arXiv subjects

Jan Hendrik Bruinier

Publications and source records attributed to Jan Hendrik Bruinier.

At least 37 records · Page 2Linked to original sources

arXiv2017

Heegner divisors in generalized Jacobians and traces of singular moduli

We prove an abstract modularity result for classes of Heegner divisors in the generalized Jacobian of a modular curve associated to a cuspidal modulus. Extending the Gross-Kohnen-Zagier theorem, we prove that the generating series of these classes is a weakly holomorphic modular form of weight 3/2. Moreover, we show that any harmonic Maass forms of weight 0 defines a functional on the generalized Jacobian. Combining these results, we obtain a unifying framework and new proofs for the Gross-Kohnen-Zagier theorem and Zagier's modularity of traces of singular moduli, together with new geometric interpretations of the traces with non-positive index.

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arXiv2015

Lattices with many Borcherds products

We prove that there are only finitely many isometry classes of even lattices $L$ of signature $(2,n)$ for which the space of cusp forms of weight $1+n/2$ for the Weil representation of the discriminant group of $L$ is trivial. We compute the list of these lattices. They have the property that every Heegner divisor for the orthogonal group of $L$ can be realized as the divisor of a Borcherds product. We obtain similar classification results in greater generality for finite quadratic modules.

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arXiv2015

Class polynomials for nonholomorphic modular functions

We give algorithms for computing the singular moduli of suitable nonholomorphic modular functions F(z). By combining the theory of isogeny volcanoes with a beautiful observation of Masser concerning the nonholomorphic Eisenstein series E_2*(z), we obtain CRT-based algorithms that compute the class polynomials H_D(F;x), whose roots are the discriminant D singular moduli for F(z). By applying these results to a specific weak Maass form F_p(z), we obtain a CRT-based algorithm for computing partition class polynomials, a sequence of polynomials whose traces give the partition numbers p(n). Under the GRH, the expected running time of this algorithm is O(n^{5/2+o(1)}). Key to these results is a fast CRT-based algorithm for computing the classical modular polynomial Phi_m(X,Y) that we obtain by extending the isogeny volcano approach previously developed for prime values of m.

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arXiv2014

Heights of Kudla-Rapoport divisors and derivatives of L-functions

We study special cycles on integral models of Shimura varieties associated with unitary similitude groups of signature (n-1,1). We construct an arithmetic theta lift from harmonic Maass forms of weight 2-n to the arithmetic Chow group of the integral model of a unitary Shimura variety, by associating to a harmonic Maass form f a linear combination of Kudla-Rapoport divisors, equipped with the Green function given by the regularized theta lift of f. Our main result is an equality of two complex numbers: (1) the height pairing of the arithmetic theta lift of f against a CM cycle, and (2) the central derivative of the convolution L-function of a weight n cusp form (depending on f) and the theta function of a positive definite hermitian lattice of rank n-1. When specialized to the case n=2, this result can be viewed as a variant of the Gross-Zagier formula for Shimura curves associated to unitary groups of signature (1,1). The proof relies on, among other things, a new method for computing improper arithmetic intersections.

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arXiv2014

Vector valued formal Fourier-Jacobi series

H. Aoki showed that any symmetric formal Fourier-Jacobi series for the symplectic group Sp_2(Z) is the Fourier-Jacobi expansion of a holomorphic Siegel modular form. We prove an analogous result for vector valued symmetric formal Fourier-Jacobi series, by combining Aoki's theorem with facts about vector valued modular forms. Recently, this result was also proved independently by M. Raum using a different approach. As an application, by means of work of W. Zhang, modularity results for special cycles of codimension 2 on Shimura varieties associated to orthogonal groups can be derived.

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arXiv2012

On the converse theorem for Borcherds products

We prove a new converse theorem for Borcherds' multiplicative theta lift which improves the previously known results. To this end we develop a newform theory for vector valued modular forms for the Weil representation, which might be of independent interest. We also derive lower bounds for the ranks of the Picard groups and the spaces of holomorphic top degree differential forms of modular varieties associated to orthogonal groups.

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arXiv2011

Harmonic Maass forms and periods

According to Waldspurger's theorem, the coefficients of half-integral weight eigenforms are given by central critical values of twisted Hecke L-functions, and therefore by periods. Here we prove that the coefficients of weight 1/2 harmonic Maass forms are determined by periods of algebraic differentials of the third kind on modular and elliptic curves.

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arXiv2011

Algebraic formulas for the coefficients of half-integral weight harmonic weak Maass forms

We prove that the coefficients of certain weight -1/2 harmonic Maass forms are traces of singular moduli for weak Maass forms. To prove this theorem, we construct a theta lift from spaces of weight -2 harmonic weak Maass forms to spaces of weight -1/2 vector-valued harmonic weak Maass forms on Mp_2(Z), a result which is of independent interest. We then prove a general theorem which guarantees (with bounded denominator) when such Maass singular moduli are algebraic. As an example of these results, we derive a formula for the partition function p(n) as a finite sum of algebraic numbers which lie in the usual discriminant -24n+1 ring class field. We indicate how these results extend to general weights. In particular, we illustrate how one can compute theta lifts for general weights by making use of the Kudla-Millson kernel and Maass differential operators.

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arXiv2011

Regularized theta lifts for orthogonal groups over totally real fields

We define a regularized theta lift from SL_2 to orthogonal groups over totally real fields. It takes harmonic `Whittaker forms' to automorphic Green functions and weakly holomorphic Whittaker forms to meromorphic modular forms on orthogonal groups with zeros and poles supported on special divisors, generalizing Borcherds' work on automorphic products. To prove our results we use the spectral expansion of the lift and study its relationship with the cohomological theta lift of Kudla and Millson.

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arXiv2010

Faltings heights of big CM cycles and derivatives of L-functions

We give a formula for the values of automorphic Green functions on the special rational 0-cycles (big CM points) attached to certain maximal tori in the Shimura varieties associated to rational quadratic spaces of signature (2d,2). Our approach depends on the fact that the Green functions in question are constructed as regularized theta lifts of harmonic weak Mass forms, and it involves the Siegel-Weil formula and the central derivatives of incoherent Eisenstein series for totally real fields. In the case of a weakly holomorphic form, the formula is an explicit combination of quantities obtained from the Fourier coefficients of the central derivative of the incoherent Eisenstein series. In the case of a general harmonic weak Maass form, there is an additional term given by the central derivative of a Rankin-Selberg type convolution.

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arXiv2008

Faltings heights of CM cycles and derivatives of L-functions

We study the Faltings height pairing of arithmetic Heegner divisors and CM cycles on Shimura varieties associated to orthogonal groups. We compute the Archimedian contribution to the height pairing and derive a conjecture relating the total pairing to the central derivative of a Rankin L-function. We prove the conjecture in certain cases where the Shimura variety has dimension 0, 1, or 2. In particular, we obtain a new proof of the Gross-Zagier formula.

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arXiv2007

Sign changes of coefficients of half integral weight modular forms

For a half integral weight modular form $f$ we study the signs of the Fourier coefficients $a(n)$. If $f$ is a Hecke eigenform of level $ N$ with real Nebentypus character, and $t$ is a fixed square-free positive integer with $a(t)\neq 0$, we show that for all but finitely many primes $p$ the sequence $(a(tp^{2m}))_{m}$ has infinitely many signs changes. Moreover, we prove similar (partly conditional) results for arbitrary cusp forms $f$ which are not necessarily Hecke eigenforms.

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arXiv2006

Hilbert modular forms and their applications

The present notes contain the material of the lectures given by the author at the summer school on ``Modular Forms and their Applications'' at the Sophus Lie Conference Center in the summer of 2004.

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arXiv2006

On the injectivity of the Kudla-Millson lift and surjectivity of the Borcherds lift

We consider the Kudla-Millson lift from elliptic modular forms of weight (p+q)/2 to closed q-forms on locally symmetric spaces corresponding to the orthogonal group O(p,q). We study the L^2-norm of the lift following the Rallis inner product formula. We compute the contribution at the Archimedian place. For locally symmetric spaces associated to even unimodular lattices, we obtain an explicit formula for the L^2-norm of the lift, which often implies that the lift is injective. For O(p,2) we discuss how such injectivity results imply the surjectivity of the Borcherds lift.

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arXiv2005

Twisted Borcherds products on Hilbert modular surfaces and their CM values

We construct a natural family of rational functions $\tildeΨ_m$ on a Hilbert modular surface from the classical $j$-invariant and its Hecke translates. These functions are obtained by means of a multiplicative analogue of the Doi-Naganuma lifting and can be viewed as twisted Borcherds products. We then study when the value of $\tildeΨ_m$ at a CM point associated to a non-biquadratic quartic CM field generates the `CM class field' of the reflex field. For the real quadratic field $\Q(\sqrt{5})$, we factorize the norm of some of these CM values to $\Q(\sqrt 5)$ numerically.

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arXiv2004

Traces of CM values of modular functions

Zagier proved that the traces of singular moduli, i.e., the sums of the values of the classical j-invariant over quadratic irrationalities, are the Fourier coefficients of a modular form of weight 3/2 with poles at the cusps. Using the theta correspondence, we generalize this result to traces of CM values of (weakly holomorphic) modular functions on modular curves of arbitrary genus. We also study the theta lift for the weight 0 Eisenstein series for SL(2,Z) and realize a certain generating series of arithmetic intersection numbers as the derivative of Zagier's Eisenstein series of weight 3/2. This recovers a result of Kudla, Rapoport and Yang.

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arXiv2004

Infinite products in number theory and geometry

We give an introduction to the theory of Borcherds products and to some number theoretic and geometric applications. In particular, we discuss how the theory can be used to study the geometry of Hilbert modular surfaces.

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arXiv2003

On Two Geometric Theta Lifts

The theta correspondence has been an important tool in studying cycles in locally symmetric spaces of orthogonal type. In this paper, we establish for O(p,2) an adjointness result between Borcherds' singular theta lift and the Kudla-Millson theta lift. We extend this result to arbitrary signature by introducing a new Borcherds lift for O(p,q). On the geometric side, this lift can be interpreted as a differential character in the sense of Cheeger and Simons.

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