Is There an Analytic Theory of Automorphic Functions for Complex Algebraic Curves?
The geometric Langlands correspondence for complex algebraic curves differs from the original Langlands correspondence for number fields in that it is formulated in terms of sheaves rather than functions (in the intermediate case of curves over finite fields, both formulations are possible). In a re...
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| Veröffentlicht in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Datum: | 2020 |
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| Sprache: | English |
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| Zitieren: | Is There an Analytic Theory of Automorphic Functions for Complex Algebraic Curves?. Edward Frenkel. SIGMA 16 (2020), 042, 31 pages |
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Frenkel, Edward 2025-12-15T15:25:43Z 2020 Is There an Analytic Theory of Automorphic Functions for Complex Algebraic Curves?. Edward Frenkel. SIGMA 16 (2020), 042, 31 pages 1815-0659 2020 Mathematics Subject Classification: 14D24; 17B67; 22E57 arXiv:1812.08160 https://nasplib.isofts.kiev.ua/handle/123456789/210708 https://doi.org/10.3842/SIGMA.2020.042 The geometric Langlands correspondence for complex algebraic curves differs from the original Langlands correspondence for number fields in that it is formulated in terms of sheaves rather than functions (in the intermediate case of curves over finite fields, both formulations are possible). In a recent preprint, Robert Langlands made a proposal for developing an analytic theory of automorphic forms on the moduli space of G-bundles on a complex algebraic curve. Langlands envisioned these forms as eigenfunctions of some analogues of Hecke operators. In these notes, I show that if G is an abelian group, then there are well-defined Hecke operators, and I give a complete description of their eigenfunctions and eigenvalues. For non-abelian G, Hecke operators involve integration, which presents some difficulties. However, there is an alternative approach to developing an analytic theory of automorphic forms, based on the existence of a large commutative algebra of global differential operators acting on half-densities on the moduli stack of G-bundles. This approach (which implements some ideas of Joerg Teschner) is outlined here, as a preview of a joint work with Pavel Etingof and David Kazhdan. The first version of this paper was based on the notes of my talk at the 6th Abel Conference, University of Minnesota, November 2018. I thank Roberto Alvarenga, Julia Gordon, Ivan Fesenko, David Kazhdan, and Raven Waller for valuable discussions. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Is There an Analytic Theory of Automorphic Functions for Complex Algebraic Curves? Article published earlier |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| title |
Is There an Analytic Theory of Automorphic Functions for Complex Algebraic Curves? |
| spellingShingle |
Is There an Analytic Theory of Automorphic Functions for Complex Algebraic Curves? Frenkel, Edward |
| title_short |
Is There an Analytic Theory of Automorphic Functions for Complex Algebraic Curves? |
| title_full |
Is There an Analytic Theory of Automorphic Functions for Complex Algebraic Curves? |
| title_fullStr |
Is There an Analytic Theory of Automorphic Functions for Complex Algebraic Curves? |
| title_full_unstemmed |
Is There an Analytic Theory of Automorphic Functions for Complex Algebraic Curves? |
| title_sort |
is there an analytic theory of automorphic functions for complex algebraic curves? |
| author |
Frenkel, Edward |
| author_facet |
Frenkel, Edward |
| publishDate |
2020 |
| language |
English |
| container_title |
Symmetry, Integrability and Geometry: Methods and Applications |
| publisher |
Інститут математики НАН України |
| format |
Article |
| description |
The geometric Langlands correspondence for complex algebraic curves differs from the original Langlands correspondence for number fields in that it is formulated in terms of sheaves rather than functions (in the intermediate case of curves over finite fields, both formulations are possible). In a recent preprint, Robert Langlands made a proposal for developing an analytic theory of automorphic forms on the moduli space of G-bundles on a complex algebraic curve. Langlands envisioned these forms as eigenfunctions of some analogues of Hecke operators. In these notes, I show that if G is an abelian group, then there are well-defined Hecke operators, and I give a complete description of their eigenfunctions and eigenvalues. For non-abelian G, Hecke operators involve integration, which presents some difficulties. However, there is an alternative approach to developing an analytic theory of automorphic forms, based on the existence of a large commutative algebra of global differential operators acting on half-densities on the moduli stack of G-bundles. This approach (which implements some ideas of Joerg Teschner) is outlined here, as a preview of a joint work with Pavel Etingof and David Kazhdan.
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| issn |
1815-0659 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/210708 |
| citation_txt |
Is There an Analytic Theory of Automorphic Functions for Complex Algebraic Curves?. Edward Frenkel. SIGMA 16 (2020), 042, 31 pages |
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AT frenkeledward isthereananalytictheoryofautomorphicfunctionsforcomplexalgebraiccurves |
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2025-12-17T12:04:33Z |
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2025-12-17T12:04:33Z |
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