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Jan Hendrik Bruinier

Publications and source records attributed to Jan Hendrik Bruinier.

At least 19 recordsLinked to original sources
arXiv2026

Bad Reduction of Genus Two Curves with complex multiplication

We study genus two curves whose Jacobians have complex multiplication by a biquadratic CM field. In this setting the Jacobian decomposes as a product of two elliptic curves with complex multiplication by orders in the same imaginary quadratic field. We derive upper bounds for the primes of stable bad reduction of such curves in terms of the discriminants of the CM orders. To do so, we give an explicit description of principally polarized abelian surfaces with CM by a biquadratic field via small CM cycles on an orthogonal Shimura variety of signature (3,2). We use it to prove an arithmetic intersection formula between such CM cycles with Humbert surfaces in terms of coefficients of incoherent Eisenstein series and representation numbers of ternary positive definite quadratic forms. Finally, employing the arithmetic properties of higher Green functions, we show that primes of stable bad reduction always exist.

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arXiv2026

Lefschetz decompositions of Kudla-Millson theta functions

In the 1980's Kudla and Millson introduced a theta function in two variables. It behaves as a Siegel modular form with respect to the first variable, and is a closed differential form on an orthogonal Shimura variety with respect to the other variable. We prove that the Lefschetz decomposition of the cohomology class of that theta function is also its modular decomposition in Eisenstein, Klingen and cuspidal parts. We also show that the image of the Kudla-Millson lift is contained in the primitive cohomology of X. As an application, we deduce dimension formulas for the cohomology of X in low degree. Our results cover the cases of moduli spaces of compact hyperkähler manifolds.

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arXiv2026

The Siegel-Weil formula in geometry and arithmetic

The present paper is an extended version of the lecture notes of a course given by the first author at the summer school on Formulas of Siegel and Weil (Bielefeld, September 2025). We survey three perspectives on the Siegel-Weil formula: classical, geometric, and arithmetic. We first recall the Siegel-Weil formula for elliptic theta series arising from positive definite lattices. We next discuss the higher genus case for lattices of arbitrary signature from an adelic viewpoint. After introducing orthogonal Shimura varieties, we present the geometric Siegel-Weil formula, in which the generating series of volumes of special cycles is shown to be an Eisenstein series. We conclude with the (partly conjectural) arithmetic Siegel-Weil formula, relating degrees of special cycles in arithmetic Chow groups to central derivatives of Eisenstein series.

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arXiv2026

Intersections of special cycles on Shimura curves and Siegel Maass forms

We show that the generating series of the number of pairs of geodesics on a compact Shimura curve with given discriminants and intersection angle are coefficients of a non-holomorphic Siegel modular form, a theta lift of the constant function. This retrieves and generalizes counting results of Rickards via the Siegel-Weil formula. More generally, we study the genus two theta lift of Maass forms on this Shimura curve and prove a Fourier-Taylor expansion in terms of some generalized Whittaker functions. We also provide a geometric interpretation of all Fourier coefficients of these theta lifts in terms of averages of geodesic Taylor coefficients over special cycles.

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arXiv2024

Arithmetic volumes of unitary Shimura varieties

The integral model of a $\mathrm{GU}(n-1,1)$ Shimura variety carries a universal abelian scheme over it, and the dual top exterior power of its Lie algebra carries a natural hermitian metric. We express the arithmetic volume of this metrized line bundle, defined as an iterated self-intersection in the arithmetic Chow ring, in terms of logarithmic derivatives of Dirichlet $L$-functions. We also determine the arithmetic volumes of Kudla-Rapoport divisors and relate them to coefficients of Eisenstein series.

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arXiv2024

Formal Siegel modular forms for arithmetic subgroups

The notion of formal Siegel modular forms for an arithmetic subgroup $Γ$ of the symplectic group of genus $n$ is a generalization of symmetric formal Fourier-Jacobi series. Assuming an upper bound on the affine covering number of the Siegel modular variety associated with $Γ$, we prove that all formal Siegel modular forms are given by Fourier-Jacobi expansions of classical holomorphic Siegel modular forms. We also show that the required upper bound is always met if $2\leq n \leq 4$. As an application we consider the case of the paramodular group of squarefree level and genus $2$.

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arXiv2022

Harmonic weak Maass forms and periods II

In this paper we investigate the Fourier coefficients of harmonic Maass forms of negative half-integral weight. We relate the algebraicity of these coefficients to the algebraicity of the coefficients of certain canonical meromorphic modular forms of positive even weight with poles at Heegner divisors. Moreover, we give an explicit formula for the coefficients of harmonic Maass forms in terms of periods of certain meromorphic modular forms with algebraic coefficients.

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arXiv2022

Kudla's Modularity Conjecture and Formal Fourier-Jacobi Series

We prove modularity of formal series of Jacobi forms that satisfy a natural symmetry condition. They are formal analogues of Fourier-Jacobi expansions of Siegel modular forms. From our result and a theorem of Wei Zhang, we deduce Kudla's conjecture on the modularity of generating series of special cycles of arbitrary codimension and for all orthogonal Shimura varieties.

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arXiv2021

Special Cycles on Toroidal Compactifications of Orthogonal Shimura Varieties

We determine the behavior of automorphic Green functions along the boundary components of toroidal compactifications of orthogonal Shimura varieties. We use this analysis to define boundary components of special divisors and prove that the generating series of the resulting special divisors on a toroidal compactification is modular.

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arXiv2021

CM values of higher automorphic Green functions for orthogonal groups

Gross and Zagier conjectured that the CM values (of certain Hecke translates) of the automorphic Green function $G_s(z_1,z_2)$ for the elliptic modular group at positive integral spectral parameter $s$ are given by logarithms of algebraic numbers in suitable class fields. We prove a partial average version of this conjecture, where we sum in the first variable $z_1$ over all CM points of a fixed discriminant $d_1$ (twisted by a genus character), and allow in the second variable the evaluation at individual CM points of discriminant $d_2$. This result is deduced from more general statements for automorphic Green functions on Shimura varieties associated with the group $\mathrm{GSpin}(n,2)$. We also use our approach to prove a Gross-Kohnen-Zagier theorem for higher Heegner divisors on Kuga-Sato varieties over modular curves.

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arXiv2020

Theta lifts for Lorentzian lattices and coefficients of mock theta functions

We evaluate regularized theta lifts for Lorentzian lattices in three different ways. In particular, we obtain formulas for their values at special points involving coefficients of mock theta functions. By comparing the different evaluations, we derive recurrences for the coefficients of mock theta functions, such as Hurwitz class numbers, Andrews' spt-function, and Ramanujan's mock theta functions.

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arXiv2019

Arithmetic degrees of special cycles and derivatives of Siegel Eisenstein series

Let V be a rational quadratic space of signature (m,2). A conjecture of Kudla relates the arithmetic degrees of top degree special cycles on an integral model of a Shimura variety associated with SO(V) to the coefficients of the central derivative of an incoherent Siegel Eisenstein series of genus m+1. We prove this conjecture for the coefficients of non-singular index T when T is not positive definite. We also prove it when T is positive definite and the corresponding special cycle has dimension 0. To obtain these results, we establish new local arithmetic Siegel-Weil formulas at the archimedean and non-archimedian places.

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arXiv2018

A converse theorem for Borcherds products on $X_0(N)$

We show that every Fricke invariant meromorphic modular form for $Γ_0(N)$ whose divisor on $X_0(N)$ is defined over $\mathbb{Q}$ and supported on Heegner divisors and the cusps is a generalized Borcherds product associated to a harmonic Maass form of weight $1/2$. Further, we derive a criterion for the finiteness of the multiplier systems of generalized Borcherds products in terms of the vanishing of the central derivatives of $L$-function of certain weight $2$ newforms. We also prove similar results for twisted Borcherds products.

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arXiv2017

Borcherds products with prescribed divisor

Given an infinite set of special divisors satisfying a mild regularity condition, we prove the existence of a Borcherds product of non-zero weight whose divisor is supported on these special divisors. We also show that every meromorphic Borcherds product is the quotient of two holomorphic ones. The proofs of both results rely on the properties of vector valued Eisenstein series for the Weil representation.

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arXiv2017

Cones of Heegner divisors

We show that the cone of primitive Heegner divisors is finitely generated for many orthogonal Shimura varieties, including the moduli space of polarized K3-surfaces. The proof relies on the growth of coefficients of modular forms.

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arXiv2017

Algebraic formulas for the coefficients of mock theta functions and Weyl vectors of Borcherds products

We present some applications of the Kudla-Millson and the Millson theta lift. The two lifts map weakly holomorphic modular functions to vector valued harmonic Maass forms of weight $3/2$ and $1/2$, respectively. We give finite algebraic formulas for the coefficients of Ramanujan's mock theta functions $f(q)$ and $ω(q)$ in terms of traces of CM-values of a weakly holomorphic modular function. Further, we construct vector valued harmonic Maass forms whose shadows are unary theta functions, and whose holomorphic parts have rational coefficients. This yields a rationality result for the coefficients of mock theta functions, i.e., harmonic Maass forms whose shadows lie in the space of unary theta functions. Moreover, the harmonic Maass forms we construct can be used to evaluate the Petersson inner products of unary theta functions with harmonic Maass forms, giving formulas and rationality results for the Weyl vectors of Borcherds products.

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