Universität zu Köln

Ehlen, Stephan ORCID: 0000-0003-2029-6219 and Sankaran, Siddarth (2018). On two arithmetic theta lifts. Compos. Math., 154 (10). S. 2090 - 2150. CAMBRIDGE: CAMBRIDGE UNIV PRESS. ISSN 1570-5846

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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 Maa beta 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.

Item Type:	Journal Article
Creators:	Creators Email ORCID ORCID Put Code Ehlen, Stephan UNSPECIFIED orcid.org/0000-0003-2029-6219 UNSPECIFIED Sankaran, Siddarth UNSPECIFIED UNSPECIFIED UNSPECIFIED
URN:	urn:nbn:de:hbz:38-171173
DOI:	10.1112/S0010437X18007327
Journal or Publication Title:	Compos. Math.
Volume:	154
Number:	10
Page Range:	S. 2090 - 2150
Date:	2018
Publisher:	CAMBRIDGE UNIV PRESS
Place of Publication:	CAMBRIDGE
ISSN:	1570-5846
Language:	English
Faculty:	Faculty of Mathematics and Natural Sciences
Divisions:	Faculty of Mathematics and Natural Sciences > Department of Mathematics and Computer Science > Mathematical Institute
Subjects:	no entry
Uncontrolled Keywords:	Keywords Language SIEGEL-WEIL FORMULA; UNITARY GROUPS; BORCHERDS FORMS; HEIGHT PAIRINGS; DERIVATIVES; SERIES; CYCLES Multiple languages Mathematics Multiple languages
Refereed:	Yes
URI:	http://kups.ub.uni-koeln.de/id/eprint/17117