Alexander Polishchuk
We describe a family of $q$-series generating the space of weight 1 modular forms coming from indefinite binary quadratic forms and study linear relations between these series.
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Generalizing math.AG/0003076 we express universal triple Massey products between line bundles on elliptic curves in terms of Hecke's indefinite theta series. We show that all Hecke's indefinite theta series arise in this way.
We prove an algebraic property of the elements defining Hecke operators on period polynomials associated with modular forms, which implies that the pairing on period polynomials corresponding to the Petersson scalar product of modular forms is Hecke equivariant. As a consequence of this proof, we derive two indefinite theta series identities which can be seen as analogues of Jacobi's formula for the theta series associated with the sum of four squares.
Mock theta functions and indefinite modular forms III
该文属于数论领域，聚焦含不定二次型的一类函数，开展其模变换性质的研究工作。
Given a Siegel theta series and a prime p not dividing the level of the theta series, we apply to the theta series the n+1 Hecke operators that generate the local Hecke algebra at p. We show that the average theta series is an eigenform and we compute the eigenvalues.
We consider the action of Hecke-type operators on Hilbert-Siegel theta series attached to lattices of even rank. We show that average Hilbert-Siegel theta series are eigenforms for these operators, and we explicitly compute the eigenvalues.
We study generating functions for Lusztig's $t$-analog of weight multiplicities associated to integrable highest weight representations of the simplest affine Lie algebra $A_1^{(1)}$. At $t=1$, these reduce to the {\em string functions} of $A_1^{(1)}$, which were shown by Kac and Peterson to be related to certain Hecke indefinite modular forms. Using their methods, we obtain a description of the general $t$-string function; we show that its values can be realized as radial averages of a certain extension of the Hecke indefinite modular form.
We show that indefinite theta series on cones converge and provide an explicit modular completion. Our completion rests on a convolution of the Gaussian with a piecewise constant function supported on the cone. Our main innovation is to formulate this convolution in terms of euclidean geometry as opposed to hyperbolic geometry. This change of perspective allows us to establish essential asymptotic estimates without further difficulty.
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.
We explore a natural action of Hecke operators acting on formal sums of optimal embeddings of real quadratic orders into Eichler orders. By associating an optimal embedding to its root geodesic on the corresponding Shimura curve, we can consider the signed intersection number of pairs of embeddings. Using the Hecke operators and the intersection pairing, we construct a generating series that is demonstrated to be a classical modular form of weight two.
We develop a setting in which one can evaluate certain Hecke-Rogers series in terms of false theta functions. We apply our setting to recent false theta function identities of Chan and Kim as well as Andrews and Warnaar.
In this paper, we compute the Zwegers's modification of the mock theta functions $Φ^{[m,0] \, \ast}$ and study the modular transformation properties of the indefinite modular forms which appear in the explicit formula for the modified functions $\tildeΦ^{[m,0] \, \ast}$.
On the convergence of certain indefinite theta series
本文给出了符号为 (n,2) 的内空间的不定 theta 级数收敛性的初等证明，并证明了关联条件的必要性。
In this paper we consider an example of a Maass waveform which was constructed by Cohen from a function $σ$, studied by Andrews, Dyson and Hickerson, and it's companion $σ^*$. We put this example in a more general framework.
By developing a connection between partial theta functions and Appell-Lerch sums, we find and prove a formula which expresses Hecke-type double sums in terms of Appell-Lerch sums and theta functions. Not only does our formula prove classical Hecke-type double sum identities such as those found in work Kac and Peterson on affine Lie Algebras and Hecke modular forms, but once we have the Hecke-type forms for Ramanujan's mock theta functions our formula gives straightforward proofs of many of the classical mock theta function identities. In particular, we obtain a new proof of the mock theta conj…
We obtain four Hecke-type double sums for three of Ramanujan's third order mock theta functions. We discuss how these four are related to the new mock theta functions of Andrews' work on $q$-orthogonal polynomials and Bringmann, Hikami, and Lovejoy's work on unified Witten-Reshetikhin-Turaev invariants of certain Seifert manifolds. We then prove identities between these new mock theta functions by first expressing them in terms of the universal mock theta function.
Köhler's Conjecture on Hecke Theta Series of Weight One
通过从二次域特征诱导的二维Galois表示推导，证明了Günter Köhler关于Hecke theta级数中二次域歧义的猜想。
We derive explicit formulas for the action of the Hecke operator $T(p)$ on the genus theta series of a positive definite integral quadratic form and prove a theorem on the generation of spaces of Eisenstein series by genus theta series. We also discuss connections of our results with Kudla's matching principle for theta integrals.
We study the properties of Eisenstein-Kronecker numbers, which are related to special values of Hecke $L$-function of imaginary quadratic fields. We prove that the generating function of these numbers is a reduced (normalized or canonical in some literature) theta function associated to the Poincare bundle of an elliptic curve. We introduce general methods to study the algebraic and $p$-adic properties of reduced theta functions for CM abelian varieties. As a corollary, when the prime $p$ is ordinary, we give a new construction of the two-variable $p$-adic measure interpolating special values …
It was shown in previous work that the one-variable $\widehatμ$-function defined by Zwegers (and Zagier) and his indefinite theta series attached to lattices of signature $(r\!+\!1,1)$ are both Heisenberg harmonic Maaß-Jacobi forms. We extend the concept of Heisenberg harmonicity to Maaß-Jacobi forms of arbitrary many elliptic variables, and produce indefinite theta series of "product type" for non-degenerate lattices of signature $(r\!+\!s,s)$. We thus obtain a clean generalization of $\widehatμ$ to these negative definite lattices. From restrictions to torsion points of Heisenberg harmonic M…
We obtain two-variable Hecke-Rogers identities for three universal mock theta functions. This implies that many of Ramanujan's mock theta functions, including all the third order functions, have a Hecke-Rogers-type double sum representation. We find new generating function identities for the Dyson rank function, the overpartition rank function, the M2-rank function and related spt-crank functions. Results are proved using the theory of basic hypergeometric functions.
Theta series for lattices with indefinite signature $(n_+,n_-)$ arise in many areas of mathematics including representation theory and enumerative algebraic geometry. Their modular properties are well understood in the Lorentzian case ($n_+=1$), but have remained obscure when $n_+\geq 2$. Using a higher-dimensional generalization of the usual (complementary) error function, discovered in an independent physics project, we construct the modular completion of a class of `conformal' holomorphic theta series ($n_+=2$). As an application, we determine the modular properties of a generalized Appell-…
Mock theta functions and indefinite modular forms
本文研究从$D(2,1;a)$余根格得到的带符号mock theta函数的模变换性质，利用Zwegers修正理论证明其线性张成在$SL_2(\mathbf{Z})$下不变。
We study the Hecke algebra modules arising from theta correspondence between certain Harish-Chandra series for type I dual pairs over finite fields. For the product of the pair of Hecke algebras under consideration, we show that there is a generic Hecke algebra module whose specializations at prime powers give the Hecke algebra modules and whose specialization at $1$ can be explicitly described. As an application, we prove the conservation relation on the first occurrence indices for all irreducible representations. As another application, we generalize the results of Aubert-Michel-Rouquier an…
Indefinite theta functions arising from affine Lie superalgebras and sums of triangular numbers
本文通过推广 Roehrig 和 Zwegers 的不定 theta 函数理论，证明若干幂级数为模形式，并由此导出三角数生成函数幂的恒等式，进而揭示其与仿射李超代数分母恒等式的联系。
Witten-Reshetikhin-Turaev invariants and indefinite false theta functions for plumbing indefinite H-graphs
本文构造无定假theta函数作为非弱负定管道3-流形的同调块候选，并证明在庞加莱同调球情形与原始同调块一致。
Theta functions for definite signature lattices constitute a rich source of modular forms. A natural question is then their generalization to indefinite signature lattices. One way to ensure a convergent theta series while keeping the holomorphicity property of definite signature theta series is to restrict the sum over lattice points to a proper subset. Although such series do not have the modular properties that a definite signature theta function has, as shown by Zwegers for signature $(1,n-1)$ lattices, they can be completed to a function that has these modular properties by compromising o…
Recently, Garvan obtained two-variable Hecke-Rogers identities for three universal mock theta functions $g_2(z;q),\,g_3(z;q),\,K(z;q)$ by using basic hypergeometric functions, and he proposed a problem of finding direct proofs of these identities by using Bailey pair technology. In this paper, we give proofs of Garvan's identities by applying Bailey's transform with the conjugate Bailey pair of Warnaar and three Bailey pairs deduced from two special cases of $_6ψ_6$ given by Slater. In particular, we obtain a compact form of two-variable Hecke-Rogers identity related to $g_3(z;q)$, which imply…
A general formula for Hecke-type false theta functions
本文提出一个通用公式，将Hecke型双和分解为θ函数和虚假θ函数，涵盖Matsusaka的分解，并与$b^2-ac>0$情况下的分解结构相似。
Gauss' classical reduction theory for indefinite binary quadratic forms over $\mathbb{Z}$ has originally been proven by means of purely algebraic and arithmetic considerations. It was later discovered that this reduction theory is closely related to a certain symbolic dynamics for the geodesic flow on the modular surface, and hence can also be deduced geometrically. In this article, we use certain symbolic dynamics for the geodesic flow on Hecke triangle surfaces (also the non-arithmetic ones) to develop reduction theories for the indefinite binary quadratic forms associated to Hecke triangle …
Precision Asymptotics for Partitions Featuring False-Indefinite Theta Functions
本文利用模性质研究假不定 theta 函数相关划分的傅里叶系数渐近行为，通过具体例子推导出包含所有指数增长项的精确渐近展开式。
Denominator identity for the affine Lie superalgebra $\widehat{\mathfrak{spo}}(2m,2m+1)$ and indefinite theta functions
本文为 Kac 和 Wakimoto 提出的三角数生成函数幂级数恒等式提供了一种新证明，核心是通过运用 Roehrig 和 Zwegers 的技术，证明涉及球面多项式的不定 theta 函数是模形式。
Theta lifts to certain cohomological representations of indefinite orthogonal groups
本文利用 Borcherds 方法将全纯 Mp₂(ℝ) 尖点形式 theta 提升到不定正交群的上同调表示 Π 中，并通过 Fourier 展开计算与尖点支集分析确认所得自守形式的平方可积性。
Indefinite theta vectors and generalized Nahm sums of arbitrary rank
本文为任意秩的带符号广义Nahm和构造不定theta表示，确定其模变换法则，并给出相关乘积向量分量的多重和表示。
Generalizing a result of~\cite{Z1991} for modular forms of level~one, we give a closed formula for the sum of all Hecke eigenforms on $Γ_0(N)$, 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 of all Hecke eigenforms of all weights on $Γ_0(N)$.
Proof of a conjecture of Andrews and Bachraoui on a Hecke sum
本文证明了安德鲁斯和巴赫拉乌伊关于两种颜色划分生成函数与Hecke型双和关系的猜想，采用Zwegers的不定theta函数理论及模假theta函数的模变换性质。
In a recent preprint, arXiv:1606.05495v1, Alexandrov, Banerjee, Manschot and Pioline introduced generalized error functions and used them to construct indefinite theta series associated to quadratic lattices L of signature (n-2,2). These series are generalizations of those constructed by Zwegers for lattices of signature (n-1,1) and are shown to be `modular completions' of certain nice $q$-series. In this paper, we show that the ABMP-indefinite theta series for signature (n-2,2) can also be obtained as integrals of the form valued theta series introduced in joint work with J. Millson in 1986. …
We describe a family of indefinite theta functions of signature $(1,1)$ that can be expressed in terms of trace functions of vertex algebras built from cones in lattices. The family of indefinite theta functions considered has interesting connections with mock theta functions and Appell-Lerch sums. We use these relations to write the McKay-Thompson series of umbral moonshine at lambency $\ell=8,12,16$ in terms of trace functions of vertex algebras modules, and thereby provide the modules for these instances of umbral moonshine.
We show that the Eigenvariety attached to Hilbert modular forms over a totally real field $F$ is smooth at the points corresponding to certain classical weight one theta series and we give a precise criterion for etaleness over the weight space at those points. In the case where the theta series has real multiplication, we construct a non-classical overconvergent generalised eigenform and compute its Fourier coefficients in terms of $p$-adic logarithms of algebraic numbers. Our approach uses deformations and pseudo-deformations of Galois representations.
We introduce an alternate set of generators for the Hecka algebra, and give an explicit formula for the action of these operators on Fourier coefficients. With this, we compute the eigenvalues of Hecke operators acting on average Siegel theta series with half-integral weight (provided the prime associated to the operators does not divide the level of the theta series). Next, we bound the eigenvalues of these operators in terms of bounds on Fourier coefficients. Then we show that the half-integral weight Kitaoka subspace is stable under all Hecke operators. Finally, we observe that an obvious i…
We obtain some Bailey pairs associated with indefinite quadratic forms with the $β_n$ connected to a finite sum. A new general identity is given, which provides identities for $q$-hypergeometric series, including mock theta functions.
We calculate the action of some Hecke operators on spaces of modular forms spanned by the Siegel theta-series of certain genera of strongly modular lattices closely related to the Leech lattice. Their eigenforms provide explicit examples of Siegel cusp forms.
We find the Hecke-Rogers type series representations of generating functions of the Hurwitz class numbers which is very close to certain mock theta functions. We also prove two combinatorial interpretation of Hurwitz class numbers appeared on OEIS.
Since the study by Jacobi and Hecke, Hecke-type series have received a lot of attention. Unlike such series associated with indefinite quadratic forms, identities on Hecke-type series associated with definite quadratic forms are quite rare in the literature. Motivated by the works of Liu, we first establish many parameterized identities with two parameters by employing different $q$-transformation formulas and then deduce various Hecke-type identities associated with definite quadratic forms by specializing the choice of these two parameters. As applications, we utilize some of these Hecke-typ…
We study the action of the derived Hecke algebra in the setting of dihedral weight one forms, and prove a conjecture of the second- and fourth- named authors relating this action to certain Stark units associated to the symmetric square L-function. The proof exploits the theta correspondence between various Hecke modules as well as ideas of Merel and Lecouturier on higher Eisenstein elements.
In this paper, we consider natural geometric objects coming from Lagrangian Floer theory and mirror symmetry. Lau and Zhou showed that some of the explicit Gromov-Witten potentials computed by Cho, Hong, Kim, and Lau are essentially classical modular forms. Recent work by Zwegers and two of the authors determined modularity properties of several simpler pieces of the last, and most mysterious, function by developing several identities between functions with properties generalizing those of the mock modular forms in Zwegers' thesis. Here, we complete the analysis of all pieces of Cho, Hong, Kim…
Hecke structure of quaternionic modular forms mod $p$
本文建立了无限四元数代数上 Shimura 曲线模 p 模形式与确定四元数代数上代数模形式的 Hecke 特征值系统之间的双射，并显式构造了后者空间的 Hecke 结构，作为 Serre 致 Tate 信的四元数类比。
On Hecke-type double-sums and general string functions for the affine Lie algebra $A_{1}^{(1)}$
本文利用 Hecke 型双重和与 theta 函数、Appell–Lerch 函数的关系，为仿射李代数 $A_{1}^{(1)}$ 在能级 $N=1,2,3,4$ 构造了一般弦函数公式。
The modular transformations of Ramanujan's tenth order mock theta functions are computed, beginning from Choi's Hecke-type identites and using Zwegers' results on indefinite theta series. Explicit completions and shadows are found as an intermediate step. Our result for the modular transformations confirms numerical work by Gordon and McIntosh, and a recent conjecture by Cheng, Duncan, and Harvey.
For an elliptic curve E over an abelian extension k/K with CM by K of Shimura type, the L-functions of its [k:K] Galois representations are Mellin transforms of Hecke theta functions; a modular parametrization (surjective map) from a modular curve to E pulls back the 1-forms on E to give the Hecke theta functions. This article refines the study of our earlier work and shows that certain class of chiral correlation functions in Type II string theory with [E]_C (E as real analytic manifold) as a target space yield the same Hecke theta functions as objects on the modular curve. The Kahler paramet…
Cycle integrals of meromorphic modular forms and Siegel theta functions
本研究探讨正定二元二次型相关亚纯模形式的周期积分，证明其适当线性组合的周期积分为有理值，并借助Hecke不定theta函数算出符号(1,2)偶格对应的Siegel theta函数的周期积分，该副结果也具独立价值。
Depth Two Mock Modularity by Eisenstein Series Coupling
本文通过耦合两个产生深度一拟模形式的爱森斯坦级数，提出了一种构造深度二拟模形式的新方法，并在Vafa-Witten不变量背景下验证了该方法的有效性。
In this paper, we explicitly construct mock modular forms whose shadows are Eisenstein series of arbitrary integral and half-integral weight, level and character at the cusps $\infty$ and $0$. As an application, we give explicit construction of harmonic weak Maass forms which are Hecke eigenforms and are the preimages of $Θ^k, k \in \{ 3, 5, 7\}$ under the shadow operator, where $Θ$ is the classical Jacobi theta function.
Theta operators and a generic entailment for $GSp_4$
本文构造了新的模p权重移微分算子，并证明了Serre权重的通用蕴含性质。
Paramodular groups and theta series
本文研究参数模群的Hecke代数与边界分量，定义参数模theta级数，证明在平方自由水平和足够大权下生成尖形式空间，并给出Garrett分解的新几何证明。
Motivated by the works of Liu, we provide a unified approach to find Appell-Lerch series and Hecke-type series representations for mock theta functions. We establish a number of parameterized identities with two parameters $a$ and $b$. Specializing the choices of $(a,b)$, we not only give various known and new representations for the mock theta functions of orders 2, 3, 5, 6 and 8, but also present many other interesting identities. We find that some mock theta functions of different orders are related to each other, in the sense that their representations can be deduced from the same $(a,b)$-…
This article describes results of joint work with Michael Rapoport and Tonghai Yang. First, we construct an modular form ϕ(τ) of weight 3/2 valued in the arithmetic Chow group of the arithmetic surface M attached toa Shimura curve over Q. The q-expansion of this function is an analogue of the Hirzebruch-Zagier generating function for the cohomology classes of curves on a Hilbert modular surface. This`arithmetic theta function' is used to define an `arithmetic theta lift' from modular forms of weight 3/2 to the arithmetic Chow group of M. For integers t_1 and t_2 with t_1t_2 not a square, the (…
Solutions for Hecke Sum Questions of Banerjee and Bringmann
本文通过q级数和Bailey对证明了Hecke型公式，并给出了双变量 refinement，同时探讨了参数对称性和 cyclotomic 伴随性。
In this paper, we study modularity of several functions which naturally arose in a recent paper of Lau and Zhou on open Gromov-Witten potentials of elliptic orbifolds. They derived a number of examples of indefinite theta functions, and we provide modular completions for several such functions which involve more complicated objects than ordinary modular forms. In particular, we give new closed formulas for special indefinite theta functions of type $(1,2)$ in terms of products of mock modular forms. This formula is also of independent interest.
The theta series of the two unimodular even positive definite lattices of rank 16 are known to be linearly dependent in degree at most 3 and linearly independent in degree 4. In this paper we consider the next case of the 24 Niemeier lattices of rank 24. The associated theta series are linearly dependent in degree at most 11 and linearly independent in degree 12. The resulting Siegel cusp form of degree 12 and weight 12 is a Hecke eigenform which seems to have interesting properties.
Hida family of theta lift from U(1) to definite U(2)
本文构造了从代数Hecke特征到确定型酉群U(2)的theta提升的Hida族，并在中心L值模p非消失条件下证明了其本原性。
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