Maass form twisted Shintani ℒ-functions

Hough, Bob

doi:10.1090/proc/13563

Maass form twisted Shintani $\mathcal {L}$-functions
HTML articles powered by AMS MathViewer

by Bob Hough

Proc. Amer. Math. Soc. 145 (2017), 4161-4174

DOI: https://doi.org/10.1090/proc/13563

Published electronically: April 6, 2017

PDF | Request permission

Abstract:

The Maass form twisted Shintani $\mathcal {L}$-functions are introduced, and some of their analytic properties are studied. These functions contain data regarding the distribution of shapes of cubic rings.

References

Manjul Bhargava and Piper Harron, The equidistribution of lattice shapes of rings of integers in cubic, quartic, and quintic number fields, Compos. Math. 152 (2016), no. 6, 1111–1120. MR 3518306, DOI 10.1112/S0010437X16007260
Manjul Bhargava, Higher composition laws. II. On cubic analogues of Gauss composition, Ann. of Math. (2) 159 (2004), no. 2, 865–886. MR 2081442, DOI 10.4007/annals.2004.159.865
B. N. Delone and D. K. Faddeev, The theory of irrationalities of the third degree, Translations of Mathematical Monographs, Vol. 10, American Mathematical Society, Providence, R.I., 1964. MR 0160744
Wee Teck Gan, Benedict Gross, and Gordan Savin, Fourier coefficients of modular forms on $G_2$, Duke Math. J. 115 (2002), no. 1, 105–169. MR 1932327, DOI 10.1215/S0012-7094-02-11514-2
Robert Harron, The shapes of pure cubic fields, Proc. Amer. Math. Soc. 145 (2017), no. 2, 509–524. MR 3577857, DOI 10.1090/proc/13309
B. Hough, Equidistribution of bounded torsion CM points, arXiv, abs/1005.1458, 2010.
Henry H. Kim, Functoriality for the exterior square of $\textrm {GL}_4$ and the symmetric fourth of $\textrm {GL}_2$, J. Amer. Math. Soc. 16 (2003), no. 1, 139–183. With appendix 1 by Dinakar Ramakrishnan and appendix 2 by Kim and Peter Sarnak. MR 1937203, DOI 10.1090/S0894-0347-02-00410-1
Wenzhi Luo, Zeév Rudnick, and Peter Sarnak, The variance of arithmetic measures associated to closed geodesics on the modular surface, J. Mod. Dyn. 3 (2009), no. 2, 271–309. MR 2504745, DOI 10.3934/jmd.2009.3.271
Fumihiro Sat\B{o}, Zeta functions of prehomogeneous vector spaces with coefficients related to periods of automorphic forms, Proc. Indian Acad. Sci. Math. Sci. 104 (1994), no. 1, 99–135. K. G. Ramanathan memorial issue. MR 1280061, DOI 10.1007/BF02830877
Fumihiro Sato, Zeta functions of $(\textrm {SL}_2\times \textrm {SL}_2\times \textrm {GL}_2,\textbf {M}_2\oplus \textbf {M}_2)$ associated with a pair of Maass cusp forms, Comment. Math. Univ. St. Pauli 55 (2006), no. 1, 77–95. MR 2252001
A. Selberg, Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet series, J. Indian Math. Soc. (N.S.) 20 (1956), 47–87. MR 88511
Takuro Shintani, On Dirichlet series whose coefficients are class numbers of integral binary cubic forms, J. Math. Soc. Japan 24 (1972), 132–188. MR 289428, DOI 10.2969/jmsj/02410132
Takashi Taniguchi and Frank Thorne, Orbital $L$-functions for the space of binary cubic forms, Canad. J. Math. 65 (2013), no. 6, 1320–1383. MR 3121674, DOI 10.4153/CJM-2013-027-0
David Charles Terr, The distribution of shapes of cubic orders, ProQuest LLC, Ann Arbor, MI, 1997. Thesis (Ph.D.)–University of California, Berkeley. MR 2697241
Matthew P. Young, The fourth moment of Dirichlet $L$-functions, Ann. of Math. (2) 173 (2011), no. 1, 1–50. MR 2753598, DOI 10.4007/annals.2011.173.1.1