Congruence Subgroups, Cusps and Manin Symbols over Number Fields

2014 ◽  
pp. 109-127 ◽  
Author(s):  
J. E. Cremona ◽  
M. T. Aranés
Keyword(s):  
Number Fields ◽  
Congruence Subgroups

Cycles and Harmonic Forms on Locally Symmetric Spaces

1985 ◽  
Vol 28 (1) ◽  
pp. 3-38 ◽  
Author(s):  
John J. Millson
Keyword(s):  
Quadratic Forms ◽  
Symmetric Spaces ◽  
Number Fields ◽  
The Other ◽  
Weil Representation ◽  
Harmonic Forms ◽  
Congruence Subgroups ◽  
Locally Symmetric Spaces ◽  
Background Material ◽  
Totally Real

AbstractTwo constructions of cohomology classes for congruence subgroups of unit groups of quadratic forms over totally real number fields are given and shown to coincide. One is geometric, using cycles, and the other is analytic, using the oscillator (Weil) representation. Considerable background material on this representation is given.

scholarly journals Modular Abelian Varieties Over Number Fields

2014 ◽  
Vol 66 (1) ◽  
pp. 170-196 ◽  
Author(s):  
Xavier Guitart ◽  
Jordi Quer
Keyword(s):  
Cohomology Class ◽  
Abelian Varieties ◽  
Galois Cohomology ◽  
Number Fields ◽  
Abelian Surfaces ◽  
Congruence Subgroups ◽  
Galois Number ◽  
Modular Varieties

AbstractThe main result of this paper is a characterization of the abelian varieties B/K defined over Galois number fields with the property that the L-function L(B/K; s) is a product of L-functions of non-CM newforms over ℚ for congruence subgroups of the form Γ1(N). The characterization involves the structure of End(B), isogenies between the Galois conjugates of B, and a Galois cohomology class attached to B/K.We call the varieties having this property strongly modular. The last section is devoted to the study of a family of abelian surfaces with quaternionic multiplication. As an illustration of the ways in which the general results of the paper can be applied, we prove the strong modularity of some particular abelian surfaces belonging to that family, and we show how to find nontrivial examples of strongly modular varieties by twisting.

Determination of Hauptmoduls and Construction of Abelian Extensions of Quadratic Number Fields

2007 ◽  
Vol 50 (3) ◽  
pp. 334-346
Author(s):  
Hung-Jen Chiang-Hsieh ◽  
Yifan Yang
Keyword(s):  
Number Fields ◽  
Cyclic Extension ◽  
Quadratic Number ◽  
Modular Functions ◽  
Genus Zero ◽  
Congruence Subgroups ◽  
Abelian Extensions ◽  
Quadratic Number Fields

AbstractWe obtain Hauptmoduls of genus zero congruence subgroups of the type (p) := Γ0(p) + wp, where p is a prime and wp is the Atkin–Lehner involution. We then use the Hauptmoduls, along with modular functions on Γ1(p) to construct families of cyclic extensions of quadratic number fields. Further examples of cyclic extension of bi-quadratic and tri-quadratic number fields are also given.

scholarly journals Twisted 6d (2, 0) SCFTs on a circle

2021 ◽  
Vol 2021 (7) ◽  
Author(s):  
Zhihao Duan ◽  
Kimyeong Lee ◽  
June Nahmgoong ◽  
Xin Wang
Keyword(s):  
Lie Groups ◽  
Point Of View ◽  
Partition Functions ◽  
Congruence Subgroups ◽  
Gauge Groups ◽  
Simple Lie Groups ◽  
Instanton Contribution ◽  
Elliptic Genera ◽  
Supersymmetric Gauge ◽  
The One

Abstract We study twisted circle compactification of 6d (2, 0) SCFTs to 5d $$ \mathcal{N} $$ N = 2 supersymmetric gauge theories with non-simply-laced gauge groups. We provide two complementary approaches towards the BPS partition functions, reflecting the 5d and 6d point of view respectively. The first is based on the blowup equations for the instanton partition function, from which in particular we determine explicitly the one-instanton contribution for all simple Lie groups. The second is based on the modular bootstrap program, and we propose a novel modular ansatz for the twisted elliptic genera that transform under the congruence subgroups Γ0(N) of SL(2, ℤ). We conjecture a vanishing bound for the refined Gopakumar-Vafa invariants of the genus one fibered Calabi-Yau threefolds, upon which one can determine the twisted elliptic genera recursively. We use our results to obtain the 6d Cardy formulas and find universal behaviour for all simple Lie groups. In addition, the Cardy formulas remain invariant under the twist once the normalization of the compact circle is taken into account.

scholarly journals Cutting towers of number fields

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

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

Kummer theory for number fields via entanglement groups

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