We define a regularized Shintani theta lift which maps weight $2k+2$ ($k \in \Z, k \geq 0$) harmonic Maass forms for congruence subgroups to (sesqui-)harmonic Maass forms of weight $3/2+k$ for the Weil representation of an even lattice of signature $(1,2)$. We show that its Fourier coefficients are given by traces of CM values and regularized cycle integrals of the input harmonic Maass form. Further, the Shintani theta lift is related via the $ξ$-operator to the Millson theta lift studied in our earlier work. We use this connection to construct $ξ$-preimages of Zagier's weight $1/2$ generating series of singular moduli and of some of Ramanujan's mock theta functions.
A Shintani-Katok-Sarnak type correspondence for Maass cusp forms of level $N$ is shown to be derived from analytic properties of prehomogeneous zeta functions whose coefficients involve periods of Maass forms.
The Maass-form twisted Shintani $\mathscr{L}$-functions are introduced, and some of their analytic properties are studied. These functions contain data regarding the distribution of shapes of cubic rings.
Previously the authors proved subconvexity of Shintani's zeta function enumerating class numbers of binary cubic forms. Here we return to prove subconvexity of the Maass form twisted version.
We describe the construction and properties of a singular theta lift for the orthogonal group $\SO(2,1)$. We obtain locally harmonic Maass forms in the sense of Bringmann-Kane-Kohnen with singular sets along geodesics in the upper half plane. We consider these forms as currents and derive properties of the Shimura-Shintani correspondence. This work provides extensions of the theta lifts considered by Hoevel and Bringann-Kane-Viazovska.
We study a certain theta lift which maps weight $-2k$ to weight $1/2-k$ harmonic weak Maass forms for $k \in \mathbb{Z}, k \geq 0$, and which is closely related to the classical Shintani lift from weight $2k+2$ to weight $k+3/2$ cusp forms. We compute the Fourier expansion of the theta lift and show that it involves twisted traces of CM values and geodesic cycle integrals of the input function. As an application, we obtain a criterion for the non-vanishing of the central $L$-value of an integral weight newform $G$ in terms of the holomorphicity of the theta lift of a certain harmonic weak Maas…
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.
We present some applications of the Kudla-Millson and the Millson theta lift. The two lifts map weakly holomorphic modular functions to vector valued harmonic Maass forms of weight $3/2$ and $1/2$, respectively. We give finite algebraic formulas for the coefficients of Ramanujan's mock theta functions $f(q)$ and $ω(q)$ in terms of traces of CM-values of a weakly holomorphic modular function. Further, we construct vector valued harmonic Maass forms whose shadows are unary theta functions, and whose holomorphic parts have rational coefficients. This yields a rationality result for the coefficien…
We present completions of mock theta functions to harmonic weak Maass forms of weight $1/2$ and algebraic formulas for the coefficients of mock theta functions. We give several harmonic weak Maass forms of weight $1/2$ that have mock theta functions as their holomorphic part. Using these harmonic weak Maass forms and the Millson theta lift we compute finite algebraic formulas for the coefficients of the appearing mock theta functions in terms of traces of singular moduli.
We investigate so-called "higher" Siegel theta lifts on Lorentzian lattices in the spirit of Bruinier-Ehlen-Yang and Bruinier-Schwagenscheidt. We give a series representation of the lift in terms of Gauss hypergeometric functions, and evaluate the lift as the constant term of a Fourier series involving the Rankin-Cohen bracket of harmonic Maass forms and theta functions. Using the higher Siegel lifts, we obtain a vector-valued analogue of Mertens' result stating that the Rankin--Cohen bracket of the holomorphic part of a harmonic Maass form of weight $\frac{3}{2}$ and a unary theta function, p…
In this note we extend integral weight harmonic Maass forms to functions defined on the upper and lower half-planes using the method of Poincaré series. This relates to Rademacher's "expansion of zero" principle, which was recently employed by Rhoades to link mock theta functions and partial theta functions.
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…
In this paper, we investigate traces of cycle integrals of certain meromorphic modular forms. By relating them to regularised theta lifts we provide explicit formulae for them in terms of coefficients of harmonic Maass forms.
Let $f(q)=1+\sum_{n=1}^{\infty} α(n)q^n$ be the well-known third order mock theta of Ramanujan. In 1964, George Andrews proved an asymptotic formula of the form $$α(n)= \sum_{c\leq\sqrt{n}} ψ(n)+O_ε\left(n^ε\right),$$ where $ψ(n)$ is an expression involving generalized Kloosterman sums and the $I$-Bessel function. Andrews conjectured that the series converges to $α(n)$ when extended to infinity, and that it does not converge absolutely. Bringmann and Ono proved the first of these conjectures. Here we obtain a power savings bound for the error in Andrews' formula, and we also prove the second o…
In this paper, we explicitly construct harmonic Maass forms that map to the weight one theta series associated by Hecke to odd ray class group characters of real quadratic fields. From this construction, we give precise arithmetic information contained in the Fourier coefficients of the holomorphic part of the harmonic Maass form, establishing the main part of a conjecture of the second author.
Let $Γ_n(\mathcal{\scriptstyle{O}}_{\mathbb{K}})$ denote the Hermitian modular group of degree $n$ over an imaginary quadratic number field $\mathbb{K}$ and $Δ_{n,\mathbb{K}}^*$ its maximal discrete extension in the special unitary group $SU(n,n;\mathbb{C})$. In this paper we study the action of $Δ_{n,\mathbb{K}}^*$ 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 $Δ_{2,\mathbb{K}}^*$ as well as examples where this is not the case. Our second focus lies on studying two different Maass sp…
We analyze the mock modular behavior of $\bar{P}_ω(q)$, a partition function introduced by Andrews, Dixit, Schultz, and Yee. This function arose in a study of smallest parts functions related to classical third order mock theta functions, one of which is $ω(q)$. We find that the modular completion of $\bar{P}_ω(q)$ is not simply a harmonic Maass form, but is instead the derivative of a linear combination of products of various harmonic Maass forms and theta functions. We precisely describe its behavior under modular transformations and find that the image under the Maass lowering operator lies…
We extend Borcherds' singular theta lift in signature $(1,2)$ to harmonic Maass forms of weight $1/2$ whose non-holomorphic part is allowed to be of exponential growth at $i\infty$. We determine the singularities of the lift and compute its Fourier expansion. It turns out that the lift is continuous but not differentiable along certain geodesics in the upper half-plane corresponding to the non-holomorphic principal part of the input. As an application, we obtain a generalization to higher level of the weight $2$ modular integral of Duke, Imamoglu and Tóth. Further, we construct automorphic pro…
Algebraic formulas for the coefficients of half-integral weight harmonic weak Maass forms
证明了某些权-1/2调和Maass型的系数是弱Maass型的奇异模的迹。为了证明这个定理，我们构造了Mp_2（Z）上从权-2调和弱Maass型空间到权-1/2向量值调和弱Maass型空间的θ提升，这是一个独立的结果.然后，我们证明了一个一般定理，保证（有界分母）时，这种马斯奇异模代数。作为这些结果的一个例子，我们导出了分拆函数p（n）的一个公式，该配分函数p（n）是位于通常判别式-24 n +1环类域中的代数数的有限和。我们指出这些结果如何扩展到一般的权重。特别是，我们说明了如何可以计算θ电梯一般的权重，利用Kudla-Millson内核和马斯微分算子。
Modular and mock modular forms possess many striking $p$-adic properties, as studied by Bringmann, Guerzhoy, Kane, Kent, Ono, and others. Candelori developed a geometric theory of harmonic Maass forms arising from the de Rham cohomology of modular curves. In the setting of over-convergent $p$-adic modular forms, Candelori and Castella showed this leads to $p$-adic analogs of harmonic Maass forms. In this paper we take an analytic approach to construct $p$-adic analogs of harmonic Maass forms of weight $0$ %and $1/2$ with square free level. Although our approaches differ, where the two theories…
The mock theta conjectures are ten identities involving Ramanujan's fifth-order mock theta functions. The conjectures were proven by Hickerson in 1988 using q-series methods. Using methods from the theory of harmonic Maass forms, specifically work of Zwegers and Bringmann-Ono, Folsom reduced the proof of the mock theta conjectures to a finite computation. Both of these approaches involve proving the identities individually, relying on work of Andrews-Garvan. Here we give a unified proof of the mock theta conjectures by realizing them as an equality between two nonholomorphic vector-valued modu…
We study special cycles on integral models of Shimura varieties associated with unitary similitude groups of signature (n-1,1). We construct an arithmetic theta lift from harmonic Maass forms of weight 2-n to the arithmetic Chow group of the integral model of a unitary Shimura variety, by associating to a harmonic Maass form f a linear combination of Kudla-Rapoport divisors, equipped with the Green function given by the regularized theta lift of f. Our main result is an equality of two complex numbers: (1) the height pairing of the arithmetic theta lift of f against a CM cycle, and (2) the cen…
Let $f$ be a cuspidal newform (holomorphic or Maass) of arbitrary level and nebentypus and denote by $λ_f(n)$ its $n$-th Hecke eigenvalue. Let $$ r(n)=\#\left\{(n_1,n_2)\in \mathbb{Z}^2:n_1^2+n_2^2=n\right\}. $$ In this paper, we study the shifted convolution sum $$ \mathcal{S}_h(X)=\sum_{n\leq X}λ_f(n+h)r(n), \qquad 1\leq h\leq X, $$ and establish uniform bounds with respect to the shift $h$ for $\mathcal{S}_h(X)$.
We explicitly construct cusp forms on the orthogonal group of signature $(1,8n+1)$ for an arbitrary natural number $n$ as liftings from Maass cusp forms of level one. In our previous works, the fundamental tool to show the automorphy of the lifting was the converse theorem by Maass. In this paper, we use the Fourier expansion of the theta lifts by Borcherds instead. We also study cuspidal representations generated by such cusp forms and show that they are irreducible and that all of their non-archimedean local components are non-tempered while the archimedean component is tempered, if the Maas…
Recently, Funke and Hofmann constructed a singular theta lift of Borcherds type for the dual reductive pair $U(1,1)\times U(p,q)$, $p,q\geq 1$, the input functions of which are harmonic weak Maass forms of weight $k= 2-p-q$. In the present paper, we give an explicit evaluation of the Fourier-Jacobi expansion of the lift. For this purpose, we adapt a method introduced by Kudla in his paper 'Another product for a Borcherds form'. As an application, in the case $U(p,1)$ we recover a new infinite product expansion associated to a Borcherds form, analogous to the case $O(p,2)$ treated by Kudla.
The quaternionic Maass Spezialschar on split $\mathrm{SO}(8)$
University of Idaho（爱达荷大学）；The University of Oklahoma（俄克拉荷马大学）；University of Toronto（多伦多大学）；The University of California, San Diego（加州大学圣地亚哥分校）；Lafayette College（拉斐特学院）
Our aim is to clarify the relationship between Kudla's and Bruinier's Green functions attached to special cycles on Shimura varieties of orthogonal and unitary type. These functions play a key role in the arithmetic geometry of the special cycles in the context of Kudla's program. In particular, we show that the generating series obtained by taking the differences of the two families of Green functions is a modular form with trivial holomorphic projection. Along the way, we construct a section of the Maass lowering operator for moderate growth modular forms valued in the Weil representation us…
Mock modular forms, which give the theoretical framework for Ramanujan's enigmatic mock theta functions, play many roles in mathematics. We study their role in the context of modular parameterizations of elliptic curves $E/\mathbb{Q}$. We show that mock modular forms which arise from Weierstrass $ζ$-functions encode the central $L$-values and $L$-derivatives which occur in the Birch and Swinnerton-Dyer Conjecture. By defining a theta lift using a kernel recently studied by Hövel, we obtain canonical weight 1/2 harmonic Maass forms whose Fourier coefficients encode the vanishing of these values…
