Formulas for Diophantine quintuples containing two pairs of conjugates in some quadratic fields

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.

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Steinberg homology, modular forms, and real quadratic fields

Journal of Number Theory ◽

10.1016/j.jnt.2020.12.014 ◽

2021 ◽

Author(s):

Avner Ash ◽

Dan Yasaki

Keyword(s):

Modular Forms ◽

Quadratic Fields ◽

Real Quadratic Fields ◽

Real Quadratic

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

International Mathematics Research Notices ◽

10.1093/imrn/rnaa326 ◽

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.

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Hyperbolic tessellations and generators of for imaginary quadratic fields

Forum of Mathematics Sigma ◽

10.1017/fms.2021.9 ◽

2021 ◽

Vol 9 ◽

Author(s):

David Burns ◽

Rob de Jeu ◽

Herbert Gangl ◽

Alexander D. Rahm ◽

Dan Yasaki

Keyword(s):

Number Field ◽

Number Fields ◽

Quadratic Fields ◽

Quadratic Number ◽

Imaginary Quadratic Fields ◽

Quadratic Number Fields ◽

Formula Group ◽

Imaginary Quadratic Number ◽

Direct Calculations

Abstract We develop methods for constructing explicit generators, modulo torsion, of the $K_3$ -groups of imaginary quadratic number fields. These methods are based on either tessellations of hyperbolic $3$ -space or on direct calculations in suitable pre-Bloch groups and lead to the very first proven examples of explicit generators, modulo torsion, of any infinite $K_3$ -group of a number field. As part of this approach, we make several improvements to the theory of Bloch groups for $ K_3 $ of any field, predict the precise power of $2$ that should occur in the Lichtenbaum conjecture at $ -1 $ and prove that this prediction is valid for all abelian number fields.

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Prime geodesics and averages of the Zagier L-series

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004121000384 ◽

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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