scholarly journals Theta series and number fields: theorems and experiments

2021 ◽  
Author(s):  
Adrian Barquero-Sanchez ◽  
Guillermo Mantilla-Soler ◽  
Nathan C. Ryan
Keyword(s):  
Theta Series ◽  
Number Fields

ON THE CLASSIFICATION OF EVEN UNIMODULAR LATTICES WITH A COMPLEX STRUCTURE

2012 ◽  
Vol 08 (04) ◽  
pp. 983-992 ◽  
Author(s):  
MICHAEL HENTSCHEL ◽  
ALOYS KRIEG ◽  
GABRIELE NEBE
Keyword(s):  
Modular Group ◽  
Complex Structure ◽  
Theta Series ◽  
Number Fields ◽  
Quadratic Number ◽  
Imaginary Quadratic Fields ◽  
Rings Of Integers ◽  
Hermitian Modular Form ◽  
Imaginary Quadratic Number

This paper classifies the even unimodular lattices that have a structure as a Hermitian [Formula: see text]-lattice of rank r ≤ 12 for rings of integers in imaginary quadratic number fields K of class number 1. The Hermitian theta series of such a lattice is a Hermitian modular form of weight r for the full modular group, therefore we call them theta lattices. For arbitrary imaginary quadratic fields we derive a mass formula for the principal genus of theta lattices which is applied to show completeness of the classifications.

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 Cutting towers of number fields

2021 ◽  
Author(s):  
Farshid Hajir ◽  
Christian Maire ◽  
Ravi Ramakrishna
Keyword(s):  
Number Fields

Kummer theory for number fields via entanglement groups

2021 ◽  
Author(s):  
Antonella Perucca ◽  
Pietro Sgobba ◽  
Sebastiano Tronto
Keyword(s):  
Number Fields ◽  
Kummer Theory

scholarly journals Hyperbolic tessellations and generators of for imaginary quadratic fields

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.

scholarly journals The 2-rank of the class group of some real cyclic quartic number fields

2021 ◽  
Vol 131 (1) ◽  
Author(s):  
Abdelmalek Azizi ◽  
Mohammed Tamimi ◽  
Abdelkader Zekhnini
Keyword(s):  
Class Group ◽  
Number Fields
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