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Counting abelian
varieties over finite fields via Frobenius densities
Jeffrey D. Achter, S. Ali Altuğ, Luis Garcia and Julia GordonAppendix: Wen-Wei Li and Thomas Rüd |
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Vol. 17 (2023), No. 7, 1239–1280
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DOI: 10.2140/ant.2023.17.1239
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Abstract |
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Let be a principally polarized abelian variety over a finite field with commutative endomorphism ring; further suppose that either is ordinary or the field is prime. Motivated by an equidistribution heuristic, we introduce a factor for each place of , and show that the product of these factors essentially computes the size of the isogeny class of . The derivation of this mass formula depends on a formula of Kottwitz and on analysis of measures on the group of symplectic similitudes and, in particular, does not rely on a calculation of class numbers. |
Keywords
abelian variety, isogeny class, orbital integral
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Mathematical Subject Classification 2010
Primary: 11G10
Secondary: 14G15, 20G25, 22E35
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Milestones
Received: 31 July 2019
Revised: 2 May 2022
Accepted: 10 June 2022
Published: 30 May 2023
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Authors |
| Jeffrey D. Achter | |
| Department of Mathematics Colorado State University Fort Collins, CO United States |
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| S. Ali Altuğ | |
| New York, NY United States |
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| Luis Garcia | |
| Department of Mathematics University College London United Kingdom |
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| Julia Gordon | |
| Department of Mathematics University of British Columbia Vancouver, BC Canada |
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| Wen-Wei Li | |
| Beijing International Center for
Mathematical Research Peking University Beijing China |
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| Thomas Rüd | |
| Massachusetts Institute of
Technology Cambridge, MA United States |
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| © 2023 MSP (Mathematical Sciences Publishers). Distributed under the Creative Commons Attribution License 4.0 (CC BY). |
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