Moments, Exponential Sums, and Monodromy Groups
We determine the geometric monodromy groups attached to various families, both one-parameter and multi-parameter, of exponential sums over finite fields, or, more precisely, the geometric monodromy groups of the $\ell $ -adic local systems on affine spaces in characteristic $p> 0$ wh...
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Cambridge University Press
2025-01-01
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| Series: | Forum of Mathematics, Sigma |
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| Online Access: | https://www.cambridge.org/core/product/identifier/S2050509425100625/type/journal_article |
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| author | Nicholas M. Katz Pham Huu Tiep |
| author_facet | Nicholas M. Katz Pham Huu Tiep |
| author_sort | Nicholas M. Katz |
| collection | DOAJ |
| description | We determine the geometric monodromy groups attached to various families, both one-parameter and multi-parameter, of exponential sums over finite fields, or, more precisely, the geometric monodromy groups of the
$\ell $
-adic local systems on affine spaces in characteristic
$p> 0$
whose trace functions are these exponential sums. The exponential sums here are much more general than we previously were able to consider. As a byproduct, we determine the number of irreducible components of maximal dimension in certain intersections of Fermat surfaces. We also show that in any family of such local systems, say parameterized by an affine space S, there is a dense open set of S over which the geometric monodromy group of the corresponding local system is a fixed known group. |
| format | Article |
| id | doaj-art-34b29f3e67d149d09837535de3b4a653 |
| institution | Kabale University |
| issn | 2050-5094 |
| language | English |
| publishDate | 2025-01-01 |
| publisher | Cambridge University Press |
| record_format | Article |
| series | Forum of Mathematics, Sigma |
| spelling | doaj-art-34b29f3e67d149d09837535de3b4a6532025-08-20T03:27:44ZengCambridge University PressForum of Mathematics, Sigma2050-50942025-01-011310.1017/fms.2025.10062Moments, Exponential Sums, and Monodromy GroupsNicholas M. Katz0Pham Huu Tiep1Department of Mathematics, https://ror.org/00hx57361Princeton University, Princeton, NJ 08544, USA; E-mail:Department of Mathematics, https://ror.org/05vt9qd57Rutgers University, Piscataway, NJ 08854, USA;We determine the geometric monodromy groups attached to various families, both one-parameter and multi-parameter, of exponential sums over finite fields, or, more precisely, the geometric monodromy groups of the $\ell $ -adic local systems on affine spaces in characteristic $p> 0$ whose trace functions are these exponential sums. The exponential sums here are much more general than we previously were able to consider. As a byproduct, we determine the number of irreducible components of maximal dimension in certain intersections of Fermat surfaces. We also show that in any family of such local systems, say parameterized by an affine space S, there is a dense open set of S over which the geometric monodromy group of the corresponding local system is a fixed known group.https://www.cambridge.org/core/product/identifier/S2050509425100625/type/journal_article11T2320C3314J4520C1520D0620G4022E46 |
| spellingShingle | Nicholas M. Katz Pham Huu Tiep Moments, Exponential Sums, and Monodromy Groups Forum of Mathematics, Sigma 11T23 20C33 14J45 20C15 20D06 20G40 22E46 |
| title | Moments, Exponential Sums, and Monodromy Groups |
| title_full | Moments, Exponential Sums, and Monodromy Groups |
| title_fullStr | Moments, Exponential Sums, and Monodromy Groups |
| title_full_unstemmed | Moments, Exponential Sums, and Monodromy Groups |
| title_short | Moments, Exponential Sums, and Monodromy Groups |
| title_sort | moments exponential sums and monodromy groups |
| topic | 11T23 20C33 14J45 20C15 20D06 20G40 22E46 |
| url | https://www.cambridge.org/core/product/identifier/S2050509425100625/type/journal_article |
| work_keys_str_mv | AT nicholasmkatz momentsexponentialsumsandmonodromygroups AT phamhuutiep momentsexponentialsumsandmonodromygroups |