scholarly journals Modularity of generating series of divisors on unitary Shimura varieties

Astérisque ◽  
2020 ◽  
Vol 421 ◽  
pp. 7-125
Author(s):  
Jan H. Bruinier ◽  
Benjamin Howard ◽  
Stephen S. Kudla ◽  
Michael Rapoport ◽  
Tonghai Yang
Keyword(s):  
Shimura Varieties ◽  
Generating Series

scholarly journals On two arithmetic theta lifts

2018 ◽  
Vol 154 (10) ◽  
pp. 2090-2149 ◽  
Author(s):  
Stephan Ehlen ◽  
Siddarth Sankaran
Keyword(s):  
Modular Forms ◽  
Theta Functions ◽  
Chow Group ◽  
Shimura Varieties ◽  
Green Functions ◽  
Theta Lift ◽  
Moderate Growth ◽  
Generating Series ◽  
Cm Points ◽  
The Difference

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.

scholarly journals Unitary cycles on Shimura curves and the Shimura lift II

2014 ◽  
Vol 150 (12) ◽  
pp. 1963-2002
Author(s):  
Siddarth Sankaran
Keyword(s):  
Modular Form ◽  
Recent Work ◽  
Chow Groups ◽  
Shimura Varieties ◽  
Shimura Curves ◽  
Shimura Lift ◽  
Generating Series

AbstractWe consider two families of arithmetic divisors defined on integral models of Shimura curves. The first was studied by Kudla, Rapoport and Yang, who proved that if one assembles these divisors in a formal generating series, one obtains the$q$-expansion of a modular form of weight 3/2. The present work concerns the Shimura lift of this modular form: we identify the Shimura lift with a generating series comprising divisors arising in the recent work of Kudla and Rapoport regarding cycles on Shimura varieties of unitary type. In the prequel to this paper, the author considered the geometry of the two families of cycles. These results are combined with the Archimedean calculations found in this work in order to establish the theorem. In particular, we obtain new examples of modular generating series whose coefficients lie in arithmetic Chow groups of Shimura varieties.

scholarly journals The Gross–Kohnen–Zagier theorem over totally real fields

2009 ◽  
Vol 145 (5) ◽  
pp. 1147-1162 ◽  
Author(s):  
Xinyi Yuan ◽  
Shou-Wu Zhang ◽  
Wei Zhang
Keyword(s):  
Chow Groups ◽  
Product Formula ◽  
Shimura Varieties ◽  
Generating Series ◽  
Totally Real ◽  
Totally Real Fields

AbstractOn Shimura varieties of orthogonal type over totally real fields, we prove a product formula and the modularity of Kudla’s generating series of special cycles in Chow groups.

scholarly journals Special cycles on toroidal compactifications of orthogonal Shimura varieties

2021 ◽  
Author(s):  
Jan Hendrik Bruinier ◽  
Shaul Zemel
Keyword(s):  
Shimura Varieties ◽  
Green Functions ◽  
Toroidal Compactifications ◽  
Generating Series ◽  
Automorphic Green Functions

AbstractWe determine the behavior of automorphic Green functions along the boundary components of toroidal compactifications of orthogonal Shimura varieties. We use this analysis to define boundary components of special divisors and prove that the generating series of the resulting special divisors on a toroidal compactification is modular.

scholarly journals Generating series of a new class of orthogonal Shimura varieties

2020 ◽  
Vol 14 (10) ◽  
pp. 2743-2772
Author(s):  
Eugenia Rosu ◽  
Dylan Yott
Keyword(s):  
Shimura Varieties ◽  
New Class ◽  
Generating Series

scholarly journals Modularity of generating series of divisors on unitary Shimura varieties II: arithmetic applications

Astérisque ◽  
2020 ◽  
Vol 421 ◽  
pp. 127-186
Author(s):  
Jan H. Bruinier ◽  
Benjamin Howard ◽  
Stephen S. Kudla ◽  
Michael Rapoport ◽  
Tonghai Yang
Keyword(s):  
Shimura Varieties ◽  
Generating Series

scholarly journals KUDLA’S MODULARITY CONJECTURE AND FORMAL FOURIER–JACOBI SERIES

2015 ◽  
Vol 3 ◽  
Author(s):  
JAN HENDRIK BRUINIER ◽  
MARTIN WESTERHOLT-RAUM
Keyword(s):  
Modular Forms ◽  
Formal Series ◽  
Siegel Modular Forms ◽  
Shimura Varieties ◽  
Jacobi Forms ◽  
Jacobi Series ◽  
Generating Series ◽  
Arbitrary Codimension ◽  
Jacobi Expansions ◽  
Natural Symmetry

We prove modularity of formal series of Jacobi forms that satisfy a natural symmetry condition. They are formal analogs of Fourier–Jacobi expansions of Siegel modular forms. From our result and a theorem of Wei Zhang, we deduce Kudla’s conjecture on the modularity of generating series of special cycles of arbitrary codimension and for all orthogonal Shimura varieties.

scholarly journals Vertical Distribution Relations for Special Cycles on Unitary Shimura Varieties

2018 ◽  
Vol 2020 (13) ◽  
pp. 3902-3926
Author(s):  
Réda Boumasmoud ◽  
Ernest Hunter Brooks ◽  
Dimitar P Jetchev
Keyword(s):  
Vertical Distribution ◽  
Three Dimensional ◽  
Complex Multiplication ◽  
Horizontal Distribution ◽  
Unitary Groups ◽  
Shimura Varieties ◽  
Heegner Points ◽  
Universal Norm

Abstract We consider cycles on three-dimensional Shimura varieties attached to unitary groups, defined over extensions of a complex multiplication (CM) field $E$, which appear in the context of the conjectures of Gan et al. [6]. We establish a vertical distribution relation for these cycles over an anticyclotomic extension of $E$, complementing the horizontal distribution relation of [8], and use this to define a family of norm-compatible cycles over these fields, thus obtaining a universal norm construction similar to the Heegner $\Lambda $-module constructed from Heegner points.

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