Narrow Results

In this paper we provide a construction of theta series on the real symplectic group of signature (1,1) or the 4-dimensional hyperbolic space. We obtain these by considering the restriction of some vector-valued singular theta series on the unitary group of signature (2,2) to this indefinite symplectic group. Our (vector-valued) theta series are proved to have algebraic Fourier coefficients, and lead to a new explicit construction of automorphic forms generating quaternionic discrete series representations and automorphic functions on the hyperbolic space.

We study vector-valued Siegel modular forms of genus 2 on the three level 2 groups Γ[2] ◁ Γ₁[2] ◁ Γ₀[2] ⊂ Sp(4, ℤ). We give generating functions for the dimension of spaces of vector-valued modular forms, construct various vector-valued modular forms by using theta functions and describe the structure of certain modules of vector-valued modular forms over rings of scalar-valued Siegel modular forms.

For a positive integer m, let R be either the ring ℤ_2m of integers modulo 2m or the quaternionic ring Σ_2m = ℤ_2m + αℤ_2m + βℤ_2m + γℤ_2m with α = 1 + î, β = 1 + ĵ and , where are elements of the ring ℍ of real quaternions satisfying , , and . In this paper, we obtain Jacobi forms (or Siegel modular forms) of genus r from byte weight enumerators (or symmetrized byte weight enumerators) in genus r of Type I and Type II codes over R. Furthermore, we derive a functional equation for partial Epstein zeta functions, which are summands of classical Epstein zeta functions associated with quadratic forms.

Hecke proved that the theta series of a positive definite even unimodular lattice is a polynomial of the well-known Essenstein series E₄(z) and the Ramanujan series Δ₂₄(z). A natural question is what kind of polynomials in E₄(z) and Δ₂₄(z) could be the theta series of positive definite even unimodular lattices. In this paper, we find two combinatorial identities on the theta series of the root lattices of the finite-dimensional simple Lie algebras of type D_4n and the cosets in their integral duals, in terms of E₄(z) and Δ₂₄(z). Using these two identities, we prove that three families of weighted symmetric polynomials of two fixed families of polynomials of E₄(z) and Δ₂₄(z) are the theta series of certain positive definite even unimodular lattices, obtained by gluing finitely many copies of the root lattices of the finite-dimensional simple Lie algebras of type D_2n. The results also show that the full permutation groups are the hidden symmetry of the theta series of certain unimodular lattices.

We show that the theta series attached to the -lattice for any positive integer divisible by 8 can be explicitly expressed as a finite rational linear combination of products of two Eisenstein series.

We apply the Hecke operators T(p) and to a degree n theta series attached to a rank 2k ℤ-lattice L, n ≤ k, equipped with a positive definite quadratic form in the case that L/pL is hyperbolic. We show that the image of the theta series under these Hecke operators can be realized as a sum of theta series attached to certain closely related lattices, thereby generalizing the Eichler Commutation Relation (similar to some work of Freitag and of Yoshida). We then show that the average theta series (averaging over isometry classes in a given genus) is an eigenform for these operators. We show the eigenvalue for T(p) is ∊(k - n, n), and the eigenvalue for T′_j(p²) (a specific linear combination of T₀(p²),…,T_j(p²)) is p^{j(k-n)+j(j-1)/2}β(n,j)∊(k-j,j) where β(*,*), ∊(*,*) are elementary functions (defined below).

In this paper we study a family of quaternary forms which represent almost all binary forms of a certain type. The result follows from the representation number by the genus of ternary forms and a correspondence among theta series.

We apply the Hecke operators T(p)² and (1 ≤ j ≤ n ≤ 2k) to a degree n theta series attached to a rank 2k ℤ-lattice L equipped with a positive definite quadratic form in the case that L/pL is regular. We explicitly realize the image of the theta series under these Hecke operators as a sum of theta series attached to certain sublattices of , thereby generalizing the Eichler Commutation Relation. We then show that the average theta series (averaging over isometry classes in a given genus) is an eigenform for these operators. We explicitly compute the eigenvalues on the average theta series, extending previous work where we had the restrictions that χ(p) = 1 and n ≤ k. We also show that for j > k when χ(p) = 1, and for j ≥ k when χ(p) = -1, and that θ(gen L) is an eigenform for T(p)².

There are six theta constants over the Hurwitz quaternions on the quaternion half-space of degree 2. The paper describes the behavior of these theta constants under the transpose mapping, which can be derived from the Fourier expansions. The results are applied to the theta series of the first and second kind.

The main purpose of this paper is to give an intrinsic interpretation of the space Θ(D) generated by the binary theta series ϑ_f attached to the positive binary quadratic forms f whose discriminant has the form D(f) = D/t², for some integer t. It turns out that , the space of modular forms of weight 1 and of level |D| which have complex multiplication (CM) by their Nebentypus character . As an application, we obtain a structure theorem of the space . The proof of this theorem rests on the results of [The space of binary theta series, Ann. Sci. Math. Québec36 (2012) 501–534] together with a characterization of the newforms f which have CM by their Nebentypus character in terms of properties of the associated Deligne–Serre Galois representationρ_f.

We consider a generalization of Jacobi theta series and show that every such function is a quasi-Jacobi form. Under certain conditions we establish transformation laws for these functions with respect to the Jacobi group and prove such functions are Jacobi forms. In establishing these results, we construct other functions which are also Jacobi forms. These results are motivated by applications in the theory of vertex operator algebras.

In Mathematische Werke, Hecke defines the operator $T_p$ and describes their utility in conjunction with theta series of quadratic forms. In particular, he shows that the image of theta series associated to classes of binary quadratic forms in CL$(\mathrm{\Delta })$ is again a theta series associated to a collection of forms in CL$(\mathrm{\Delta })$. We state and prove an explicit formula for the action of $T_p$ on a binary quadratic form of negative discriminant.

In this work, we use the Rankin–Selberg method to obtain results on the analytic properties of the standard $L$-function attached to vector-valued Siegel modular forms. In particular we provide a detailed description of its possible poles and obtain a non-vanishing result of the twisted $L$-function beyond the usual range of absolute convergence. Our results include also the case of metaplectic Siegel modular forms. We remark that these results were known in this generality only in the case of scalar weight Siegel modular forms. As an interesting by-product of our work we establish the cuspidality of some theta series.

We consider the action of Hecke-type operators on Hilbert–Siegel theta series attached to lattices of even rank. We show that the average Hilbert–Siegel theta series are eigenforms for these operators, and we explicitly compute the eigenvalues.

The purpose of this paper is to establish several families of identities involving universal mock theta functions and express them in terms of Appell–Lerch sums. The subsequent consequences of these results are a variety of interesting identities on theta series and classical mock theta functions.

For $\left|q\right|<1$, define $f_i={\prod }_{n=1}^{\infty }(1-q^{in})$, and let $(A(q),B(q))$ be any of the pairs $(f_1^4,\frac{f_1^8}{f_2^2}),$$(f_1^4,\frac{f_1^{10}}{f_3^2}),$$(f_1^6,\frac{f_2^4}{f_1^2}),$$(f_1^6,\frac{f_1^{14}}{f_2^4}),$$(f_1^{10},\frac{f_2^6}{f_1^2}),$$(f_1^{14},\frac{f_3^5}{f_1}),(f_1^{14},\frac{f_2^8}{f_1^2}).$ For any such pair $(A(q),B(q))$, define the sequences $\{a(n)\}$ and $\{b(n)\}$ to be the coefficients of $q^n$ of $A(q)$ and $B(q)$, respectively. Then for each pair it is shown that $a(n)$ vanishes if and only if $b(n)$ vanishes. In each case, a criterion is given which states precisely when $a(n)=b(n)=0$. Moreover, for the pairs $(f_1^{26},\frac{f_3^9}{f_1}),$$(f_1^{26},\frac{f_2^{16}}{f_1^6})$ it is shown that $a(n)=b(n)=0$ if $12n+13$ satisfies a criteria of Serre for $a(n)=0$.

For a paramodular group of any degree and square free level we study the Hecke algebra and the boundary components. We define paramodular theta series and show that for square free level and large enough weight they generate the space of cusp forms (basis problem), using the doubling or pullback of Eisenstein series method. For this we give a new geometric proof of Garrett’s double coset decomposition which works in our more general situation.

Well-known results of Lagrange and Jacobi prove that every $m\in \mathbb{N}$ can be expressed as a sum of four integer squares, and the number $r(m)$ of such representations can be given by an explicit formula in $m$. In this paper, we prove that the only real quadratic number field for which the sum of four squares is universal is $\mathbb{Q}(\sqrt{5})$. We provide explicit formulas for $r(m)$ for $K=\mathbb{Q}(\sqrt{2})$ and $K=\mathbb{Q}(\sqrt{5})$. We then consider the theta series of the sum of four squares over any real quadratic number field, providing explicit upper and lower bounds for the Eisenstein coefficients. Last, we include examples of the complete theta series decomposition of the sum of four squares over $\mathbb{Q}(\sqrt{3})$, $\mathbb{Q}(\sqrt{13})$ and $\mathbb{Q}(\sqrt{17})$.

In this paper, near-miss identities for the number of representations of some integral ternary quadratic forms with congruence conditions are found and proven. The genus and spinor genus of the corresponding lattice cosets are then classified. Finally, a complete genus and spinor genus classification for all conductor 2 lattice cosets of 2-adically unimodular lattices is given.

In this paper, we consider an imaginary quadratic field $K=\mathbb{Q}(\sqrt{-m})$ with $-m\equiv 3$ (mod $4$). In particular, we study the ring of integers corresponding to the field $K$ and visualize the form of ${𝜖}_K/p{𝜖}_K$. We also consider lattices over the ring of integers ${𝜖}_K$ and discuss the theta series to see its relation with the weight enumerator. As a consequence, we will see how the theta series differs for different $m$ and $m^{\prime }$.