scholarly journals Units of Real Quadratic Fields

1971 ◽  
Vol 44 ◽  
pp. 51-55 ◽  
Author(s):  
Akira Takaku
Keyword(s):  
Positive Solution ◽  
Quadratic Fields ◽  
Pell’S Equation ◽  
Real Quadratic Fields ◽  
Real Quadratic

1. Let D be a positive square-free integer. Throughout this note we shall use the following notations;d = d(D): the discriminant of ,t0, u0: the least positive solution of Pell’s equation t2 — du2 = 4,

scholarly journals Units and Class Numbers of Real Quadratic Fields

1970 ◽  
Vol 37 ◽  
pp. 61-65
Author(s):  
Hideo Yokoi
Keyword(s):  
Quadratic Field ◽  
Integral Solution ◽  
Quadratic Fields ◽  
Class Numbers ◽  
Ideal Class ◽  
Real Quadratic Field ◽  
Pell’S Equation ◽  
Real Quadratic Fields ◽  
Real Quadratic ◽  
The Ideal

The aim of this paper is to prove the following main theorem: THEOREM. For the discriminant d>0 of a real quadratic field let (x,y) = (t,u) be the least positive integral solution of Pell’s equation x2 — dy2 = 4 and put and denote by hd the ideal class number.

scholarly journals EXTREME VALUES OF GEODESIC PERIODS ON ARITHMETIC HYPERBOLIC SURFACES

2020 ◽  
pp. 1-36
Author(s):  
Bart Michels
Keyword(s):  
Lower Bound ◽  
Closed Geodesic ◽  
Extreme Values ◽  
Theta Series ◽  
Hyperbolic Surface ◽  
Quadratic Fields ◽  
Maass Forms ◽  
Real Quadratic Fields ◽  
Central Values ◽  
Real Quadratic

Abstract Given a closed geodesic on a compact arithmetic hyperbolic surface, we show the existence of a sequence of Laplacian eigenfunctions whose integrals along the geodesic exhibit nontrivial growth. Via Waldspurger’s formula we deduce a lower bound for central values of Rankin-Selberg L-functions of Maass forms times theta series associated to real quadratic fields.

scholarly journals On the Number of Quadratic Twists with a Rational Point of Almost Minimal Height

2020 ◽  
Author(s):  
Joachim Petit
Keyword(s):  
Asymptotic Formula ◽  
Elliptic Curve ◽  
Asymptotic Estimate ◽  
Rational Point ◽  
Quadratic Fields ◽  
Real Quadratic Fields ◽  
Quadratic Twists ◽  
The Family ◽  
Minimal Height ◽  
Real Quadratic

Abstract We investigate the number of curves having a rational point of almost minimal height in the family of quadratic twists of a given elliptic curve. This problem takes its origin in the work of Hooley, who asked this question in the setting of real quadratic fields. In particular, he showed an asymptotic estimate for the number of such fields with almost minimal fundamental unit. Our main result establishes the analogue asymptotic formula in the setting of quadratic twists of a fixed elliptic curve.

scholarly journals Prime geodesics and averages of the Zagier L-series

2021 ◽  
pp. 1-24
Author(s):  
OLGA BALKANOVA ◽  
DMITRY FROLENKOV ◽  
MORTEN S. RISAGER
Keyword(s):  
Asymptotic Expansion ◽  
Asymptotic Expansions ◽  
Error Term ◽  
Quadratic Fields ◽  
Real Quadratic Fields ◽  
Prime Geodesic Theorem ◽  
Error Terms ◽  
Average Size ◽  
Real Quadratic

Abstract The Zagier L-series encode data of real quadratic fields. We study the average size of these L-series, and prove asymptotic expansions and omega results for the expansion. We then show how the error term in the asymptotic expansion can be used to obtain error terms in the prime geodesic theorem.

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