相关论文: Analogue of the theta group $\Gamma_{\theta}$

In this series of papers, we introduce higher level versions of the theta group $\Gamma_{\theta}.$ In this paper, we treat the theta group of level $5$, $\Gamma_{\theta,5},$ and construct modular forms on $\Gamma_{\theta,5}$. Moreover we…

We develop two structure theorems for vector valued Siegel modular forms for Igusa's subgroup \Gamma_2[2,4], the multiplier system induced by the theta constants and the representation Sym^2. In the proof, we identify some of these modular…

This paper is an annotated list of transformation properties and identities satisfied by the four theta functions $\theta _1$, $\theta _2$, $\theta _3$, $\theta _4$ of one complex variable, presented in a ready-to-use form. An attempt is…

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…

Let $\mathbb{F}$ be a field of characteristic zero and let $\mathfrak{g}$ be a non-zero finite-dimensional split semisimple Lie algebra with root system $\Delta$. Let $\Gamma$ be a finite set of integral weights of $\mathfrak{g}$ containing…

In this text, we develop the theory of vectorial modular forms with values in Tate algebras introduced by the first author, in a very special case (dimension two, for a very particular representation of {\Gamma} := GL 2 (Fq[$theta$])).…

We consider the space of Siegel modular forms of genus $g$ of weight two relative to the main congruence subgroup of level 2 and to Igusa's group $\Gamma_g(4, 8)$ and $\Gamma_g(2,4)$.One of the main results of this paper is that in the case…

Let $r,h\in\mathbb{N}$ with $r\geq 7$ and let $F(x,y)\in \mathbb{Z}[x ,y]$ be a binary form such that \[ F(x , y) =(\alpha x + \beta y)^r -(\gamma x + \delta y)^r, \] where $\alpha$, $\beta$, $\gamma$ and $\delta$ are algebraic constants…

We fix data $(K/F, E)$ consisting of a Galois extension $K/F$ of characteristic $p$ global fields with arbitrary abelian Galois group $G$ and a Drinfeld module $E$ defined over a certain Dedekind subring of $F$. For this data, we define a…

False theta functions form a family of functions with intriguing modular properties and connections to mock modular forms. In this paper, we take the first step towards investigating modular transformations of higher rank false theta…

We present a new proof of the transformation law of $\vartheta_1$ under the action of the generator of the full modular group $\Gamma$ using Siegel's method.

We revisit Beukers' modular-form proof of the irrationality of $\zeta(3)$ from the point of view of the auxiliary weight two modular form. For the Fricke group $\Gamma_0(6)^\star$, we show that Beukers' choice is not isolated: it belongs to…

In this paper we study deformations of mod $p$ Galois representations $\tau$ (over an imaginary quadratic field $F$) of dimension $2$ whose semi-simplification is the direct sum of two characters $\tau_1$ and $\tau_2$. As opposed to our…

After recalling the notion of higher roots (or hyper-roots) associated with "quantum modules" of type $(G, k)$, for $G$ a semi-simple Lie group and $k$ a positive integer, following the definition given by A. Ocneanu in 2000, we study the…

In this short paper, we find the transformation formula for the theta series under the action of the Jacobi modular group on the Siegel-Jacobi space. This formula generalizes the formula (5.1) obtained by Mumford in his book[p.189, Tata…

In previous work, we defined certain virtual fundamental classes for special cycles on the moduli stack of Hermitian shtukas, and related them to the higher derivatives of non-singular Fourier coefficients of Siegel-Eisenstein series. In…

Let $\Gamma=\langle \alpha, \beta \rangle$ be a numerical semigroup. In this article we consider the dual $\Delta^*$ of a $\Gamma$-semimodule $\Delta$; in particular we deduce a formula that expresses the minimal set of generators of…

We study over rings of scalar valued Siegel modular forms. modules of vector valued modular forms of degree two. For the two simplest representations, standard and Sym^2, appears rather natural consider the cases of the group $\Gamma[4,8] $…

Using different Lubin-Tate groups, we compare $(\phi, \Gamma)$ modules associated to a Galois representation via Fontaine's theory.

An algorithm to obtain equations between theta functions with integral characteristics evaluated at $\tau$ and $p\tau$ for $g>1$ is presented.