated from the Noetherian parameters and $${\varepsilon}$$ . We show that many functions of interest in arithmetic geometry fall within the Noetherian class, including elliptic and abelian functions, modular functions and universal covers of compact Riemann surfaces, Jacobi theta functions, periods of algebraic integrals, and the uniformizing map of the Siegel modular variety $${\mathcal{A}_{g}}$$ . We thus effectivize the (geometric side of) Pila–Zannier strategy for unlikely intersections in various contexts.

Periods of modular forms on $\Gamma_0(N)$ and products of Jacobi theta functions

Generalizing a result of [15] for modular forms of level one, we give a closed formula for the sum of all Hecke eigenforms on \Gamma_0(N) , multiplied by their odd period polynomials in two variables, as a single product of Jacobi theta series for any squarefree level N . We also show that for N = 2, 3 and 5 this formula completely determines the Fourier expansions all Hecke eigenforms of all weights on \Gamma_0(N) .

On the Asymptotics of Associated Sigma-Functions and Jacobi Theta-Functions

For the associated sigma-functions, Jacobi theta-functions, and their logarithmic derivatives, we present asymptotic formulas valid outside the efficiently constructed exceptional sets of disks.

The asymptotic formulas for coefficients and algebraicity of Jacobi forms expressed by infinite product

We determine asymptotic formulas for the Fourier coefficients of Jacobi forms expressed by infinite products with Jacobi theta functions and the Dedekind eta function. These are generalizations of results about the growth of the Fourier coefficients of Jacobi forms given by an inverse of Jacobi theta function to derive the asymptotic behavior of the Betti numbers of the Hilbert scheme of points on an algebraic surface by Bringmann–Manschot and about the asymptotic behavior of the χ y -genera of Hilbert schemes of points on K3 surfaces by Manschot–Rolon. We also get the algebraicity of the generating functions given by Göttsche for the Hilbert schemes associated to general algebraic surfaces.

Algebraic results for certain values of the Jacobi theta-constant $\theta_3(\tau)$

In its most elaborate form, the Jacobi theta function is defined for two complex variables $z$ and τ by $\theta (z|\tau ) =\sum _{\nu =-\infty }^{\infty } e^{\pi i\nu ^2\tau + 2\pi i\nu z}$, which converges for all complex number $z$, and τ in the upper half-plane. The special case \[ \theta _3(\tau ):=\theta (0|\tau )= 1+2\sum _{\nu =1}^{\infty } e^{\pi i\nu ^2 \tau } \] is called a Jacobi theta-constant or Thetanullwert of the Jacobi theta function $\theta (z|\tau )$. In this paper, we prove the algebraic independence results for the values of the Jacobi theta-constant $\theta _3(\tau )$. For example, the three values $\theta _3(\tau )$, $\theta _3(n\tau )$, and $D\theta _3(\tau )$ are algebraically independent over $\mathbb{Q} $ for any τ such that $q=e^{\pi i\tau }$ is an algebraic number, where $n\geq 2$ is an integer and $D:=(\pi i)^{-1}{d}/{d\tau }$ is a differential operator. This generalizes a result of the first author, who proved the algebraic independence of the two values $\theta _3(\tau )$ and $\theta _3(2^m\tau )$ for $m\geq 1$. As an application of our main theorem, the algebraic dependence over $\mathbb{Q} $ of the three values $\theta _3(\ell \tau )$, $\theta _3(m\tau )$, and $\theta _3(n\tau )$ for integers $\ell ,m,n\geq 1$ is also presented.

Infinite sums for Fibonacci polynomials and Lucas polynomials

In this paper we establish certain infinite sums involving many arithmetical functions and the Fibonacci polynomials or the Lucas polynomials. Several of the sums are given explicitly in Jacobi theta functions.

Two-species diffusion-annihilation process on the fully-connected lattice: probability distributions and extreme value statistics

We study the two-species diffusion-annihilation process, Ø, on the fully-connected lattice. Probability distributions for the number of particles and the reaction time are obtained for a finite-size system using a master equation approach. Mean values and variances are deduced from generating functions. When the reaction is far from complete, i.e. for a large number of particles of each species, mean-field theory is exact and the fluctuations are Gaussian. In the scaling limit the reaction time displays extreme-value statistics in the vicinity of the absorbing states. A generalized Gumbel distribution is obtained for unequal initial densities, . For equal or almost equal initial densities, , the fluctuations of the reaction time near the absorbing state are governed by a probability density involving derivatives of , the Jacobi theta function.

Hyperquasipolynomials for the Theta-Function

Let g be a linear combination with quasipolynomial coefficients of shifts of the Jacobi theta function and its derivatives in the argument. All entire functions f: ℂ → ℂ satisfying f(x+y)g(x−y) = α1(x)β1(y)+· · ·+αr(x)βr(y) for some r ∈ ℕ and αj, βj: ℂ → ℂ are described.

Solutions of a class of multiplicatively advanced differential equations

The multiplicatively advanced differential equations (MADEs) of form f(n)(t)=αf(βt) with α≠0, β>1 are studied along with a class of their solutions of type fμ,λ(t) defined on [0,∞). For λ∈Q+,μ∈R, the solutions fμ,λ(t) are extended to (−∞,∞) in a non-unique manner to obtain Schwartz wavelet solutions Fμ,λ(t) of the original MADE, with all moments of Fμ,λ(t) vanishing. Examples are studied in detail. The Fourier transform of each Fμ,λ(t) is computed and, in a number of examples, is related to the Jacobi theta function. Additional conditions sufficient for the uniqueness of certain MADE initial value problems are given. Conditions for decay and non-decay at −∞ are obtained. Decay rates at ±∞ in terms of familiar functions are established.

Zeros of combinations of the Riemann Ξ-function and the confluent hypergeometric function on bounded vertical shifts

In 1914, Hardy proved that infinitely many non-trivial zeros of the Riemann zeta function lie on the critical line using the transformation formula of the Jacobi theta function. Recently the first author obtained an integral representation involving the Riemann Ξ-function and the confluent hypergeometric function linked to the general theta transformation. Using this result, we show that a series consisting of bounded vertical shifts of a product of the Riemann Ξ-function and the real part of a confluent hypergeometric function has infinitely many zeros on the critical line, thereby generalizing a previous result due to the first and the last authors along with Roy and Robles. The latter itself is a generalization of Hardy's theorem.

Fermionic one-loop amplitudes of the RNS superstring

We investigate massless n-point one-loop amplitudes of the open RNS superstring with two external fermions and determine their worldsheet integrands. The contributing correlation functions involving spin-1/2 and spin-3/2 operators from the fermion vertices are evaluated to any multiplicity. Moreover, we introduce techniques to sum these correlators over the spin structures of the worldsheet fermions such as to manifest all cancellations due to spacetime supersymmetry. These spin sums require generalizations of the Riemann identities among Jacobi theta functions, and the results can be expressed in terms of doubly-periodic functions known from the mathematics literature on elliptic multiple zeta values. On the boundary of moduli space, our spin-summed correlators specialize to compact representations of fermionic one-loop integrands for ambitwistor strings.