Multiplicative excellent families of elliptic surfaces of type E7 or E8

Kumar, Abhinav; Shioda, Tetsuji

doi:10.2140/ant.2013.7.1613

Download this article
	For screen
	For printing

Recent Issues

The Journal

About the journal

Ethics and policies

Peer-review process

Submission guidelines

ISSN 1944-7833 (online)

ISSN 1937-0652 (print)

Author index

To appear

Other MSP journals

Multiplicative excellent families of elliptic surfaces of type $E_7$ or $E_8$

Abhinav Kumar and Tetsuji Shioda

Vol. 7 (2013), No. 7, 1613–1641

DOI: 10.2140/ant.2013.7.1613

Abstract

We describe explicit multiplicative excellent families of rational elliptic surfaces with Galois group isomorphic to the Weyl group of the root lattices $E_{7}$ or $E_{8}$ . 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 $ℚ$ 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 $W (E_{7})$ or $W (E_{8})$ .

Keywords

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

Mathematical Subject Classification 2010

Primary: 14J27

Secondary: 11G05, 12F10, 13A50

Milestones

Received: 8 April 2012

Revised: 13 December 2012

Accepted: 10 January 2013

Published: 12 October 2013

Authors

Abhinav Kumar
Department of Mathematics Massachusetts Institute of Technology 77 Massachusetts Avenue Cambridge, MA 02139 United States

Tetsuji Shioda
Department of Mathematics Rikkyo University 3-34-1 Nishi-Ikebukuro Toshima-ku Tokyo 171-8501 Japan
http://www.rkmath.rikkyo.ac.jp/math/shioda/

Vol. 7, No. 7, 2013