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Abstract Markoff modâ graphs are conjectured to be connected for all primes . In this paper, we use results of Chen and Bourgain, Gamburd, and Sarnak to confirm the conjecture for all . We also provide a method that quickly verifies connectivity for many primes below this bound. In our study of Markoff modâ graphs, we introduce the notion ofmaximal divisorsof a number. We prove sharp asymptotic and explicit upper bounds on the number of maximal divisors, which ultimately improves the Markoff graph âbound by roughly 140 orders of magnitude as compared with an approach using all divisors.
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Differentiating Siegel Modular Forms and the Moving Slope of đ g
Abstract We study the cone of moving divisors on the moduli space $${\mathcal{A}}_{g}$$ of principally polarized abelian varieties. Partly motivated by the generalized RankinâCohen bracket, we construct a non-linear holomorphic differential operator that sends Siegel modular forms to Siegel modular forms, and we apply it to produce new modular forms. Our construction recovers the known divisors of minimal moving slope on $${\mathcal{A}}_{g}$$ for $$g\leq 4$$, and gives an explicit upper bound for the moving slope of $${\mathcal{A}}_{5}$$ and a conjectural upper bound for the moving slope of $${\mathcal{A}}_{6}$$.
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- Award ID(s):
- 2101631
- PAR ID:
- 10589004
- Publisher / Repository:
- Oxford University Press
- Date Published:
- Journal Name:
- International Mathematics Research Notices
- Volume:
- 2024
- Issue:
- 4
- ISSN:
- 1073-7928
- Page Range / eLocation ID:
- 3442 to 3486
- 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/10.1093/imrn/rnad203
