Abstract
Let k be a field, let G be a reductive group, and let V be a linear representation of G. Let \(V/\!/G =\mathop{ \mathrm{Spec}}\nolimits ({\mathrm{Sym}}^{{\ast}}(V ^{{\ast}}))^{G}\) denote the geometric quotient and let \(\pi: V \rightarrow V/\!/G\) denote the quotient map. Arithmetic invariant theory studies the map π on the level of k-rational points. In this article, which is a continuation of the results of our earlier paper “Arithmetic invariant theory”, we provide necessary and sufficient conditions for a rational element of \(V/\!\!/G\) to lie in the image of π, assuming that generic stabilizers are abelian. We illustrate the various scenarios that can occur with some recent examples of arithmetic interest.
To David Vogan on his 60th birthday
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
M. Bhargava, Higher composition laws I: A new view on Gauss composition, and quadratic generalizations, Ann. of Math. 159 (2004), no. 1, 217–250.
M. Bhargava, Higher composition laws II: On cubic analogues of Gauss composition, Ann. of Math. 159 (2004), no. 2, 865–886.
M. Bhargava, Most hyperelliptic curves over \(\mathbb{Q}\) have no rational points, (2013)http://arxiv.org/abs/1308.0395.
M. Bhargava and B. Gross, Arithmetic invariant theory in Symmetry: Representation Theory and Its Applications; In Honor of Nolan R. Wallach, Progress in Mathematics Vol. 257 (2014), 33–54.
M. Bhargava and B. Gross, The average size of the 2-Selmer group of the Jacobians of hyperelliptic curves with a rational Weierstrass point in Automorphic Representations and L-functions, TIFR Studies in Math 22 (2013), 23–91.
M. Bhargava, B. Gross, and X. Wang, A positive proportion of locally soluble hyperelliptic curves over \(\mathbb{Q}\) have no point over any old degree extension, to appear in JAMS.
M. Bhargava and W. Ho, Coregular spaces and genus one curves (2013), http://arxiv.org/abs/1306.4424.
B. J. Birch and J. R. Merriman, Finiteness theorems for binary forms, Proc. London Math. Soc. s3-24 (1972), 385–394.
N. Bruin, B. Poonen, M. Stoll, Generalized explicit descent and its application to curves of genus 3 (2012), http://arxiv.org/abs/1205.4456.
J. Giraud, Cohomologie non abélienne, Die Grundlehren der math. Wissenschaften, Band 179, Springer-Verlag 1971.
B. Gross, On Bhargava’s representations and Vinberg’s invariant theory, in: Frontiers of Mathematical Sciences, International Press (2011), 317–321.
W-T. Gan, B. Gross, and D. Prasad, Sur les conjectures de Gross et Prasad, Astérisque 346, 2012.
A. Knus, A. Merkurjev, M. Rost, and J.-P. Tignol, The book of involutions, AMS Colloquium Publications 44, 1998.
H. Jacquet and S. Rallis, On the Gross-Prasad conjecture for unitary groups, Clay Math. Proc. 13, 205–264, Amer. Math. Soc., Providence, RI, 2011.
G. Laumon and L. Moret-Bailly, Champs algébriques, Springer-Verlag, 2000.
J. Milnor and D. Husemoller, Symmetric Bilinear Forms, Springer Ergebnisse 73, 1970.
J. Nakagawa, Binary forms and orders of algebraic number fields, Invent. Math. 97 (1989), 219–235.
D. Panyushev, On invariant theory of \(\theta\) -groups, J. Algebra 283 (2005), 655–670.
B. Poonen and E. F. Schaefer, Explicit descent for Jacobians of cyclic covers of the projective line, J. Reine Angew. Math. 488 (1997), 141–188.
V. L. Popov and E. B. Vinberg, Invariant Theory, in Algebraic Geometry IV, Encylopaedia of Mathematical Sciences, Vol. 55, Springer-Verlag, 1994.
S. Rallis and G. Shiffman, Multiplicity one conjectures (2007), http://arxiv.org/abs/0705.2168.
E. F. Schaefer, 2-descent on the Jacobians of hyperelliptic curves, J. Number Theory 51 (1995), 219–232.
J-P. Serre, Galois Cohomology, Springer Monographs in Mathematics, 2002.
J-P. Serre, Local Fields, Springer Graduate Texts in Mathematics, Vol. 67, 1995.
A. Shankar and X. Wang, Average size of the 2-Selmer group for monic even hyperelliptic curves (2013), http://arxiv.org/abs/1307.3531.
J. Thorne, The arithmetic of simple singularities, Ph.D. Thesis, Harvard University, 2012.
X. Wang, Maximal linear spaces contained in the the base loci of pencils of quadrics (2013), http://arxiv.org/abs/1302.2385.
M. M. Wood, Rings and ideals parametrized by binary n-ic forms, J. London Math. Soc. (2) 83 (2011), 208–231.
M. M. Wood, Parametrization of ideal classes in rings associated to binary forms, J. reine angew. Math. (Crelle), Vol. 2014, Issue 689, 169–199, DOI: 10.1515/crelle-2012-0058.
W. Zhang, Fourier transform and the global Gan–Gross–Prasad conjecture for unitary groups, Ann. of Math. Vol. 180 (2014), Issue 3, 971–1049.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2015 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Bhargava, M., Gross, B.H., Wang, X. (2015). Arithmetic invariant theory II: Pure inner forms and obstructions to the existence of orbits. In: Nevins, M., Trapa, P. (eds) Representations of Reductive Groups. Progress in Mathematics, vol 312. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-23443-4_5
Download citation
DOI: https://doi.org/10.1007/978-3-319-23443-4_5
Published:
Publisher Name: Birkhäuser, Cham
Print ISBN: 978-3-319-23442-7
Online ISBN: 978-3-319-23443-4
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)