Bisymmetric functions, Macdonald polynomials and sl3 basic hypergeometric series

Warnaar, S. O. (2008) Bisymmetric functions, Macdonald polynomials and sl3 basic hypergeometric series. Compositio Mathematica, 144 02: 271-303. doi:10.1112/S0010437X07003211

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Author Warnaar, S. O.
Title Bisymmetric functions, Macdonald polynomials and sl3 basic hypergeometric series
Formatted title
Bisymmetric functions, Macdonald polynomials and sl3 basic hypergeometric series
Journal name Compositio Mathematica   Check publisher's open access policy
ISSN 1570-5846
0010-437X
Publication date 2008-03-01
Sub-type Article (original research)
DOI 10.1112/S0010437X07003211
Volume 144
Issue 02
Start page 271
End page 303
Total pages 33
Editor B. Edixhoven
B. Moonen
Place of publication Dordrecht, The Netherlands
Publisher Cambridge University Press
Language eng
Subject 010101 Algebra and Number Theory
010104 Combinatorics and Discrete Mathematics (excl. Physical Combinatorics)
0101 Pure Mathematics
Abstract Given a curve of genus 3 with an unramified double cover, we give an explicit description of the associated Prym variety. We also describe how an unramified double cover of a non-hyperelliptic genus 3 curve can be mapped into the Jacobian of a curve of genus 2 over its field of definition and how this can be used to perform Chabauty- and Brauer–Manin-type calculations for curves of genus 5 with an fixed-point-free involution. As an application, we determine the rational points on a smooth plane quartic and give examples of curves of genus 3 and 5 violating the Hasse principle. The methods are, in principle, applicable to any genus 3 curve with a double cover. We also show how these constructions can be used to design smooth plane quartics with specific arithmetic properties. As an example, we give a smooth plane quartic with all 28 bitangents defined over $\mathbb {Q}(t)$. By specialization, this also gives examples over $\mathbb {Q}$.
Keyword Prym varieties
Chabauty methods
Rational points on curves
Covering techniques
Brauer–Manin
Smooth plane quartics
Q-Index Code C1
Q-Index Status Confirmed Code

 
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