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Abstract We consider the generating series of special cycles on , with full level structure, valued in the cohomology of degree . The modularity theorem of Kudla–Millson for locally symmetric spaces implies that these series are modular. When , the images of these loci in are the ‐elliptic Noether–Lefschetz loci, which are conjectured to be modular. In the Appendix, it is shown that the resulting modular forms are nonzero for when and .
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Definite orthogonal modular forms: computations, excursions, and discoveries
Abstract We consider spaces of modular forms attached to definite orthogonal groups of low even rank and nontrivial level, equipped with Hecke operators defined by Kneser neighbours. After reviewing algorithms to compute with these spaces, we investigate endoscopy using theta series and a theorem of Rallis. Along the way, we exhibit many examples and pose several conjectures. As a first application, we express counts of Kneser neighbours in terms of coefficients of classical or Siegel modular forms, complementing work of Chenevier–Lannes. As a second application, we prove new instances of Eisenstein congruences of Ramanujan and Kurokawa–Mizumoto type.
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- Award ID(s):
- 1946311
- PAR ID:
- 10420429
- Date Published:
- Journal Name:
- Research in Number Theory
- Volume:
- 8
- Issue:
- 4
- ISSN:
- 2522-0160
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1007/s40993-022-00373-2
