Borcherds products and arithmetic intersection theory on Hilbert modular surfaces

Bruinier, Jan Hendrik; Gil, José Ignacio Burgos; Kühn, Ulf

doi:10.1215/s0012-7094-07-13911-5

Cited by 63 publications

(87 citation statements)

References 44 publications

(70 reference statements)

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“…7 The Hilbert modular surface H is connected and hence we may use Res F Q SL 2 instead of G to study the complex points. Notice that our lattice is different from the default choice in [BBGK07]. Using their notation, we work with Γ(O F ⊕ ad F ).…”

Section: Hirzebruch-zagier Divisors and Hecke Orbitsmentioning

confidence: 99%

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Exceptional splitting of reductions of abelian surfaces

Shankar¹,

Tang²

2020

Duke Math. J.

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Heuristics based on the Sato-Tate conjecture suggest that an abelian surface defined over a number field has infinitely many places of split reduction. We prove this result for abelian surfaces with real multiplication. As in [Cha14] and [Elk89], this shows that a density-zero set of primes pertaining to the reduction of abelian varieties is infinite. The proof relies on the Arakelov intersection theory on Hilbert modular surfaces.

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Section: Hirzebruch-zagier Divisors and Hecke Orbitsmentioning

confidence: 99%

“…5.1]. Any endomorphism of B is 8 This is the definition in [BBGK07]. The lattice in [vdG88] differs by a multiple of the scalar matrix √ D · I so these two definitions of T (r) coincide.…”

Section: Hirzebruch-zagier Divisors and Hecke Orbitsmentioning

confidence: 99%

Exceptional splitting of reductions of abelian surfaces

Shankar¹,

Tang²

2020

Duke Math. J.

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“…iii) The functionˆ ;r .z; f / is a Green function for the divisor Z ;r .f / C C ;r .f / in the sense of [BGKK07], [BBGK07]. Here C ;r .f / is a divisor on X 0 .N / supported at the cusps; see also (5.12).…”

mentioning

confidence: 99%

Heegner divisors,L-functions and harmonic weak Maass forms

2010

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Recent works, mostly related to Ramanujan's mock theta functions, make use of the fact that harmonic weak Maass forms can be combinatorial generating functions. Generalizing works of Waldspurger, Kohnen and Zagier, we prove that such forms also serve as "generating functions" for central values and derivatives of quadratic twists of weight 2 modular L-functions. To obtain these results, we construct differentials of the third kind with twisted Heegner divisor by suitably generalizing the Borcherds lift to harmonic weak Maass forms. The connection with periods, Fourier coefficients, derivatives of L-functions, and points in the Jacobian of modular curves is obtained by analyzing the properties of these differentials using works of Scholl, Waldschmidt, and Gross and Zagier.

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“…A slight variant (see Theorem 3.7), together with [10, Theorem 6.3], can be employed to give a different proof of the converse theorem for Borcherds products [4,Theorem 5.12] for lattices that split two hyperbolic planes over Z. Similar results as Theorems 1.1 and 1.3 were obtained in [5,Section 4] for the special case of Hilbert modular surfaces for the full Hilbert modular group.…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 80%

“…Proof of Proposition We generalize the argument of [, Lemma 4.11]. According to Theorem , the space

M_{2 - κ}^{!} false(ρ_{L^{-}} false)

has a basis of weakly holomorphic modular forms with integral coefficients.…”

Section: Proofsmentioning

confidence: 94%

Borcherds products with prescribed divisor

Bruinier

2017

Bull. London Math. Soc.

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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.Comment: Proof of Theorem 2.1 added in v

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Borcherds products and arithmetic intersection theory on Hilbert modular surfaces

Cited by 63 publications

References 44 publications

Exceptional splitting of reductions of abelian surfaces

Exceptional splitting of reductions of abelian surfaces

Heegner divisors,L-functions and harmonic weak Maass forms

Borcherds products with prescribed divisor

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