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On two arithmetic theta lifts

Part of: Arithmetic algebraic geometry Discontinuous groups and automorphic forms

Published online by Cambridge University Press:  07 September 2018

Stephan Ehlen  and
Siddarth Sankaran
Stephan Ehlen
Affiliation:
Mathematisches Institut, University of Cologne, Weyertal 86-90, 50931 Cologne, Germany email stephan.ehlen@math.uni-koeln.de
Siddarth Sankaran
Affiliation:
Department of Mathematics, University of Manitoba, 420 Machray Hall, Winnipeg, Canada email siddarth.sankaran@umanitoba.ca
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Abstract

Our aim is to clarify the relationship between Kudla’s and Bruinier’s Green functions attached to special cycles on Shimura varieties of orthogonal and unitary type, which play a key role in the arithmetic geometry of these cycles in the context of Kudla’s program. In particular, we show that the generating series obtained by taking the differences of the two families of Green functions is a non-holomorphic modular form and has trivial (cuspidal) holomorphic projection. Along the way, we construct a section of the Maaß lowering operator for moderate growth forms valued in the Weil representation using a regularized theta lift, which may be of independent interest, as it in particular has applications to mock modular forms. We also consider arithmetic-geometric applications to integral models of $U(n,1)$ Shimura varieties. Each family of Green functions gives rise to a formal arithmetic theta function, valued in an arithmetic Chow group, that is conjectured to be modular; our main result is the modularity of the difference of the two arithmetic theta functions. Finally, we relate the arithmetic heights of the special cycles to special derivatives of Eisenstein series, as predicted by Kudla’s conjecture, and describe a refinement of a theorem of Bruinier, Howard and Yang on arithmetic intersections against CM points.

MSC classification

Primary: 11F12: Automorphic forms, one variable 11F27: Theta series; Weil representation; theta correspondences 11G18: Arithmetic aspects of modular and Shimura varieties 11G15: Complex multiplication and moduli of abelian varieties
Type
Research Article
Information
Compositio Mathematica , Volume 154 , Issue 10 , October 2018 , pp. 2090 - 2149
Copyright
© The Authors 2018 

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