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IMPROPER INTERSECTIONS OF KUDLA–RAPOPORT DIVISORS AND EISENSTEIN SERIES

Part of: Arithmetic algebraic geometry Arithmetic problems. Diophantine geometry Discontinuous groups and automorphic forms

Published online by Cambridge University Press:  23 April 2015

Siddarth Sankaran
Siddarth Sankaran*
Affiliation:
Department of Mathematics, McGill University, Montreal, Canada (siddarth.sankaran@mcgill.ca)
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Abstract

We consider a certain family of Kudla–Rapoport cycles on an integral model of a Shimura variety attached to a unitary group of signature (1, 1), and prove that the arithmetic degrees of these cycles are Fourier coefficients of the central derivative of an Eisenstein series of genus 2. The integral model in question parameterizes abelian surfaces equipped with a non-principal polarization and an action of an imaginary quadratic number ring, and in this setting the cycles are degenerate: they may contain components of positive dimension. This result can be viewed as confirmation, in the degenerate setting and for dimension 2, of conjectures of Kudla and Kudla–Rapoport that predict relations between the intersection numbers of special cycles and the Fourier coefficients of automorphic forms.

Keywords

Shimura varietiesarithmetic cyclesEisenstein seriesKudla programme

MSC classification

Primary: 14G35: Modular and Shimura varieties
Secondary: 11G18: Arithmetic aspects of modular and Shimura varieties 11F27: Theta series; Weil representation; theta correspondences
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 16 , Issue 5 , November 2017 , pp. 899 - 945
Copyright
© Cambridge University Press 2015 

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