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The Eichler-Selberg trace formulas express the traces of Hecke operators on a spaces of cusp forms in terms of weighted sums of Hurwitz-Kronecker class numbers. For cusp forms on $$\text {\rm SL}_2(\mathbb{Z}),$$ Zagier proved these formulas by cleverly making use of the weight 3/2 nonholomorphic Eisenstein series he discovered in the 1970s. The holomorphic part of this form, its so-called {\it mock modular form}, is the generating function for these class numbers. In this expository note we revisit Zagier's method, and we show how to obtain such formulas for congruence subgroups, working out the details for $$\Gamma_0(2)$$ and $$\Gamma_0(4).$$ The trace formulas fall out naturally from the computation of the Rankin-Cohen brackets of Zagier's mock modular form with specific theta functions.
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Iwasawa–Greenberg main conjecture for nonordinary modular forms and Eisenstein congruences on GU(3,1)
Abstract In this paper, we prove one divisibility of the Iwasawa–Greenberg main conjecture for the Rankin–Selberg product of a weight two cusp form and an ordinary complex multiplication form of higher weight, using congruences between Klingen Eisenstein series and cusp forms on $$\mathrm {GU}(3,1)$$ , generalizing an earlier result of the third-named author to allow nonordinary cusp forms. The main result is a key input in the third-named author’s proof of Kobayashi’s $$\pm $$ -main conjecture for supersingular elliptic curves. The new ingredient here is developing a semiordinary Hida theory along an appropriate smaller weight space and a study of the semiordinary Eisenstein family.
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- PAR ID:
- 10424678
- Date Published:
- Journal Name:
- Forum of Mathematics, Sigma
- Volume:
- 10
- ISSN:
- 2050-5094
- 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/10.1017/fms.2022.95
