Hough, Robert D. The local zeta function in enumerating quartic fields. (English) Zbl 1454.11175 J. Number Theory 210, 1-131 (2020). Summary: An exact formula is obtained for the Fourier transform of the local condition of maximality modulo primes \(p > 3\) in the prehomogeneous vector space \(2 \otimes \operatorname{Sym}^2( \mathbb{Z}_p^3)\) parametrizing quartic fields, thus solving the local ‘quartic case’ in enumerating quartic fields. MSC: 11N45 Asymptotic results on counting functions for algebraic and topological structures 11R16 Cubic and quartic extensions 11S90 Prehomogeneous vector spaces 11T23 Exponential sums Keywords:quartic number fields; prehomogeneous vector space; group action; orbit decomposition; exponential sum Software:Magma; Mathematica PDFBibTeX XMLCite \textit{R. D. Hough}, J. Number Theory 210, 1--131 (2020; Zbl 1454.11175) Full Text: DOI arXiv References: [1] Bhargava, Manjul, Higher composition laws. II. On cubic analogues of Gauss composition, Ann. Math. (2), 159, 2, 865-886 (2004) · Zbl 1169.11044 [2] Bhargava, Manjul, Higher composition laws. III. The parametrization of quartic rings, Ann. Math. (2), 159, 3, 1329-1360 (2004) · Zbl 1169.11045 [3] Bhargava, Manjul, Higher composition laws. IV. The parametrization of quintic rings, Ann. Math. (2), 167, 1, 53-94 (2008) · Zbl 1173.11058 [4] Bosma, Wieb; Cannon, John; Playoust, Catherine, The Magma algebra system. I. The user language, J. Symb. Comput., 24, 235-265 (1997) · Zbl 0898.68039 [5] Hough, Robert, The shape of quartic fields (2017), arXiv preprint · Zbl 1470.11273 [6] Sato, Mikio; Shintani, Takuro, On zeta functions associated with prehomogeneous vector spaces, Ann. Math. (2), 100, 131-170 (1974) · Zbl 0309.10014 [7] Shintani, Takuro, On Dirichlet series whose coefficients are class numbers of integral binary cubic forms, J. Math. Soc. Jpn., 24, 132-188 (1972) · Zbl 0227.10031 [8] Taniguchi, Takashi; Thorne, Frank, Orbital L-functions for the space of binary cubic forms, Can. J. Math., 65, 6, 1320-1383 (2013) · Zbl 1370.11107 [9] Taniguchi, Takashi; Thorne, Frank, Secondary terms in counting functions for cubic fields, Duke Math. J., 162, 13, 2451-2508 (2013) · Zbl 1294.11192 [10] Taniguchi, Takashi; Thorne, Frank, Orbital exponential sums for prehomogeneous vector space (2016) · Zbl 1506.11148 [11] Wolfram Research, Inc., Mathematica, Version 11.3 (2018), Champaign, IL [12] Wright, D. J.; Yukie, A., Prehomogeneous vector spaces and field extensions, Invent. Math., 110, 2, 283-314 (1992) · Zbl 0803.12004 [13] Yukie, A., Shintani Zeta Functions, London Mathematical Society Lecture Note Series, vol. 183 (1993), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0801.11021 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.