Theta Series Research Articles

Eichler orders and Jacobi forms of squarefree level

For all Eichler orders with a same squarefree level in a definite quaternion algebra over the field of rational numbers, we prove that a weighted sum of Jacobi theta series associated to these orders is a Jacobi Eisenstein series. Multiply the Fourier coefficients of the latter by a power of 2 to get our modified Hurwitz class numbers. As its corollaries, the modified Hurwitz class numbers can be used to calculate the traces of Brandt matrices and the type number of the Eichler orders.

Siegel theta series for indefinite quadratic forms

The modular transformation behavior of theta series for indefinite quadratic forms is well understood in the case of elliptic modular forms due to Vignéras, who deduced that solving a differential equation of second order serves as a criterion for modularity. In this paper, we will give a generalization of this result to Siegel theta series.

On the modular forms of weight 1/2 over algebraic number fields

Serre and Stark succeeded in deciding a basis of the space of modular forms of weight 1/2 over the rational number field. Achimescu and Saha generalized their result to the case of modular forms of weight 1/2 over totally real algebraic number fields. Gove also solved this problem in the case of modular forms of weight 1/2 over imaginary quadratic fields.In this paper, we determine an explicit basis of the space of modular forms of weight 1/2, level c and character ψ over algebraic number fields. We prove our assertion using their arguments and Shimura's transformation formula of theta series over algebraic number fields.

Hecke operators on Hilbert–Siegel 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.

Geometry of the eigencurve at CM points and trivial zeros of Katz p-adic L-functions

The primary goal of this paper is to investigate the geometry of the p-adic eigencurve at a point f corresponding to a weight one cuspidal CM theta series θψ irregular at the prime number p. We show that f belongs to exactly three or four irreducible components and study their intersection multiplicities. In particular, we introduce an anti-cyclotomic L-invariant L-(ψ-) and show that the congruence ideal of the CM component Θψ has a simple zero at f if and only if L-(ψ-) does not vanish. Further, using Roy's Strong Six Exponential Theorem we show that at least one amongst L-(ψ-) and L-(ψ-−1) is non-zero. Combined with a divisibility proved by Hida and Tilouine, we deduce that the anti-cyclotomic Katz p-adic L-function of ψ- has a simple (trivial) zero at s=0 if L-(ψ-) is non-zero, which can be seen as an anti-cyclotomic analogue of a result of Ferrero and Greenberg. Finally, we propose a formula for the linear term of the two-variable Katz p-adic L-function of ψ- at s=0 extending a conjecture of Gross.

Linear inequalities concerning partitions into distinct parts

Linear inequalities involving Euler’s partition function p(n) have been the subject of recent studies. In this article, we consider the partition function Q(n) counting the partitions of n into distinct parts. Using truncated theta series, we provide four infinite families of linear inequalities for Q(n) and partition theoretic interpretations for these results.

Congruences for level $1$ cusp forms of half-integral weight

Suppose that $\ell \geq 5$ is prime. For a positive integer $N$ with $4 \mid N$, previous works studied properties of half-integral weight modular forms on $\Gamma_0(N)$ which are supported on finitely many square classes modulo $\ell$, in some cases proving that these forms are congruent to the image of a single variable theta series under some number of iterations of the Ramanujan $\Theta$-operator. Here, we study the analogous problem for modular forms of half-integral weight on $\operatorname{SL}_{2}(\mathbb{Z})$. Let $\eta$ be the Dedekind eta function. For a wide range of weights, we prove that every half-integral weight modular form on $\operatorname{SL}_{2}(\mathbb{Z})$ which is supported on finitely many square classes modulo $\ell$ can be written modulo $\ell$ in terms of $\eta^{\ell}$ and an iterated derivative of $\eta$.

Self-dual codes over $${\mathbb {F}}_2[u]/\langle u^4 \rangle $$ and Jacobi forms over a totally real subfield of $${\mathbb {Q}}(\zeta _8)$$

Let $$K={\mathbb {Q}}(\zeta _8)$$ be the cyclotomic field over $${\mathbb {Q}}$$ of the extension degree 4. We give an integral lattice construction on $${\mathbb {Q}}(\zeta _8)$$ induced from codes over the ring $${\mathcal {R}}= {\mathbb {F}}_2[u]/\langle u^4 \rangle $$ . We define a theta series using these lattices and discuss its relation with the complete weight enumerator of a code over $${\mathcal {R}}$$ . If C is a Type II code of length l, we find that the complete weight enumerator of C gives a Jacobi form of weight l and the index 2l over the maximal totally real subfield $$k={\mathbb {Q}}(\zeta _8+\zeta _8^{-1})$$ of K. Also, we see that Hilbert–Siegel modular form of weight n and genus g can be seen in terms of the complete joint weight enumerator of codes $$C_j$$ , for $$1\le j\le g$$ over $${\mathcal {R}}$$ .

Theta series and number fields: theorems and experiments

Let d and n be positive integers and let K be a totally real number field of discriminant d and degree n. We construct a theta series $$\theta _{K} \in {\mathcal {M}}_{d,n}$$ , where $${\mathcal {M}}_{d,n}$$ is a space of modular forms defined in terms of n and d. Moreover, if d is square free and n is at most 4 then $$\theta _{K}$$ is a complete invariant for K. We also investigate whether or not the collection of $$\theta $$ -series, associated to the set of isomorphism classes of quartic number fields of a fixed squarefree discriminant d, is a linearly independent subset of $${\mathcal {M}}_{d,4}$$ . This is known to be true if the degree of the number field is less than or equal to 3. We give computational and heuristic evidence suggesting that in degree 4 these theta series should be independent as well.

Harmonic theta series and the Kodaira dimension of 𝒜6

We construct a basis of the space ${\text S}_{14}({\text{Sp}}_{12}({\mathbb Z}))$ of Siegel cusp forms of degree $6$ and weight $14$ consisting of harmonic theta series. One of these functions has vanishing order $2$ at the boundary which implies that the Kodaira dimension of $\mathcal{A}_6$ is non-negative.

Automorphic forms for some even unimodular lattices

We look at genera of even unimodular lattices of rank 12 over the ring of integers of \({{\mathbb {Q}}}(\sqrt{5})\) and of rank 8 over the ring of integers of \({{\mathbb {Q}}}(\sqrt{3})\), using Kneser neighbours to diagonalise spaces of scalar-valued algebraic modular forms. We conjecture most of the global Arthur parameters, and prove several of them using theta series, in the manner of Ikeda and Yamana. We find instances of congruences for non-parallel weight Hilbert modular forms. Turning to the genus of Hermitian lattices of rank 12 over the Eisenstein integers, even and unimodular over \({{\mathbb {Z}}}\), we prove a conjecture of Hentschel, Krieg and Nebe, identifying a certain linear combination of theta series as an Hermitian Ikeda lift, and we prove that another is an Hermitian Miyawaki lift.

Twist Formulas for One-Row Colored A2 Webs and $\mathfrak {s}\mathfrak {l}_{3}$ Tails of (2, 2m)-Torus Links

The $\mathfrak {s}\mathfrak {l}_{3}$ colored Jones polynomial $J_{\lambda }^{\mathfrak {s}\mathfrak {l}_{3}}(L)$ is obtained by coloring the link components with two-row Young diagram λ. Although it is difficult to compute $J_{\lambda }^{\mathfrak {s}\mathfrak {l}_{3}}(L)$ in general, we can calculate it by using Kuperberg’s A2 skein relation. In this paper, we show some formulas for twisted two strands colored by one-row Young diagram in A2 web space and compute $J_{(n,0)}^{\mathfrak {s}\mathfrak {l}_{3}}(T(2,2m))$ for an oriented (2,2m)-torus link. These explicit formulas derives the $\mathfrak {s}\mathfrak {l}_{3}$ tail of T(2,2m). They also give explicit descriptions of the $\mathfrak {s}\mathfrak {l}_{3}$ false theta series with one-row coloring because the $\mathfrak {s}\mathfrak {l}_{2}$ tail of T(2,2m) is known as the false theta series.

Quantum modularity of partial theta series with periodic coefficients

Abstract We explicitly prove the quantum modularity of partial theta series with even or odd periodic coefficients. As an application, we show that the Kontsevich–Zagier series ℱ t ⁢ ( q ) {\mathscr{F}_{t}(q)} which matches (at a root of unity) the colored Jones polynomial for the family of torus knots T ⁢ ( 3 , 2 t ) {T(3,2^{t})} , t ≥ 2 {t\geq 2} , is a weight 3 2 {\frac{3}{2}} quantum modular form. This generalizes Zagier’s result on the quantum modularity for the “strange” series F ⁢ ( q ) {F(q)} .

On the space of theta functions whose levels are square-free

Hecke conjectured that an explicit set of theta series obtained from a quaternion algebra defined over ℚ ramified at a prime N is a basis of a space of holomorphicmodular forms of weight 2 for the Hecke congruence group Γ0(N). However, Eichler noticed that Hecke’s conjecture is not true in general. Hence it is natural to ask the dimension of the subspace of M2(Γ0(N)) spanned by the theta series, and this question is called Hecke’s basis problem, which we have shown an answer in [K. Sugiyama, On the space of theta functions for a prime level, Comment. Math. Univ. St. Pauli, 67(1):66–81, 2019]. In this paper, we generalize the results for a square-free positive integer N.

On the standard L-function attached to quaternionic modular forms

In this paper we study the analytic properties of the standard L-function attached to vector valued quaternionic modular forms using the Rankin-Selberg method. This involves the construction of vector valued theta series, which we obtain by applying some differential operators on Jacobi-theta series studied by Krieg. Such differential operators are obtained from the Howe-Weyl duality for the pair Spn(C)×GL2n(C).

On the Rankin–Selberg method for vector-valued Siegel modular forms

In this work, we use the Rankin–Selberg method to obtain results on the analytic properties of the standard [Formula: see text]-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 [Formula: see text]-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.

Classical Theta Lifts for Higher Metaplectic Covering Groups

The classical theta correspondence establishes a relationship between automorphic representations on special orthogonal groups and automorphic representations on symplectic groups or their double covers. This correspondence is achieved by using as integral kernel a theta series on the metaplectic double cover of a symplectic group that is constructed from the Weil representation. There is also an analogous local correspondence. In this work we present an extension of the classical theta correspondence to higher degree metaplectic covers of symplectic and special orthogonal groups. The key issue here is that for higher degree covers there is no analogue of the Weil representation, and additional ingredients are needed. Our work reflects a broader paradigm: constructions in automorphic forms that work for algebraic groups or their double covers should often extend to higher degree metaplectic covers.

Tutte Polynomial, Complete Invariant, and Theta Series

In this study, we present two results that relate Tutte polynomials. First, we provide new and complete polynomial invariants for graphs. We note that the number of variables of our polynomials is one. Second, let $$L_1$$ and $$L_2$$ be two non-isomorphic lattices. We state that $$L_1$$ and $$L_2$$ are theta series equivalent if those theta series are the same. The problem of identifying theta series equivalent lattices is discussed in Prof. Conway’s book The Sensual (Quadratic) Form with the title “Can You Hear the Shape of a Lattice?” In this study, we present a method to find theta series equivalent lattices using matroids and their Tutte polynomials.

Hermitian Theta Series and Maa\xdf Spaces Under the Action of the Maximal Discrete Extension of the Hermitian Modular Group

Let Gamma _n(mathcal {scriptstyle {O}}_{mathbb {K}}) denote the Hermitian modular group of degree n over an imaginary quadratic number field mathbb {K} and Delta _{n,mathbb {K}}^* its maximal discrete extension in the special unitary group SU(n,n;mathbb {C}). In this paper we study the action of Delta _{n,mathbb {K}}^* on Hermitian theta series and Maaß spaces. For n=2 we will find theta lattices such that the corresponding theta series are modular forms with respect to Delta _{2,mathbb {K}}^* as well as examples where this is not the case. Our second focus lies on studying two different Maaß spaces. We will see that the new found group Delta _{2,mathbb {K}}^* consolidates the different definitions of the spaces.

On q-series identities for false theta series

We prove several infinite families of q-series identities for false theta functions and related series. These identities are motivated by considerations of characters of modules of vertex operator superalgebras and of quantum dilogarithms. We also obtain closely related modular identities of the Göllnitz-Gordon-Andrews type. As a byproduct of our identities, we establish several identities for the Rogers dilogarithm function coming from multi q-hypergeometric series with “double poles”.