Related papers: Modular forms for the $A_{1}$-tower

We work out the exact relationship between algebraic modular forms for a two-by-two general unitary group over a definite quaternion algebra, and those arising from genera of positive-definite quinary lattices, relating stabilisers of local…

The theta-block conjecture proposed by Gritsenko--Poor--Yuen in 2013 characterizes Siegel paramodular forms which are simultaneously Borcherds products and additive Jacobi lifts. In this paper, we prove this conjecture for two new infinite…

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…

In two previous papers we used the geometry of complete polyhedral fans to construct a subring $\TTT (l)$ of the modular forms on $\Gamma_{1} (l)$, and showed that for weight two the cuspidal part of $\TTT (l)$ coincides with the space of…

We compute generators and relations for the graded rings of paramodular forms of degree two and levels 5 and 7. The generators are expressed as quotients of Gritsenko lifts and Borcherds products. The computation is made possible by a…

We study $p$-adic integral models of certain PEL Shimura varieties with level subgroup at $p$ related to the $\Gamma_1(p)$-level subgroup in the case of modular curves. We will consider two cases: the case of Shimura varieties associated…

We prove several dimension formulas for spaces of scalar-valued Siegel modular forms of degree $2$ with respect to certain congruence subgroups of level $4$. In case of cusp forms, all modular forms considered originate from cuspidal…

We study algebras of meromorphic modular forms whose poles lie on Heegner divisors for orthogonal and unitary groups associated to root lattices. We give a uniform construction of $147$ hyperplane arrangements on type IV symmetric domains…

We examine the cell modules for the category of type An webs and their natural cellular forms. We modify the bases of these modules, as described by Elias, to obtain an orthogonal basis of each cell module. Hence, we calculate the…

Siegel modular forms in the space of the mod $p$ kernel of the theta operator are constructed by the Eisenstein series in some odd-degree cases. Additionally, a similar result in the case of Hermitian modular forms is given.

We construct a maximal discrete extension of the paramodular group with a full level-2 structure. The corresponding Siegel variety parametrizes (birationally) the space of Kummer surfaces associated to (1,p)-polarized abelian surfaces with…

We study the cone of moving divisors on the moduli space ${\mathcal A}_g$ of principally polarized abelian varieties. Partly motivated by the generalized Rankin-Cohen bracket, we construct a non-linear holomorphic differential operator that…

We present an explicit and computationally actionable blueprint for constructing vector-valued Siegel modular forms associated to real multiplication (RM) abelian surfaces, leveraging the theta correspondence for the unitary dual pair…

Suppose that p > 5 is a rational prime. Starting from a well-known p-adic analytic family of ordinary elliptic cusp forms of level p due to Hida, we construct a certain p-adic analytic family of holomorphic Siegel cusp forms of arbitrary…

The Shimura correspondence is a fundamental tool in the study of half-integral weight modular forms. In this paper, we prove a Shimura-type correspondence for spaces of half-integral weight cusp forms which transform with a power of the…

We use the orientation underlying the Hirzebruch genus of level three to map the beta family at the prime p=2 into the ring of divided congruences. This procedure, which may be thought of as the elliptic greek letter beta construction,…

In this paper we define a new type of modular object and construct explicit examples of such functions. Our functions are closely related to cusp forms constructed by Zagier which played an important role in the construction by Kohnen and…

We show that the ring of modular forms with characters for $\mathrm{O}(2,4;\mathbb{Z})$ is generated by forms of weights 4, 4, 6, 8, 10, 10, 12, and 30 with three relations of weights 8, 20, and 60. The proof is based on the study of a…

The purpose of this article is to give a simple and explicit construction of mock modular forms whose shadows are Eisenstein series of arbitrary integral weight, level, and character. As application, we construct forms whose shadows are…

This paper studies modular forms of rank four and level one. There are two possiblities for the isomorphism type of the space of modular forms that can arise from an irreducible representation of the modular group of rank four, and we…