David W. Farmer, Sarah Zubairy
It is believed that Dirichlet series with a functional equation and Euler product of a particular form are associated to holomorphic newforms on a Hecke congruence group. We perform computer algebra experiments which find that in certain cases one can associate a kind of ``higher order modular form'' to such Dirichlet series. This suggests a possible approach to a proof of the conjecture.
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We construct bases for the spaces of higher order modular forms of all orders and weights. We also provide a cohomological interpretation of these forms.
We study modular polynomials classifying cyclic isogenies between Drinfeld modules of arbitrary rank over the ring F_q[T].
We calculate the effect of simple Hecke operators on u-expansions of higher rank Drinfeld modular forms, the eigenvalue for the Drinfeld discriminant function $Δ_t$ and show that a certain natural class of Hecke operators is completely multiplicative.
Prime detecting quasi-modular forms in higher level
本文研究高等层级素数检测拟模形式的结构，通过分析拟模尖点形式的符号变化，揭示其在不同算术进展中检测素数的性质，并给出一级情况的解析证明。
In this work we study the analytic properties of the standard L-function attached to Siegel-Jacobi modular forms of higher index, generalizing previous results of Arakawa and Murase. Furthermore, we obtain algebraicity results on special L-values in the spirit of Deligne's Period Conjectures.
We apply differential operators to modular forms on orthogonal groups $\mathrm{O}(2, \ell)$ to construct infinite families of modular forms on special cycles. These operators generalize the quasi-pullback. The subspaces of theta lifts are preserved; in particular, the higher pullbacks of the lift of a (lattice-index) Jacobi form $ϕ$ are theta lifts of partial development coefficients of $ϕ$. For certain lattices of signature (2, 2) and (2, 3), for which there are interpretations as Hilbert-Siegel modular forms, we observe that the higher pullbacks coincide with differential operators introduce…
Zagier proved that the traces of singular values of the classical j-invariant are the Fourier coefficients of a weight 3/2 modular form and Duke provided a new proof of the result by establishing an exact formula for the traces using Niebur's work on a certain class of non-holomorphic modular forms. In this short note, by utilizing Niebur's work again, we generalize Duke's result to exact formulas for traces of singular moduli of higher level modular functions.
This is a sequel to the paper [F. Breuer, H.-G. Rück, Drinfeld modular polynomials in higher rank, J. Number Theory 129 (2009), 59-83.], in which we introduced Drinfeld modular polynomials of higher rank, using an analytic construction. These polynomials relate the isomorphism invariants of Drinfeld F_q[T]-modules of rank r\geq 2 linked by isogenies of a specified type. In the current paper, we give an algebraic construction of greater generality, and prove a generalization of the Kronecker congruences relations, which describe what happens when modular polynomials associated to P-isogenies ar…
In this paper we study generalizations of quadratic form Poincaré series, which naturally occur as outputs of theta lifts. Integrating against them yields evaluations of higher Green's functions. For this we require a new regularized inner product, which is of independent interest.
For positive integers $g$ and $N$, let $\mathcal{F}_N$ be the field of meromorphic Siegel modular functions of genus $g$ and level $N$ whose Fourier coefficients belong to the $N$th cyclotomic field. We present explicit generators of $\mathcal{F}_N$ over $\mathcal{F}_1$ in terms of quotient of theta constants, when $g\geq2$ and $N\geq 3$.
Classical mock modular and quantum modular forms are known to have an intimate relationship with Mordell integrals thanks to Zwegers' groundbreaking PhD thesis. More recently, generalisations of mock/quantum modular forms to so-called "higher depth" versions have been intensively studied. In essence, a mock/quantum modular form of depth $d$ is such that the error of modularity transforms as another mock/quantum modular form of depth $d - 1$. In this short note we use techniques of Bringmann, Kaszian, and Milas to show that the double Eichler integrals of a family of depth two quantum modular f…
In Monstrous moonshine, genus 0 property and the notion of replicability are strongly connected. With regards to recent developments of moonshine, we investigate a higher genus generalization of replicability for a general automorphic form. Specifically, we extend the definitions of replicates and a Hecke operator to harmonic Maass functions on modular curves of higher genera to obtain number theoretic generalizations of important results in Monstrous moonshine. Furthermore, we show the utility of the extended notions in yielding uniform proofs for numerous arithmetic properties of Fourier coe…
In two previous papers we used the geometry of complete polyhedral fans to construct a subring $\TTT (l)$ of the modular forms on $Γ_{1} (l)$, and showed that for weight two the cuspidal part of $\TTT (l)$ coincides with the space of cusp forms of analytic rank zero. In this paper we show that in weights greater than two, the cuspidal part of $\TTT (l)$ coincides with the space of all cusp forms.
We show that the absolute value $|f|$ of an invertible holomorphic function $f$ on the Drinfeld symmetric space $\OM^r$ $(r \geq 2)$ is constant on fibers of the building map to the Bruhat-Tits building $\MB\MT$. Its logarithm $\log|f|$ is an affine map on the realization of $\MB\MT$. These results are used to study the vanishing loci of modular forms (coefficient forms, Eisenstein series, para-Eisenstein series) and to determine their images in $\MB\MT$.
Continuing the work of \cite{7} and \cite{8}, we derive an analogue of the classical "$k/12$-formula" for Drinfeld modular forms of rank $r \geq 2$. Here the vanishing order $ν_ω(f)$ of one modular form at some point $ω$ of the complex upper half-plane is replaced by the intersection multiplicity $ν_{\bo}(f_1,\ldots,f_{r-1})$ of $r-1$ independent Drinfeld modular forms at some point $\bo$ of the Drinfeld symmetric space $\OM^r$. We apply the formula to determine the common zeroes of $r-1$ consecutive Eisenstein series $E_{q^{i}-1}$, where $n-r<i<n$ for some $n \geq r$.
We construct and study a natural compactification $\overline{M}^r(N)$ of the moduli scheme $M^r(N)$ for rank-$r$ Drinfeld $\F_q[T]$-modules with a structure of level $N \in \F_q[T]$. Namely, $\overline{M}^r(N) = {\rm Proj}\,{\bf Eis}(N)$, the projective variety associated with the graded ring ${\bf Eis}(N)$ generated by the Eisenstein series of rank $r$ and level $N$. We use this to define the ring ${\bf Mod}(N)$ of all modular forms of rank $r$ and level $N$. It equals the integral closure of ${\bf Eis}(N)$ in their common quotient field $\widetilde{\MF}_r(N)$. Modular forms are characterized…
Higher Hida and Coleman theories on the modular curve
本文为模曲线上自守线丛的 0 次和 1 次上同调构造 Hida 与 Coleman 理论，并建立二者间的 p-adic 对偶配对。
We construct an explicit family of modular iterated integrals which involves cusp forms. This leads to a new method of producing "invariant versions" of iterated integrals of modular forms. The construction will be based on an extension of higher-order modular forms which, in contrast to the standard higher-order forms, applies to general Fuchsian groups of the first kind and, as such, is of independent interest.
In the present paper, we introduce a special function on the Drinfeld period domain $Ω^{r}$ for $r\geq 2$ which gives the false Eisenstein series of Gekeler when $r=2$. We also study its functional equation and relation with quasi-periodic functions of a Drinfeld module as well as transcendence of its values at CM points.
We construct several examples of higher-dimensional Calabi-Yau manifolds and prove their modularity.
Let $E(z,s)$ be the non-holomorphic Eisenstein series for the modular group $SL(2,{\mathbb Z})$. The classical Kronecker limit formula shows that the second term in the Laurent expansion at $s=1$ of $E(z,s)$ is essentially the logarithm of the Dedekind eta function. This eta function is a weight $1/2$ modular form and Dedekind expressed its multiplier system in terms of Dedekind sums. Building on work of Goldstein, we extend these results from the modular group to more general Fuchsian groups $Γ$. The analogue of the eta function has a multiplier system that may be expressed in terms of a map …
On Drinfeld modular forms of higher rank VII: Expansions at the boundary
本文研究秩r≥2的Drinfeld模形式沿模簇边界的展开，建立判别式乘积公式，用部分zeta函数描述消失阶，并通过Eisenstein级数的线性无关性分解模形式空间，刻画边界条件。
The coefficient forms \( {}_{a} \ell_{k} \) and the para-Eisenstein series \(α_{k}\) are simplicial Drinfeld modular forms. We study the attached simplicial complexes \(\mathcal{BT}^{r}( {}_{a} \ell_{k})\) and \(\mathcal{BT}^{r}(α_{k})\), which are full subcomplexes of the Bruhat-Tits building \(\mathcal{BT}^{r}\) of \( \mathrm{PGL}(r, K_{\infty})\). They are connected (if the rank \(r\) is larger than 2), strongly equidimensional of codimension 1 in \(\mathcal{BT}^{r}\), boundaryless, and satisfy a symmetry property under the non-trivial involution of the Dynkin diagram \(A_{r-1}\).
Drinfeld Quasi-Modular Forms of Higher Level
该研究探讨同余子群的Drinfeld拟模形式向量空间结构，通过E展开和超导数表示其结构，引入双斜线算子定义Hecke算子，刻画本征形式并给出Γ₀(𝔫)子群下Hecke作用于E展开的显式公式。
Bianchi modular forms are automorphic forms over an imaginary quadratic field, associated to a Bianchi group. Even though modern studies of Bianchi modular forms go back to the mid 1960's, most of the fundamental problems surrounding their theory are still wide open. Only for certain types of Bianchi modular forms, which we will call non-genuine, it is possible at present to develop dimension formulas: They are (twists of) those forms which arise from elliptic cuspidal modular forms via the Langlands Base-Change procedure, or arise from a quadratic extension of the imaginary quadratic field vi…
Higher Hida theory for Drinfeld modular curves
受Boxer和Pilloni启发，为Drinfeld模曲线上的线丛上同调建立高阶Hida理论，并插值Serre对偶。
In this paper we introduce a certain space of higher order modular forms of weight 0 and show that it has a Hodge structure coming from the geometry of the fundamental group of a modular curve. This generalizes the usual structure on classical weight 2 forms coming from the cohomology of the modular curve. Further we construct some higher order Poincare series to get higher order higher weight forms and using them we define a space of higher weight, higher order forms which has a mixed Hodge structure as well.
Modularity of Higher Theta Series III: Proof of the Modularity Conjecture
该研究证明了埃尔米特 shtuka 模堆栈上高阶θ级数的模性猜想，确立一般线性 shtuka 的超模性，还证明低余秩希钦堆栈的迹猜想，实现特殊周期的虚基本类为范畴迹。
In this paper, we prove vector-valued higher depth quantum modular properties arising from characters of certain vertex algebras. We then find two-dimensional Mordell integral representations for their errors of modularity.
Higher genus modular graph tensors map Feynman graphs to functions on the Torelli space of genus-$h$ compact Riemann surfaces which transform as tensors under the modular group $Sp(2h , \mathbb Z)$, thereby generalizing a construction of Kawazumi. An infinite family of algebraic identities between one-loop and tree-level modular graph tensors are proven for arbitrary genus and arbitrary tensorial rank. We also derive a family of identities that apply to modular graph tensors of higher loop order.
The goal of this paper is to improve existing bounds for Fourier coefficients of higher genus Siegel modular forms of small weight.
Modular graph forms are a class of modular covariant functions which appear in the genus-one contribution to the low-energy expansion of closed string scattering amplitudes. Modular graph forms with holomorphic subgraphs enjoy the simplifying property that they may be reduced to sums of products of modular graph forms of strictly lower loop order. In the particular case of dihedral modular graph forms, a closed form expression for this holomorphic subgraph reduction was obtained previously by D'Hoker and Green. In the current work, we extend these results to trihedral modular graph forms. Doin…
We construct plane models of the modular curve $X_H(\ell)$, and use their explicit equations to compute Galois representations associated to modular forms for values of $\ell$ that are significantly higher than in prior works.
Siegel modular forms arising from higher Chow cycles
本文证明了一般阿贝尔簇上高阶Chow环的无穷小不变量可构造出具有有界奇点的亚纯Siegel模形式，且该构造与秩1退化下的Siegel算子具有函子性。
We generalize the work of Bertolini and Darmon on the anticyclotomic main conjecture for weight two forms to higher weight modular forms.
On the central derivatives of l-functions and modularity of heenger cycles
本文建立高权重中心L导数的算术交集公式，证明一般亏格形式的导数可通过高阶海根循环的全局高度配对表示，为格罗斯-扎格ier-张公式及其推广提供框架，并研究海根循环生成函数的模性，证明弱猜想并将其简化为消失猜想。
Drinfeld modular forms of higher rank from a lattice-oriented point of view
本文从格的视角引入带水平结构的秩r Drinfeld模空间$\u2112_N^r$，研究其结构、度量与完备化，定义相应Drinfeld模形式并延拓群作用，证明其尖点形式与已有研究结果一致。
We prove that Heegner cycles of codimension m+1 inside Kuga-Sato type varieties of dimension 2m+1 are coefficients of modular forms of weight 3/2+m in the appropriate quotient group. The main technical tool for generating the necessary relations is a Borcherds style theta lift with polynomials. We also show how this lift defines a new singular Shimura-type correspondence from weakly holomorphic modular forms of weight 1/2-m to meromorphic modular forms of weight 2m+2.
In this article, we study the nature of zeros of weakly holomorphic modular forms. In particular, we prove results about transcendental zeros of modular forms of higher levels and for certain Fricke groups which extend a work of Kohnen. Furthermore, we investigate the algebraic independence of values of weakly holomorphic modular forms.
We show that a weakly holomorphic modular function can be written as a sum of modular units of higher level. We further find a necessary and sufficient condition for a Siegel modular function of degree $g$ to have neither zero nor pole on the domain when restricted to certain subset of the Siegel upper half-space $\mathbb{H}_g$.
Higher-Weight Limiting Modular Symbols
本文通过对 Shokurov 的高权模符号进行极限过程，定义了端点为无理实数的高权极限模符号，并证明该极限等价于连分数展开逼近，方法借鉴了权 2 情形下的编码空间转移策略。
We study the relationship between recent conjectures on slopes of overconvergent p-adic modular forms "near the boundary" of p-adic weight space. We also prove in tame level 1 that the coefficients of the Fredholm series of the U_p operator never vanish modulo p, a phenomenon that fails at higher level. In higher level, we do check that infinitely many coefficients are non-zero modulo p using a modular interpretation of the mod p reduction of the Fredholm series recently discovered by Andreatta, Iovita and Pilloni.
In this paper we prove a theorem about the coefficients in a block of a half integral weight modular form. We show that the result of Serre and Stark for weight 1/2 forms does not generalize to higher higher weights. Let f be a half integral weight cusp form of weight at leat 3/2. We consider blocks of coefficients of f and prove that, under some weak assumptions on f, if such a coefficient block is periodic when considered as a function of in the square root of its index, then it must vanish completely. The proof is analytic in nature and uses Shimura's lifting theorem together with estimates…
Higher moments for symmetric powers of modular forms
本文研究了模形式对称幂的高阶矩，建立了渐近公式并改进了先前结果。
In [4] and [5], Folsom presents a family of modular units as higher-level analogues of the Rogers-Ramanujan $q$-continued fraction. These units are constructed from analytic solutions to the higher-order $q$-recurrence equations of Selberg. Here, we consider another family of modular units, which are quotients of Hall-Littlewood $q$-series that appear in the generalized Rogers-Ramanujan type identities of [6]. In analogy with the results of Folsom, we provide a formula for the rank of the subgroup these units generate and show that their specializations at the cusp $0$ generate a subgroup of t…
Rational points on ellipsoids and modular forms
该研究运用模形式理论与球谐分析，在球面和高维椭球有理点的计数与等分布问题上得到新的最优界，展现了模形式的强大应用价值。
The Birch and Swinnerton-Dyer conjecture has been numerically verified for the Jacobians of 32 modular hyperelliptic curves of genus 2 by Flynn, Leprévost, Schaefer, Stein, Stoll and Wetherell, using modular methods. In the calculation of the real period, there is a slight inaccuracy, which might give problems for curves with non-reduced components in the special fibre of their Néron model. In this present paper we explain how the real period can be computed, and how the verification has been extended to many more hyperelliptic curves, some of genus 3, 4 and 5, without using modular methods.
Under a non-torsion assumption on Heegner points, results of Kolyvagin describe the structure of Shafarevich-Tate groups of elliptic curves. In this paper we prove analogous results for ($p$-primary) Shafarevich-Tate groups associated with higher weight modular forms over imaginary quadratic fields satisfying a "Heegner hypothesis". More precisely, we show that the structure of Shafarevich-Tate groups is controlled by cohomology classes built out of Nekovář's Heegner cycles on Kuga-Sato varieties. As an application of our main theorem, we improve on a result of Besser giving a bound on the ord…
\begin{document} \begin This paper continues work of the earlier articles with the same title. For two classes of modular forms $f$: \begin{itemize} \item para-Eisenstein series $α_{k}$ and \item coefficient forms ${}_a \ell_{k}$, where $k \in \mathbb{N}$ and $a$ is a non-constant element of $\mathbb{F}_{q}[T]$, \end{itemize} the growth behavior on the fundamental domain and the zero loci $Ω(f)$ as well as their images $\mathcal{BT}(f)$ in the Bruhat-Tits building $\mathcal{BT}$ are studied. We obtain a complete description for $f = α_{k}$ and for those of the forms ${}_{a}\ell_{k}$ where $k \…
False theta functions are functions that are closely related to classical theta functions and mock theta functions. In this paper, we study their modular properties at all ranks by forming modular completions analogous to modular completions of indefinite theta functions of any signature and thereby develop a structure parallel to the recently developed theory of higher depth mock modular forms. We then demonstrate this theoretical base on a number of examples up to depth three coming from characters of modules for the vertex algebra $W^0(p)_{A_n}$, $1 \leq n \leq 3$, and from $\hat{Z}$-invari…
Norm relations for CM points on modular curves
本文研究了模曲线上的CM点的规范关系，通过欧拉系统方法探讨Selmer群结构，并为更高维推广提供基础。
The theory of elliptic modular forms has gained significant momentum from the discovery of relaxed yet well-behaved notions of modularity, such as mock modular forms, higher order modular forms, and iterated Eichler-Shimura integrals. Applications beyond number theory range from combinatorics, geometry, and representation theory to string theory and conformal field theory. We unify these relaxed notions in the framework of vector-valued modular forms by introducing a new class of $\mathrm{SL}_{2}(\mathbb{Z})$-representations: virtually real-arithmetic types. The key point of the paper is that …
Schertz style class invariants for higher degree CM fields
本文将 Schertz 的经典类不变量构造推广到 Siegel 模函数，利用 Γ^0(N) 与 N-system 给出高次 CM 域的更小类不变量及其共轭构造。
We introduce and study higher depth quantum modular forms. We construct two families of examples coming from rank two false theta functions, whose "companions" in the lower half-plane can be also realized both as double Eichler integrals and as non-holomorphic theta series having values of "double error" functions as coefficients. In particular, we prove that the false theta of $\frak{sl}_3$, appearing in the character of the vertex algebra $W^0(p)_{A_2}$, can be written as the sum of two depth two quantum modular forms of positive integral weight.
In this paper we formulate a conjecture which partially generalizes the Gross-Kohnen-Zagier theorem to higher weight modular forms. For f in S_k(N) satisfying certain conditions, we construct a map from the Heegner points of level N to a complex torus defined by f. We define higher weight analogues of Heegner divisors on this torus. We conjecture they all lie on a line, and their positions are given by the coefficients of a certain Jacobi form corresponding to f. In weight 2, our map is the modular parametrization map (restricted to Heegner points), and our conjectures are implied by Gross-Koh…
Each finite $p$-perfect group $G$ ($p$ a prime) has a universal central $p$-extension. For a perfect group these central extensions come from its {\sl Schur multiplier}. Serre gave a Stiefel-Whitney class approach to analyzing spin covers of alternating groups ($p=2$) aimed at geometric covering space problems. This included the regular version of the Inverse Galois Problem. Every finite simple group with order divisible by $p$ has an infinite string of perfect centerless group covers exhibiting nontrivial Schur multipliers for the prime $p$. Sequences of moduli spaces of curves attached to $G…
The genus of projective curves discretely separates decidedly different two variable algebraic relations. So, we can focus on the connected moduli M_g of genus g curves. Yet, modern applications require a data variable (function) on such curves. The resulting spaces are versions, depending on our need from this data variable, of Hurwitz spaces. A Nielsen class is a set defined by r \ge 3 conjugacy classes C in the data variable monodromy G. It gives a striking genus analog. Using Frattini covers of G, every Nielsen class produces a projective system of related Nielsen classes for any prime p d…
Higher Hida theory for Hilbert modular varieties in the totally split case
研究全实域F中素数p完全分裂时，Hilbert模簇X上自守层的凝聚上同调的p进性质，构造了p进插值普通部分的模M^q。
Nearly higher Coleman theory and p-adic L-functions for $\mathrm{GSp}(4) \times \mathrm{GL}(2)$ and $\mathrm{GSp}(4) \times \mathrm{GL}(2) \times \mathrm{GL}(2)$
本文构造了Siegel模形式与模形式扭曲的$p$-adic $L$-函数，利用近过收敛模形式空间和Gan-Gross-Prasad周期，在Coleman族中实现插值。
The definitions of the $n^{th}$ Gauss sum and the associated $n^{th}$ central charge are introduced for premodular categories $\mathcal{C}$ and $n\in\mathbb{Z}$. We first derive an expression of the $n^{th}$ Gauss sum of a modular category $\mathcal{C}$, for any integer $n$ coprime to the order of the T-matrix of $\mathcal{C}$, in terms of the first Gauss sum, the global dimension, the twist and their Galois conjugates. As a consequence, we show for these $n$, the higher Gauss sums are $d$-numbers and the associated central charges are roots of unity. In particular, if $\mathcal{C}$ is the Dri…
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