scholarly journals Even positive definite unimodular quadratic forms over real quadratic fields

1989 ◽  
Vol 19 (3) ◽  
pp. 725-734 ◽  
Author(s):  
J.S. Hsia
Keyword(s):  
Quadratic Forms ◽  
Positive Definite ◽  
Quadratic Fields ◽  
Real Quadratic Fields ◽  
Real Quadratic

scholarly journals UNIVERSAL QUADRATIC FORMS AND ELEMENTS OF SMALL NORM IN REAL QUADRATIC FIELDS

2016 ◽  
Vol 94 (1) ◽  
pp. 7-14 ◽  
Author(s):  
VÍTĚZSLAV KALA
Keyword(s):  
Positive Integer ◽  
Quadratic Forms ◽  
Quadratic Fields ◽  
Real Quadratic Fields ◽  
Small Norm ◽  
Real Quadratic

For any positive integer $M$ we show that there are infinitely many real quadratic fields that do not admit $M$-ary universal quadratic forms (without any restriction on the parity of their cross coefficients).

scholarly journals Number fields without n-ary universal quadratic forms

2015 ◽  
Vol 159 (2) ◽  
pp. 239-252 ◽  
Author(s):  
VALENTIN BLOMER ◽  
VÍTĚZSLAV KALA
Keyword(s):  
Positive Integer ◽  
Quadratic Forms ◽  
Number Fields ◽  
Quadratic Fields ◽  
Real Quadratic Fields ◽  
Real Quadratic

AbstractGiven any positive integer M, we show that there are infinitely many real quadratic fields that do not admit universal quadratic forms with even cross coefficients in M variables.

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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