Sato–Tate distributions and Galois endomorphism modules in genus 2
For an abelian surface A over a number field k, we study the limiting distribution of the normalized Euler factors of the L-function of A. This distribution is expected to correspond to taking characteristic polynomials of a uniform random matrix in some closed subgroup of USp(4); this Sato–Tate gro...
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| Format: | Article |
| Language: | en_US |
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Cambridge University Press
2013
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| Online Access: | http://hdl.handle.net/1721.1/80369 |
| _version_ | 1811075232785498112 |
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| author | Kedlaya, Kiran S. Sutherland, Andrew Victor Fite, Francesc Rotger, Victor |
| author2 | Massachusetts Institute of Technology. Department of Mathematics |
| author_facet | Massachusetts Institute of Technology. Department of Mathematics Kedlaya, Kiran S. Sutherland, Andrew Victor Fite, Francesc Rotger, Victor |
| author_sort | Kedlaya, Kiran S. |
| collection | MIT |
| description | For an abelian surface A over a number field k, we study the limiting distribution of the normalized Euler factors of the L-function of A. This distribution is expected to correspond to taking characteristic polynomials of a uniform random matrix in some closed subgroup of USp(4); this Sato–Tate group may be obtained from the Galois action on any Tate module of A. We show that the Sato–Tate group is limited to a particular list of 55 groups up to conjugacy. We then classify A according to the Galois module structure on the ℝ-algebra generated by endomorphisms of [superscript A][line over Q] (the Galois type), and establish a matching with the classification of Sato–Tate groups; this shows that there are at most 52 groups up to conjugacy which occur as Sato–Tate groups for suitable A and k, of which 34 can occur for k=ℚ. Finally, we present examples of Jacobians of hyperelliptic curves exhibiting each Galois type (over ℚ whenever possible), and observe numerical agreement with the expected Sato–Tate distribution by comparing moment statistics. |
| first_indexed | 2024-09-23T10:02:51Z |
| format | Article |
| id | mit-1721.1/80369 |
| institution | Massachusetts Institute of Technology |
| language | en_US |
| last_indexed | 2024-09-23T10:02:51Z |
| publishDate | 2013 |
| publisher | Cambridge University Press |
| record_format | dspace |
| spelling | mit-1721.1/803692022-09-30T18:32:43Z Sato–Tate distributions and Galois endomorphism modules in genus 2 Kedlaya, Kiran S. Sutherland, Andrew Victor Fite, Francesc Rotger, Victor Massachusetts Institute of Technology. Department of Mathematics Kedlaya, Kiran S. Sutherland, Andrew Victor For an abelian surface A over a number field k, we study the limiting distribution of the normalized Euler factors of the L-function of A. This distribution is expected to correspond to taking characteristic polynomials of a uniform random matrix in some closed subgroup of USp(4); this Sato–Tate group may be obtained from the Galois action on any Tate module of A. We show that the Sato–Tate group is limited to a particular list of 55 groups up to conjugacy. We then classify A according to the Galois module structure on the ℝ-algebra generated by endomorphisms of [superscript A][line over Q] (the Galois type), and establish a matching with the classification of Sato–Tate groups; this shows that there are at most 52 groups up to conjugacy which occur as Sato–Tate groups for suitable A and k, of which 34 can occur for k=ℚ. Finally, we present examples of Jacobians of hyperelliptic curves exhibiting each Galois type (over ℚ whenever possible), and observe numerical agreement with the expected Sato–Tate distribution by comparing moment statistics. National Science Foundation (U.S.) (CAREER Grant DMS-0545904) National Science Foundation (U.S.) (Grant DMS-1101343) United States. Defense Advanced Research Projects Agency (Grant HR0011-09-1-0048) NEC Research Support Fund Ida M. Green Fellowship University of California, San Diego (Warschawski Professorship) National Science Foundation (U.S.) (Grant DMS-1115455) 2013-09-06T17:03:27Z 2013-09-06T17:03:27Z 2012-07 2011-11 Article http://purl.org/eprint/type/JournalArticle 0010-437X 1570-5846 http://hdl.handle.net/1721.1/80369 Fite, Francesc, Kiran S. Kedlaya, Victor Rotger, and Andrew V. Sutherland. “Sato–Tate distributions and Galois endomorphism modules in genus 2.” Compositio Mathematica 148, no. 05 (September 25, 2012): 1390-1442. © Foundation Compositio Mathematica 2012 en_US http://dx.doi.org/10.1112/s0010437x12000279 Compositio Mathematica Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use. application/pdf Cambridge University Press Other University Web Domain |
| spellingShingle | Kedlaya, Kiran S. Sutherland, Andrew Victor Fite, Francesc Rotger, Victor Sato–Tate distributions and Galois endomorphism modules in genus 2 |
| title | Sato–Tate distributions and Galois endomorphism modules in genus 2 |
| title_full | Sato–Tate distributions and Galois endomorphism modules in genus 2 |
| title_fullStr | Sato–Tate distributions and Galois endomorphism modules in genus 2 |
| title_full_unstemmed | Sato–Tate distributions and Galois endomorphism modules in genus 2 |
| title_short | Sato–Tate distributions and Galois endomorphism modules in genus 2 |
| title_sort | sato tate distributions and galois endomorphism modules in genus 2 |
| url | http://hdl.handle.net/1721.1/80369 |
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