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2013 Multiplicative excellent families of elliptic surfaces of type $E_7$ or $E_8$
Abhinav Kumar, Tetsuji Shioda
Algebra Number Theory 7(7): 1613-1641 (2013). DOI: 10.2140/ant.2013.7.1613


We describe explicit multiplicative excellent families of rational elliptic surfaces with Galois group isomorphic to the Weyl group of the root lattices E7 or E8. The Weierstrass coefficients of each family are related by an invertible polynomial transformation to the generators of the multiplicative invariant ring of the associated Weyl group, given by the fundamental characters of the corresponding Lie group. As an application, we give examples of elliptic surfaces with multiplicative reduction and all sections defined over for most of the entries of fiber configurations and Mordell–Weil lattices described by Oguiso and Shioda, as well as examples of explicit polynomials with Galois group W(E7) or W(E8).


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Abhinav Kumar. Tetsuji Shioda. "Multiplicative excellent families of elliptic surfaces of type $E_7$ or $E_8$." Algebra Number Theory 7 (7) 1613 - 1641, 2013.


Received: 8 April 2012; Revised: 13 December 2012; Accepted: 10 January 2013; Published: 2013
First available in Project Euclid: 20 December 2017

zbMATH: 1281.14030
MathSciNet: MR3117502
Digital Object Identifier: 10.2140/ant.2013.7.1613

Primary: 14J27
Secondary: 11G05 , 12F10 , 13A50

Keywords: inverse Galois problem , Mordell–Weil group , multiplicative invariants , rational elliptic surfaces , Weyl group

Rights: Copyright © 2013 Mathematical Sciences Publishers

Vol.7 • No. 7 • 2013
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