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Manuel K. -H. Müller

Publications and source records attributed to Manuel K. -H. Müller.

6 recordsLinked to original sources
arXiv2026

Eisenstein series for orthogonal groups and Kneser neighbours

We study Eisenstein series for orthogonal groups of even unimodular lattices of signature $(n,2)$ which arise as lifts of elliptic modular forms for $\mathrm{SL}_2(\mathbb{Z})$ on 1-dimensional cusps. When the weight is small they have to be defined by analytic continuation. In general, they will be smooth but non-holomorphic after regularization. We show exactly in which cases we obtain holomorphic modular forms by computing the intertwining factors appearing in Langlands' constant term formula in terms of standard parameters and Rankin-Selberg $L$-values. This is related to Kneser's neighbour theory of positive-definite even lattices, which also governs the eigenvalues of these Eisenstein series. In particular, we describe the space of singular weight modular forms and construct the function $Φ_{12}$, the denominator function of the Fake Monster Algebra, as an Eisenstein series. We apply these results to show that the corresponding orthogonal modular varieties are of general type.

math.NT↗
arXiv2025

The Picard group of the Baily--Borel compactification of the moduli space of quasi-polarized K3 surfaces and generalizations

In this paper, we investigate the Picard group of the Baily--Borel compactification of orthogonal Shimura varieties. As a key result, we determine the Picard group of the Baily--Borel compactification of the moduli space of quasi-polarized K3 surfaces, proving that it is isomorphic to $\mathbb{Z}$. Notably, this contrasts with the moduli space of smooth curves, where the Picard group exhibits a more complex structure after natural compactification. Our result follows from a general theorem for orthogonal Shimura varieties: for even lattices $M$ of signature $(2,n)$ with $n > 8$ satisfying specific arithmetic conditions (e.g. K3 type or $p$-elementary), the rational Picard group of $\overline{\operatorname{Sh}}_Γ(M)$ with $Γ$ containing the stable orthogonal group is $1$-dimensional. The core of our proof lies in constructing an arithmetic obstruction space that governs the extension of Heegner divisors to the generic points of the boundary in orthogonal Shimura varieties. We further establish a connection between this obstruction space and the space of theta series, demonstrating that the obstruction space is maximal under our conditions.

math.AG↗
arXiv2024

The invariants of the Weil representation of $\mathrm{SL}_2(\mathbb{Z})$

The transformation behaviour of the vector valued theta function of a positive-definite even lattice under the metaplectic group $\mathrm{Mp}_2(\mathbb{Z})$ is described by the Weil representation. We show that the invariants of this representation are induced from $5$ fundamental invariants. As an application we give simple generating sets for Jacobi forms of singular weight.

math.NT↗
arXiv2024

The basis problem for modular forms for the Weil representation

The vector valued theta series of a positive-definite even lattice is a modular form for the Weil representation of $\mathrm{SL}_2(\mathbb{Z})$. We show that the space of cusp forms for the Weil representation is generated by such functions. This gives a positive answer to Eichler's basis problem in this case. As applications we derive Waldspurger's result on the basis problem for scalar valued modular forms and give a new proof of the surjectivity of the Borcherds lift based on the analysis of local Picard groups.

math.NT↗
arXiv2024

Modular forms for the Weil representation induced from isotropic subgroups

For an isotropic subgroup $H$ of a discriminant form $D$ there exists a lift from modular forms for the Weil representation of the discriminant form $H^\bot/H$ to modular forms for the Weil representation of $D$. We determine a set of discriminant forms such that all modular forms for any discriminant form are induced from the discriminant forms in this set. Furthermore for any discriminant form in this set there exist modular forms that are not induced from smaller discriminant forms.

math.NT↗