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Heights of Kudla–Rapoport divisors and derivatives of \(L\)-functions

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Abstract

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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Notes

  1. More precisely, these are the harmonic weak Maass forms of [7]. For simplicity we omit the adjective ‘weak’.

  2. More precisely, there is a choice of \(i=\sqrt{-1}\) such that \(\psi _0(ix,x)\) and \(\psi (ix,x)\) are positive definite, and we choose \(\delta _{{{\varvec{k}}}}\) to lie in the same connected component of \(\mathbb {C}\backslash {\mathbb {R}}\) as \(i.\)

  3. In the sense of [11], so \(G_{\ell }\) is the unique deformation of \(G\), with its action of \(\mathcal {O}_{{{\varvec{k}}},\mathfrak {p}}\), to \(R_{\ell }\).

  4. As both \({\mathcal Z}\) and \({\mathcal Y}\) are Cohen–Macaulay, the finite intersection multiplicity agrees with the more natural Serre intersection multiplicity defined as in [33, p. 11] or [16, Section 3.1]. This follows from [32, p. 111].

  5. There is a mild abuse of notation: we are using \(A_0^{\mathrm {univ}}\) to denote both the universal elliptic curve over \({\mathcal M}_{{\mathbb {L}}}\), and the universal elliptic curve over \({\mathcal Y}_{({\mathbb {L}}_0,\Lambda )}\).

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Acknowledgments

We thank the referee for his/her careful reading of our manuscript and for the insightful comments.

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Authors and Affiliations

  1. Fachbereich Mathematik, Technische Universität Darmstadt, Schlossgartenstr. 7, 64289, Darmstadt, Germany

    Jan Hendrik Bruinier

  2. Department of Mathematics, Boston College, 140 Commonwealth Ave, Chestnut Hill, MA, 02467, USA

    Benjamin Howard

  3. Department of Mathematics, University of Wisconsin Madison, Van Vleck Hall, Madison, WI, 53706, USA

    Tonghai Yang

Authors
  1. Jan Hendrik Bruinier
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  2. Benjamin Howard
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Corresponding author

Correspondence to Jan Hendrik Bruinier.

Additional information

J. H. Bruinier is partially supported by DFG Grant BR-2163/4-1. B. Howard is partially supported by NSF Grant DMS-1201480. T. Yang is partially supported by a NSF Grant DMS-1200380 and a Chinese grant.

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Bruinier, J.H., Howard, B. & Yang, T. Heights of Kudla–Rapoport divisors and derivatives of \(L\)-functions. Invent. math. 201, 1–95 (2015). https://doi.org/10.1007/s00222-014-0545-9

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  • DOI: https://doi.org/10.1007/s00222-014-0545-9

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