The determinant of the Eisenstein matrix and Hilbert class fields

Isaac Efrat, Peter Sarnak

Type: Article

Publication Date: 1985-01-01

Citations: 24

DOI: https://doi.org/10.1090/s0002-9947-1985-0792829-1

Abstract

We compute the determinant of the Eisenstein matrix associated to the Hilbert-Blumenthal modular group <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="PSL Subscript 2 Baseline left-parenthesis script upper O Subscript k Baseline right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mtext>PSL</mml:mtext> <mml:mn>2</mml:mn> </mml:msub> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">O</mml:mi> </mml:mrow> <mml:mi>k</mml:mi> </mml:msub> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">{\text {PSL}_2}({\mathcal {O}_k})</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, and express it in terms of the zeta function of the Hilbert class field of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K"> <mml:semantics> <mml:mi>K</mml:mi> <mml:annotation encoding="application/x-tex">K</mml:annotation> </mml:semantics> </mml:math> </inline-formula>.

Locations

Transactions of the American Mathematical Society - View - PDF
Transactions of the American Mathematical Society - View - PDF

Similar Works

Action	Title	Year	Authors
+ PDF	The Determinant of the Eisenstein Matrix and Hilbert Class Fields	1985	I. Efrat P. Sarnak
+ PDF	Determinant expression of Selberg zeta functions. II	1992	Shin-ya Koyama
+	On 𝑝-adic Hermitian Eisenstein series	2006	Shōyū Nagaoka
+ PDF	The scattering matrix for the Hilbert modular group	2009	Riad Masri
+ PDF	Petersson's trace formula and the Hecke eigenvalues of Hilbert modular forms	2008	Andrew Knightly Charles Li
+	Vector-valued Modular Forms, Computational Considerations	2020	Tobias Magnusson
+ PDF	Determinant expression of Selberg zeta functions. III	1991	Shin-ya Koyama
+	Eisenstein’s Program and Modular Forms	2020	A. L. Smirnov
+	Codes over rings of size four, Hermitian lattices, and corresponding theta functions	2007	Tony Shaska G. S. Wijesiri
+ PDF	An explicit modular equation in two variables for 𝑄(√3)	1988	Harvey Cohn Jesse Ira Deutsch
+ PDF	Computing isogeny covariant differential modular forms	2004	Chris Hurlburt
+ PDF	Fourier expansion of Eisenstein series on the Hilbert modular group and Hilbert class fields	2002	Claus Sorensen
+ PDF	The Dedekind-Mertens formula and determinantal rings	1999	Winfried Bruns Anna Guerrieri
+	Eisenstein series and elliptic functions on <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"><mml:msub><mml:mi>Γ</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mn>10</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:math>	2010	S. Barry Cooper Heung Yeung Lam
+	Modularity of residual Galois extensions and the Eisenstein ideal	2019	Tobias Berger Krzysztof Klosin
+	Hilbert modular forms and Galois representations	2024	Ajith Nair Ajmain Yamin
+ PDF	An explicit modular equation in two variables and Hilbert’s twelfth problem	1982	Harvey Cohn
+ PDF Chat	Eisenstein series of even weight 𝑘≥2 and integral binary quadratic forms	2021	Andreas Mono
+ PDF	Exponential sums of Lerch’s zeta functions	1985	Kai Wang
+	The Chowla-Selberg formula for quartic Abelian CM fields	2015	Robert Cass

Works That Cite This (24)

Action	Title	Year	Authors
+	Explicit descriptions of spectral properties of Laplacians on spheres $${\mathbb {S}}^{N}\,(N\ge 1)$$: a review	2020	Richard Olu Awonusika
+	Zeta-functions of binary Hermitian forms and special values of Eisenstein series on three-dimensional hyperbolic space	1987	J�rgen Elstrodt Fritz Grunewald J. Mennicke
+	Eisenstein matrix and existence of cusp forms in rank one symmetric spaces	1993	Andrei Reznikov
+ PDF	Determinant expression of Selberg zeta functions. I	1991	Shin-ya Koyama
+ PDF	Hilbert-Asai Eisenstein series, regularized products, and heat kernels	1999	Jay Jorgenson Serge Lang
+	Prime geodesic theorem for hyperbolic 3-manifolds: general cofinite cases	2004	Maki Nakasuji
+ PDF Chat	Statistical mechanics of 2+1 gravity from Riemann zeta function and Alexander polynomial: exact results	2001	Arkady L. Kholodenko
+	Hilbert-Asai Eisenstein Series, Regularized Products, and Heat Kernels	2001	Jay Jorgenson Serge Lang
+	Functional determinant of Laplacian on Cayley projective plane $${\mathbf {P}}^\mathbf{2 }$$($${\mathbf {Cay}}$$)	2019	Richard Olu Awonusika
+	Constructing 1-cusped isospectral non-isometric hyperbolic 3-manifolds	2015	Stavros Garoufalidis Alan W. Reid

Works Cited by This (5)

Action	Title	Year	Authors
+ PDF	The arithmetic and geometry of some hyperbolic three manifolds	1983	Peter Sarnak
+	Lectures on the Theory of Algebraic Numbers	1981	E. Hecke
+	The Selberg trace formula for 𝑃𝑆𝐿₂(𝑅)ⁿ	1987	Isaac Efrat
+ PDF	Algebraic number theory	1999	Kazuya Katô Nobushige Kurokawa Takeshi Saito
+ PDF	The Selberg Trace Formula for PSL(2,ℝ)	1983	Dennis A. Hejhal