Related papers: Siegel modular forms of degree two and level five

We introduce a new class of $\mathfrak{sl}_2$-triples in a complex simple Lie algebra $\mathfrak{g}$, which we call magical. Such an $\mathfrak{sl}_2$-triple canonically defines a real form and various decompositions of $\mathfrak{g}$.…

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…

Assuming the Riemann hypothesis for $L$-functions attached to primitive Dirichlet characters, modular cusp forms, and their tensor products and symmetric squares, we write down explicit finite sets of Hecke operators that span the Hecke…

We prove an arithmetic Hilbert-Samuel type theorem for semi-positive singular hermitian line bundles of finite height. In particular, the theorem applies to the log-singular metrics of Burgos-Kramer-K\"uhn. Our theorem is thus suitable for…

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…

Highest weight modules of the double affine Lie algebra $\widehat{\widehat{\mathfrak{sl}}}_{2}$ are studied under a new triangular decomposition. Singular vectors of Verma modules are determined using a similar condition with horizontal…

We describe (in a representation theoretic setting) a simple comparison of trace formulas, which implies that the conjugate of a Hilbert modular form $f$ by an automorphism of ${\Bbb C}$ again is a Hilbert modular form of the same level and…

In this paper we answer a question posed by Horikawa in 1978, who showed that the above moduli space is composed of 11 locally closed strata building up 4 irreducible components and having at most 3 connected components. We prove that the…

We describe an algorithmic method to determine the image of restriction maps for Siegel modular forms with \textit{arbitrary} characters and arbitrary weight. A program has been implemented in the mathematical software \texttt{Java} to…

In this paper we study the maximal extension $\Gamma_t^*$ of the subgroup $\Gamma_t$ of $\operatorname{Sp}_4 (\bq)$ which is conjugate to the paramodular group. The index of this extension is $2^{\nu(t)}$ where $\nu(t)$ is the number of…

In this article, we provide bounds on systoles associated to a holomorphic $1$-form $\omega$ on a Riemann surface $X$. In particular, we show that if $X$ has genus two, then, up to homotopy, there are at most $10$ systolic loops on…

We prove that the graded ring of modular forms of weight divisible by 3 is naturally isomorphic to a certain log canonical ring of the corresponding elliptic modular surface.

In this paper we extend some results of Norman and Oort and of de Jong, and give an explicit description of the geometry of the Siegel modular threefold with paramodular level structure. We also discuss advantages and restrictions of three…

Parabolic triples of the form $(E_*,\theta,\sigma)$ are considered, where $(E_*,\theta)$ is a parabolic Higgs bundle on a given compact Riemann surface $X$ with parabolic structure on a fixed divisor $S$, and $\sigma$ is a nonzero section…

In this article, we explicitly construct a canonical basis for the space of certain weakly holomorphic Drinfeld modular forms for $\Gamma_0(T)$ (resp., for $\Gamma_0^+(T)$) and compute the generating function satisfied by the basis…

A weight ring in type A is the coordinate ring of the GIT quotient of the variety of flags in $\C^n$ modulo a twisted action of the maximal torus in $\SL(n,\C)$. We show that any weight ring in type A is generated by elements of degree…

Using the framework relating hypergeometric motives to modular forms, we define an explicit family of weight 2 Hecke eigenforms with complex multiplication. We use the theory of ${}_2F_1(1)$ hypergeometric series and Ramanujan's theory of…

It is known that the Selberg zeta function for the modular group has an expression in terms of the class numbers and the fundamental units of the indefinite binary quadratic forms. In the present paper, we generalize such a expression to…

It is well known that, fixed an even, unimodular, positive definite quadratic form, one can construct a modular form in each genus; this form is called the theta series associated to the quadratic form. Varying the quadratic form, one…

We determine conditions for the existence and non-existence of Ramanujan-type congruences for Jacobi forms. We extend these results to Siegel modular forms of degree 2 and as an application, we establish Ramanujan-type congruences for…