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We compute the Fourier expansion of vector valued Eisenstein series for the Weil representation associated to an even lattice. To this end, we define certain twists by Dirichlet characters of the usual Eisenstein series associated to isotropic elements in the discriminant form of the underlying lattice. These twisted functions still form a generating system for the space of Eisenstein series but their Fourier coefficients have better multiplicative properties than the coefficients of the individual Eisenstein series. We adapt a method of Bruinier and Kuss to obtain algebraic formulas for the Fourier coefficients of the twisted Eisenstein series in terms of special values of Dirichlet L -functions and representation numbers modulo prime powers of the underlying lattice. In particular, we obtain that the Fourier coefficients of the individual Eisenstein series are rational numbers. Additionally, we show that the twisted Eisenstein series are eigenforms of the Hecke operators on vector valued modular forms introduced by Bruinier and Stein. [ABSTRACT FROM AUTHOR]

For a positive integer $d$ and a nonnegative number $\unicode[STIX]{x1D709}$ , let $N(d,\unicode[STIX]{x1D709})$ be the number of $\unicode[STIX]{x1D6FC}\in \overline{\mathbb{Q}}$ of degree at most $d$ and Weil height at most $\unicode[STIX]{x1D709}$. We prove upper and lower bounds on $N(d,\unicode[STIX]{x1D709})$. For each fixed $\unicode[STIX]{x1D709}>0$ , these imply the asymptotic formula $\log N(d,\unicode[STIX]{x1D709})\sim \unicode[STIX]{x1D709}d^{2}$ as $d\rightarrow \infty$ , which was conjectured in a question at Mathoverflow [ https://mathoverflow.net/questions/177206/ ]. [ABSTRACT FROM AUTHOR]

We re-write the character formulae of Adler and the first author in a form amenable to explicit computations in p-adic harmonic analysis, and use them to prove the stability of character sums for a modification of Reeder’s conjectural positive-depth, unramified, toral supercuspidal L-packets. [ABSTRACT FROM AUTHOR]

There are few constructions of square-integrable automorphic functions on metaplectic groups. Such functions may be obtained by the residues of certain Eisenstein series on covers of groups, "theta functions," but the Fourier coefficients of these residues are not well-understood, even for low degree covers of GL2. Patterson and Chinta--Friedberg--Hoffstein proposed conjectured relations for the Fourier coefficients of the GL2 quartic and sextic theta functions (resp.), each obtained from a conjectured equality of non-Eulerian Dirichlet series. In this article we propose a new framework for constructing specific L² metaplectic functions and for understanding these conjectures: descent integrals. We study descent integrals which begin with theta functions on covers of larger rank classical groups and use them to construct certain L² metaplectic functions on covers related to GL2. We then establish information about the Fourier coefficients of these metaplectic automorphic functions, properties which are consistent with the conjectures of Patterson and Chinta--Friedberg--Hoffstein. In particular, we prove that Fourier coefficients of the descent functions are arithmetic for infinitely many primes p. We also show that they generate a representation with nonzero projection to the space of theta.We conjecture that the descents may be used to realize the quartic and sextic theta functions. Moreover, this framework suggests that each of the conjectures of Patterson and Chinta--Friedberg--Hoffstein is the first in a series of relations between certain Fourier coefficients of two automorphic forms on different covering groups. [ABSTRACT FROM AUTHOR]

This paper is a significant part of a general project aimed to classify all irreducible representations of finite quasi-simple groups over an algebraically closed field, in which the image of at least one element is represented by an almost cyclic matrix (that is, a square matrix M of size n over a field F with the property that there exists α∈F such that M is similar to , where M1 is cyclic and 0≤k≤n). The paper focuses on the Weil representations of finite classical groups, as there is strong evidence that these representations play a key role in the general picture. [ABSTRACT FROM AUTHOR]

Over a p-adic field of odd residual characteristic, Gan and Savin proved a correspondence between the Bernstein components of the even and odd Weil representations of the metaplectic group and the components of the trivial representation of the equal rank odd orthogonal groups. In this paper, we extend their result to the case of even residual characteristic [ABSTRACT FROM AUTHOR]

The purpose of this paper is to investigate the eight-dimensional non-associative unital division algebras {0}≠Aover finite fields, admitting elementary abelian 2-group of automorphismsE≤Aut(A), of order 8, using tools from algebraic geometry. [ABSTRACT FROM PUBLISHER]

In this paper using the (p,k)-parametrization of theta functions and Eisenstein Series, developed by Alaca, Alaca and Williams, we obtain some new theta function identities and then use them to derive explicit formulae for the number of representations of a positive integer n by certain quadratic forms 6Σk=1 ak(x22k-1+x2k-1x2k+x22k/ in twelve variables where ak ∊ {1,2,4}. [ABSTRACT FROM AUTHOR]

We consider the minimization of theta functions θΛ(α)=∑p∈Λe−πα∣p∣2 amongst periodic configurations Λ⊂ℝd, by reducing the dimension of the problem, following as a motivation the case d = 3, where minimizers are supposed to be either the body-centered cubic or the face-centered cubic lattices. A first way to reduce dimension is by considering layered lattices, and minimize either among competitors presenting different sequences of repetitions of the layers, or among competitors presenting different shifts of the layers with respect to each other. The second case presents the problem of minimizing theta functions also on translated lattices, namely, minimizing (Λ,u)↦θΛ+u(α), relevant to the study of two-component Bose-Einstein condensates, Wigner bilayers and of general crystals. Another way to reduce dimension is by considering lattices with a product structure or by successively minimizing over concentric layers. The first direction leads to the question of minimization amongst orthorhombic lattices, whereas the second is relevant for asymptotics questions, which we study in detail in two dimensions. [ABSTRACT FROM AUTHOR]

Weil pairing plays an important role in elliptic curve cryptosystems, when we discuss about discrete log problem on an elliptic curve E defined over Fq, where q is a prime or a prime power. However, its calculation is very complex. In this paper, we give some property when we calculate the Weil pairing of an elliptic curve. [ABSTRACT FROM AUTHOR]

Humans are fast and accurate in categorizing complex natural images. It is, however, unclear what features of visual information are exploited by brain to perceive the images with such speed and accuracy. It has been shown that low-level contrast statistics of natural scenes can explain the variance of amplitude of event-related potentials (ERP) in response to rapidly presented images. In this study, we investigated the effect of these statistics on frequency content of ERPs. We recorded ERPs from human subjects, while they viewed natural images each presented for 70 ms. Our results showed that Weibull contrast statistics, as a biologically plausible model, explained the variance of ERPs the best, compared to other image statistics that we assessed. Our time-frequency analysis revealed a significant correlation between these statistics and ERPs' power within theta frequency band (~3-7 Hz). This is interesting, as theta band is believed to be involved in context updating and semantic encoding. This correlation became significant at~110 ms after stimulus onset, and peaked at 138 ms. Our results show that not only the amplitude but also the frequency of neural responses can be modulated with low-level contrast statistics of natural images and highlights their potential role in scene perception. [ABSTRACT FROM AUTHOR]

We prove some new theta-function identities for two continued fractions of Ramanujan which are analogous to those of Ramanujan-Göllnitz-Gordon continued fraction. Then these identities are used to prove new general theorems for the explicit evaluations of the continued fractions. [ABSTRACT FROM AUTHOR]

We study the space of vector valued theta functions for theWeil representation of a positive definite even lattice of rank two with fundamental discriminant. We work out the relation of this space to the corresponding scalar valued theta functions of weight one and determine an orthogonal basis with respect to the Petersson inner product. Moreover, we give an explicit formula for the Petersson norms of the elements of this basis. [ABSTRACT FROM AUTHOR]

Motivated by mathematical structures which arise in string vacua and gauge theories with $${{\mathcal{N}=2}}$$ supersymmetry, we study the properties of certain generalized theta series which appear as Fourier coefficients of functions on a twisted torus. In Calabi-Yau string vacua, such theta series encode instanton corrections from k Neveu-Schwarz five-branes. The theta series are determined by vector-valued wave-functions, and in this work we obtain the transformation of these wave-functions induced by Kontsevich-Soibelman symplectomorphisms. This effectively provides a quantum version of these transformations, where the quantization parameter is inversely proportional to the five-brane charge k. Consistency with wall-crossing implies a new five-term relation for Faddeev's quantum dilogarithm $${\Phi_b}$$ at b = 1, which we prove. By allowing the torus to be non-commutative, we obtain a more general five-term relation valid for arbitrary b and k, which may be relevant for the physics of five-branes at finite chemical potential for angular momentum. [ABSTRACT FROM AUTHOR]

We prove that there are only finitely many isometry classes of even lattices L of signature (2, n) for which the space of cusp forms of weight 1 + n/2 for the Weil representation of the discriminant group of L is trivial. We compute the list of these lattices. They have the property that every Heegner divisor for the orthogonal group of L can be realized as the divisor of a Borcherds product. We obtain similar classification results in greater generality for finite quadratic modules. [ABSTRACT FROM AUTHOR]

We develop an algorithm to compute Fourier expansions of vector valued modular forms for Weil representations. As an application, we compute explicit linear equivalences of special divisors on modular varieties of orthogonal type. We define three families of Hecke operators for Jacobi forms, and analyze the induced action on vector valued modular forms. The newspaces attached to one of these families are used to give a more memory efficient version of our algorithm. [ABSTRACT FROM AUTHOR]

In 1940, Weil [Sur les fonctions algébriques à corps de constantes finis, C. R. Acad. Sci. Paris 210 (1940) 592-594] proved the Riemann hypothesis for curves over finite fields. It follows from the Castelnuovo-Severi defect inequality concerning correspondences between algebraic curves (see [A. Mattuck and J. Tate, On the inequality of Castelnuovo-Severi, Abh. Math. Sem. Univ. Hamburg 22 (1958) 295-299]). An important step in the proof of Castelnuovo-Severi's defect inequality is the invariance of the Castelnuovo-Severi defect under trivial correspondences, so that the degree of divisors can be modified by adding multiples of trivial correspondences. In the number field case, the Weil distribution Δ(h) (see [A. Weil, Sur les formules explicites de la théorie des nombres, Izv. Akad. Nauk SSSR Ser. Mat. 36 (1972) 3-18]) corresponds to the Castelnuovo-Severi defect. Functions of the form ∑ξ∈K* f(ξx) with f in the Schwartz-Bruhat space S(픸)0 correspond to trivial correspondences. In this paper, we show that the two terms and in the Weil distribution can be chosen to be zero by adding "trivial correspondences" to h while keeping the Weil distribution essentially unchanged. As an application of this result, the Weil distribution is expressed as the spectral trace of an operator on a Hilbert space. [ABSTRACT FROM AUTHOR]

Given a discriminant form D of level N, there is a natural lifting which maps elliptic modular forms of level N to vector-valued modular forms for the Weil representation associated to D. We show that in some cases the zero component of a lifting of a newform f is just a scalar multiple of f. In order to do so, we split the lifting map into certain partial liftings corresponding to the prime powers exactly dividing N and then proceed to compute the zero components of these partial maps explicitly. As an application, we show that the L-function L풜(f, s) of a newform f and an ideal class 풜 as defined by Gross and Zagier can be written as a certain L-series of the lifting of f. [ABSTRACT FROM AUTHOR]

Let F he the field GF(q²) of q² elements, q odd, and let V he an W-vector space endowed with a nonsingular Hermitian form (p. Let a he the adjoint involutory antiautomorphism o/EndFV associated to the form, and let U(φ) he the corresponding unitary group. We ask whether the restrictions of the Weil representation of U(φ) to certain subgroups are multiplicity-free. These subgroups consist of the members of U(φ) in suhalgehras of the form FI+ N, where N is a a-stable commutative nilpotent suhalgehra of EndF V with the further property that N contains its annihilator. We give a necessary condition for multiplicity-freeness that depends on the dimensions of N and that annihilator. Moreover, the case that N is conjugate to its regular representation is completely settled. Several other classes of suhalgehra arc discussed in detail. [ABSTRACT FROM AUTHOR]

Let F be a finite field of odd cardinality, and let G = GL 2 ( F ) . The group G × G × G acts on F 2 ⊗ F 2 ⊗ F 2 via symplectic similitudes, and has a natural Weil representation. To answer a question raised by V. Drinfeld, we decompose that representation into irreducibles. We also decompose the analogous representation of GL 2 ( A ) , where A is a cubic algebra over F . [ABSTRACT FROM AUTHOR]

GivenFa locally compact, nondiscrete, non-archimedean field of characteristic ≠ 2 andRan integral domain such that a non-trivial smooth character χ:F → R×exists, we construct the (reduced) metaplectic group attached to χ andR. We show that it is in the expected cases a double cover of the symplectic group overF. Finally, we define a faithful infinite dimensionalR-representation of the metaplectic group analogue to the Weil representation in the complex case. [ABSTRACT FROM AUTHOR]

We continue the study of the Hrushovski-Kazhdan integration theory and consider exponential integrals. The Grothendieck ring is enlarged via a tautological additive character and hence can receive such integrals. We then define the Fourier transform in our integration theory and establish some fundamental properties of it. Thereafter, a basic theory of distributions is also developed. We construct the Weil representations in the end as an application. The results are completely parallel to the classical ones. [ABSTRACT FROM AUTHOR]

We prove a generalization of the classical converse theorem of Weil, allowing the twists by nontrivial Dirichlet characters to have arbitrary poles. [ABSTRACT FROM AUTHOR]

A new method for deriving formulas for the Jacobi theta functions is considered. [ABSTRACT FROM AUTHOR]

In this paper, we establish some new modular equations of degree 9. We also establish several new P-Q mixed modular equations involving theta-functions which are similar to those recorded by Ramanujan in his notebooks. As an application, we establish some new general formulas for explicit evaluations of a Remarkable product of theta-functions. [ABSTRACT FROM AUTHOR]

Abstract: We prove a new converse theorem for Borcherdsʼ multiplicative theta lift which improves the previously known results. To this end we develop a newform theory for vector valued modular forms for the Weil representation, which might be of independent interest. We also derive lower bounds for the ranks of the Picard groups and the spaces of holomorphic top degree differential forms of modular varieties associated to orthogonal groups. [Copyright &y& Elsevier]

In this paper we are interested in the geometric local theta correspondence at the Iwahori level for dual reductive pairs (G,H) of type II over a non-Archimedean field of characteristic p ≠ 2 in the framework of the geometric Langlands program. We consider the geometric version of the IH ×IGinvariants of the Weil representation SIH×IG as a bimodule under the action of Iwahori-Hecke algebras HIG and HIH and we give some partial geometric description of the corresponding category under the action of Hecke functors. We also define geometric Jacquet functors for any connected reductive group G at the Iwahori level and we show that they commute with the Hecke action of the HIL-subelgebra of HIG for a Levi subgroup L. [ABSTRACT FROM AUTHOR]

Abstract: We find all irreducible constituents of the Weil representation of a unitary group of rank m associated to a ramified quadratic extension A of a finite, commutative, local and principal ring R of odd characteristic. We show that this Weil representation is multiplicity free with monomial irreducible constituents. We also find the number of these constituents and describe them in terms of Clifford theory with respect to a congruence subgroup. We find all character degrees in the special case when R is a field. [Copyright &y& Elsevier]

Abstract: Let be the group of isometries of a symplectic vector space V over a finite field F of odd cardinality. The group possesses distinguished representations—the Weil representations. We show that they are compatible with base change in the sense of Shintani for a finite extension . The result is also true for the group of similitudes of V. [Copyright &y& Elsevier]

We study the prolongation of various geometric structures on a fibered manifold with respect to the Weil functor TA or a fiber product preserving bundle functor F on the category of all fibered manifolds with m-dimensional bases and their morphisms with local diffeomorphisms as base maps. Special attention is paid to connections, vector bundles, principal bundles and weak principal bundles. [ABSTRACT FROM AUTHOR]

Abstract: Let be the ring of 2-by-2 matrices with coefficients in a finite field and let be the subset of of symmetric matrices. In [4], M. Kuroda determined the number of solutions for the equation , (). In this paper, we consider the problem for and give a complete formula for the number of solutions of the equation. In the process, we also give an -form of for any n. [Copyright &y& Elsevier]

The mock theta functions introduced by Ramanujan have been studied by many authors both analytically and combinatorially. The combinatorial interpretations that are known for some of them are quite different in nature. In this paper we present combinatorial interpretation as two-line array for many of the classical mock theta functions. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we obtain a second term identity of the regularized Siegel–Weil formula for the unitary dual pair , and we use this to derive a Rallis inner product formula for theta lifting from to . By choosing a good Schwartz function in the inner product formula, we obtain a special value formula for some Hecke L-function at in terms of the norm of a theta function on . [Copyright &y& Elsevier]

Abstract: Let F be a p-adic field with p odd. Quadratic base change and theta-lifting are shown to be compatible for supercuspidal representations of . The argument involves the theory of types and the lattice-model of the Weil representation. [Copyright &y& Elsevier]

The metaplectic group is defined by its oscillator or Weil representation. Using the types of the Weil representations we define two Hecke algebras that govern two Bernstein's components containing the even and the odd Weil representation, respectively. [ABSTRACT FROM AUTHOR]

The Weil representation of a real symplectic group Sp(2n, ℝ) admits a canonical extension to a holomorphic representation of a certain complex semigroup consisting of Lagrangian linear relations (this semigroup includes the Olshanskii semigroup). We obtain the explicit realization of the Weil representation of this semigroup in the Cartier model, that is, in the space of smooth sections of a certain line bundle on the 2n-dimensional torus T2n. We show that operators of the representation are integral operators whose kernels are theta-functions on T4n. We also extend this construction to a functor from a certain category of Lagrangian linear relations between symplectic vector spaces of different dimensions to a category of integral operators acting on sections of line bundles on the tori. [ABSTRACT FROM AUTHOR]

We reinterpret the residue map for the Brauer group of a smooth variety using a root stack construction and Weil restriction for algebraic stacks, and apply the result to find a geometric representative of the residue of the Brauer class associated to a conic bundle. [ABSTRACT FROM AUTHOR]

This work presents a novel way to obtain the associated Romanovski functions Rn,m(x) with n ≥ m in the three separate regions in terms of n and m. We obtain the raising and lowering relations with respect to the both indices, simultaneously, in the three regions. Then, a reducible Weil representation of the real Lie algebra sp(4) is realized in the space of complex-valued smooth functions on H2 × S1 by differential forms for the Cartan-Weyl basis. Its invariant subspace is the second rare instance of the highest weight irreducible representation of sp(4) all whose weight spaces are one-dimensional. [ABSTRACT FROM AUTHOR]

This work presents a novel way to obtain the associated Romanovski functions Rn,m(x) with n ≥ m in the three separate regions in terms of n and m. We obtain the raising and lowering relations with respect to the both indices, simultaneously, in the three regions. Then, a reducible Weil representation of the real Lie algebra sp(4) is realized in the space of complex-valued smooth functions on H2 × S1 by differential forms for the Cartan-Weyl basis. Its invariant subspace is the second rare instance of the highest weight irreducible representation of sp(4) all whose weight spaces are one-dimensional. [ABSTRACT FROM AUTHOR]