Four Perspectives on Secondary Terms in the Davenport-Heilbronn Theorems

This paper is an expanded version of the author's lecture at the Integers Conference 2011. We discuss the secondary terms in the Davenport-Heilbronn theorems on cubic fields and 3-torsion in class groups of quadratic fields. Such secondary terms had been conjectured by Datskovsky-Wright and Roberts, and proofs of these or closely related secondary terms were obtained independently by Bhargava, Shankar, and Tsimerman; Hough; Zhao; and Taniguchi and the author. In this paper we discuss the history of the problem and highlight the diverse methods used to address it.

[1]  David P. Roberts.  Density of cubic field discriminants , 2001, Math. Comput..

[2]  H. Cohen,et al.  Counting cubic extensions with given quadratic resolvent , 2010, 1003.1869.

[3]  H. Cohen,et al.  Counting discriminants of number fields , 2006 .

[4]  Takashi Taniguchi,et al.  Secondary terms in counting functions for cubic fields , 2011, 1102.2914.

[5]  H. Cohen.  Enumerating quartic dihedral extensions of Q with signatures , 2003 .

[6]  David J. Wright.  Twists of the Iwasawa-Tate zeta function , 1989 .

[7]  L. Mao.  Divisibility of Class Numbers of Imaginary Quadratic Fields , 2003 .

[8]  C. Pomerance,et al.  Error estimates for the Davenport-Heilbronn theorems , 2010 .

[9]  David J. Wright.  The adelic zeta function associated to the space of binary cubic forms , 1985 .

[10]  Andrew Odlyzko,et al.  Bounds for discriminants and related estimates for class numbers, regulators and zeros of zeta functions : a survey of recent results , 1990 .

[11]  J. Tate.  Fourier analysis in number fields and Hecke's zeta-functions , 1950 .

[12]  Henri Cohen,et al.  Heuristics on class groups of number fields , 1984 .

[13]  B. Datskovsky,et al.  Density of discriminants of cubic extensions. , 1988 .

[14]  D. Faddeev,et al.  The theory of irrationalities of the third degree , 2009 .

[15]  Manjul Bhargava,et al.  The density of discriminants of quartic rings and fields , 2005, 1005.5578.

[16]  Integral points on elliptic curves and 3-torsion in class groups , 2004, math/0405180.

[17]  W. Duke.  Hyperbolic distribution problems and half-integral weight Maass forms , 1988 .

[18]  Franz Lemmermeyer,et al.  Class Field Towers , 2010 .

[19]  Andrew Marc Baily.  On the density of discriminants of quartic fields. , 1980 .

[20]  Jordan S. Ellenberg,et al.  The number of extensions of a number field with fixed degree and bounded discriminant , 2003 .

[21]  Mikio Sato,et al.  A classification of irreducible prehomogeneous vector spaces and their relative invariants , 1977, Nagoya Mathematical Journal.

[22]  Fourier coefficients of modular forms on G2 , 2002 .

[23]  T. Shintani,et al.  On zeta functions associated with prehomogeneous vector spaces. , 1972, Proceedings of the National Academy of Sciences of the United States of America.

[24]  L. Pierce,et al.  The 3‐part of Class Numbers of Quadratic Fields , 2005 .

[25]  Bjorn Poonen,et al.  Bertini theorems over finite fields , 2002, math/0204002.

[26]  David J. Wright,et al.  Prehomogeneous vector spaces and field extensions , 1992 .

[27]  M. Bhargava.  Higher composition laws and applications , 2006 .

[28]  H. Davenport,et al.  On the Density of Discriminants of Cubic Fields , 1969 .

[29]  M. Bhargava.  Mass Formulae for Extensions of Local Fields, and Conjectures on the Density of Number Field Discriminants , 2007 .

[30]  David J. Wright,et al.  The adelic zeta function associated to the space of binary cubic forms. II: Local theory. , 1986 .

[31]  Jordan S. Ellenberg,et al.  Reflection Principles and Bounds for Class Group Torsion , 2007 .

[32]  J. Tyrrell,et al.  ALGEBRAIC NUMBER THEORY , 1969 .