Theta Series of Quaternion Algebras over Function Fields

Let K be the function field over a finite field of odd order, and let H be a definite quaternion algebra over K. If Λ is an order of level M in H, we define theta series for each ideal I of Λ using the reduced norm on H. Using harmonic analysis on the completed algebra H_∞ and the arithmetic of quaternion algebras, we establish a transformation law for these theta series. We also define analogs of the classical Hecke operators and show that in general, the Hecke operators map the theta series to a linear combination of theta series attached to different ideals, a generalization of the classical Eichler Commutation Relation.     

Vector-valued Maass-Poincaré series

Shortly before his death, Ramanujan wrote about his discovery of mock theta functions, functions with interesting analytic properties. Recently, Zweger showed that mock theta functions could be ``completed' to satisfy the transformation properties of a weight real analytic vector-valued modular form. Using Maass-Poincaré series, Bringmann and Ono proved the Andrews-Dragonette conjecture, establishing an exact formula for the coefficients of Ramanujan's mock theta function . In this paper we study vector-valued Maass-Poincaré series of all weights, and give their Fourier expansions.

     

Spherical metrics with conical singularities on 2-spheres

Suppose that \(\theta _1,\theta _2,\ldots ,\theta _n\) are positive numbers and \(n\ge 3\). We want to know whether there exists a spherical metric on \(\mathbb {S}^2\) with n conical singularities of angles \(2\pi \theta _1,2\pi \theta _2,\ldots ,2\pi \theta _n\). A sufficient condition was obtained by Mondello and Panov (Int Math Res Not 2016(16):4937–4995, 2016). We show that their condition is also necessary when we assume that \(\theta _1,\theta _2,\ldots ,\theta _n \not \in \mathbb {N}\).     

A separation property of the zeros of Eisenstein series for SL(2, Z)

Let E_k(z) be the Eisenstein series with weight k for the modulargroup SL(2, ). We prove that the zeros of E_k(eⁱ) interlace withthe zeros of E_k+12(eⁱ) on /2 < < 2/3. That is, any zeroof E_k(eⁱ) lies between two consecutive zeros of E_k+12(eⁱ) on/2 < < 2/3.     

Trace operators and theta series for Γ<Subscript>0,n</Subscript>[q] and Γ<Subscript>n</Subscript>[q]

We consider the action of suitable trace operators on non homogeneous theta series that are Siegel modular forms for the principal congruence subgroups of the symplectic group of level q, _n[q]. Then, we prove that modular forms for the Hecke subgroup of level q, _0,n[q], that are linear combination of such theta series, can also be expressed as combination of (homogeneous) theta series that are modular forms with respect to _0,n[q].     

Theta series of imaginary quadratic function fields

LetL be an imaginary quadratic extension of the rational function field . We prove transformation rules for the theta series corresponding to partial zeta functions of the extension .     

Limit theorems for random walks with dynamical random transitions

Let be an ergodic and conservative non-singular transformation of (, ,m) (thedynamic environment), let _w be a random probability on a locally compact second countable groupG, and define

Higher order asymptotic theory for normalizing transformations of maximum likelihood estimators

Suppose thatX _n =(X ₁,...X _n) is a collection ofm-dimensional random vectorsX _i forming a stochastic process with a parameter . Let be the MLE of . We assume that a transformationA( ) of has thek-thorder Edgeworth expansion (k=2,3). IfA extinguishes the terms in the Edgeworth expansion up tok-th-order (k2), then we say thatA is thek-th-order normalizing transformation. In this paper, we elucidate thek-th-order asymptotics of the normalizing transformations. Some conditions forA to be thek-th-order normalizing transformation will be given. Our results are very general, and can be applied to the i.i.d. case, multivariate analysis and time series analysis. Finally, we also study thek-th-order asymptotics of a modified signed log likelihood ratio in terms of the Edgeworth approximation.Research supported by the Office of Naval Research Contract N00014-91-J-1020.     

Petersson Products of Singular and Almost Singular Theta Series

We evaluate Petersson products of pairs of Siegel theta series of degree n attached to even lattices of rank mn. This allows us to get nice expressions, involving Dirichlet series and standard L-functions, for Petersson products of Siegel modular forms that are linear combinations of such theta series. Some applications to the theory of theta-liftings are also discussed.Mathematics Subject Classification (2000): 11E45, 11E10, 11F27, 11F46     

Algebraic Independence of Certain Values of Exponential Function

The algebraic independence of e^θ1,…,e^θs is proved, where θ1,… ,θs are certain gap series or power series of algebraic numbers, or certain transcendental continued fractions with algebraic elements.     

Asymptotic properties of the likelihood ratio of independent observations

For observations of independent random quantities in the series scheme we study the asymptotic behavior of the logarithm of the likelihood ratio. We find conditions for it to be asymptotically infinitely divisible. and in the parametric case we find for it the decomposition in which () is a random variable converging in distribution to an infinitely divisible law with zero mean, finite dispersion, and Kolmogorov function foru in some subset of the real axis, while _n (u; )0 in probability.Translated fromTeoriya Sluchaínykh Protsessov, Vol. 14, pp. 76–83, 1986.     

Absolute convergence of Fourier series in eigenfunctions of an elliptic operator

In this article we investigate absolute convergence of Fourier series in eigenfunctions of an m-th order elliptic operator on functions in the Besov class B _2, ^N/2 . We show that in terms of Besov classes the theorem of Peetre on absolute convergence of series in eigenfunctions in the class B _2,1 ^N/2 is best possible. We construct a function in B _2, ^N/2 whose Fourier series is absolutely divergent at any preassigned point.Translated from Matematicheskie Zametki, Vol. 19, No. 3, pp. 435–448, March, 1976.In conclusion, the author thanks Sh. A. Alimov for guiding this work.     

On metaplectic forms over function fields

We propose a concept of half integral weight in the global function field context, and construct natural families of functions with given weight. An analogue of Shimura correspondence (between weight 2 functions and weight ${\frac{3}{2}}$ functions) via theta series from “definite” quaternion algebras over function fields is then established. From the study of Fourier coefficients of these theta series, we arrive at a Waldspurger-type formula. This formula is then applied to L-series coming from elliptic curves over function fields.     

The t-coefficient method II: A new series expansion formula of theta function products and its implications

By means of Jacobi?s triple product identity and the t  -coefficient method, we establish a general series expansion formula with five free parameters for the product of arbitrary two Jacobi theta functions. It embodies the triple, quintuple, sextuple and septuple theta function product identities and the generalized Schröter formula. As further applications, we also set up a series expansion formula for the product of three theta functions. It not only generalizes Ewell?s and Chen–Chen–Huang?s octuple product identities, but also contains three cubic theta function identities due to Farkas–Kra and Ramanujan respectively and the Macdonald identity for the root system _A2$A_2$ as special cases. In the meantime, many other new identities including a new short expression of the triple theta series of Andrews are also presented.     

Conformal CR Positive Mass Theorem

In this paper, we prove the following version of conformal CR positive mass theorem: Suppose that \((N, J,\theta )\) and \((N, J,\hat{\theta }=e^{2f}\theta )\) are three-dimensional asymptotically flat pseudohermitian manifolds such that their Tanaka-Webster curvatures satisfy \(e^{2f}\hat{R}-R\ge 0.\) Then the p-mass of \((N, J, \theta )\) and \((N, J, \hat{\theta })\) satisfy \( m(J, \hat{\theta })-m(J, \theta )\ge 0, \) and equality holds if and only if \(\hat{\theta }=\theta \). We also prove that the p-mass is independent of the choice of the sequence of coordinates spheres.     

Temps d'arrÊt d'un système dynamique

Summary The most general positive integer-valued random variable v such that for a given bijective measure preserving transformation , the transformation ^v is still bijective and measure preserving is shown to be a (generally infinite) superposition of return times.

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Equipe de Recherche n 1 «< Processus stochastiques et applications»> dépendant de la section n 2 «Théories Physiques et Probabilités» associée au C.N.R.S.     

Jacobiformen und Thetareihen

We give a characterisation of Jacobi forms by classical modular forms from which we obtain dimension formulas for the spaces of Jacobi forms in certain cases. Then we consider the ordinary theta series to the quaternary quadratic forms of discriminant q² (q an odd prime) representing 2; these possess a natural continuation to Jacobi forms for which we give a sufficient condition of linear independence. If this condition is fulfilled and if there is no cusp form of weight 4 with respect to _o(q) which vanishes at the cusp 0 with a certain order then the classical theta series are also linear independent.     

Siegel's formula for genus 2

Let S be a semi-integral, symmetric, positive-definite m x m matrix; mn1. By the Siegel fundamental formula we mean the identity between the Siegel theta series of genus n, associated with the genus of the matrix S, and the correspond ing Eisenstein-Siegel series (C. L. Siegel, Lectures on the Analytical Theory of Quadratic Forms, 3rd rev. edition, Peppmüller, Göttingen, 1963). The validity of the mentioned formula for m/2n+1 is an open problem in the general case. In this paper we prove Siegel's formula for n=2, m=6.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 76, pp. 210–215, 1978.