Effective moments of Dirichlet L-functions in Galois orbits

2019 ◽  
Vol 12 (3) ◽  
pp. 475-490
Author(s):  
Rizwanur Khan ◽  
Ruoyun Lei ◽  
Djordje Milićević
Keyword(s):  
Galois Orbits

scholarly journals Finite simple groups with short Galois orbits on conjugacy classes

2020 ◽  
Vol 544 ◽  
pp. 151-169
Author(s):  
Victor Bovdi ◽  
Thomas Breuer ◽  
Attila Maróti
Keyword(s):  
Conjugacy Classes ◽  
Simple Groups ◽  
Finite Simple Groups ◽  
Galois Orbits

RÉPARTITION GALOISIENNE D'UNE CLASSE D'ISOGÉNIE DE COURBES ELLIPTIQUES

2012 ◽  
Vol 09 (02) ◽  
pp. 517-543
Author(s):  
RODOLPHE RICHARD
Keyword(s):  
New York ◽  
Elliptic Curves ◽  
Ergodic Theory ◽  
Mathematical Society ◽  
Princeton University ◽  
Galois Orbits ◽  
Theoretic Proof ◽  
Arithmetic Theory ◽  
Invent Math ◽  
Automorphic Functions

Dans cet article, on montre que les orbites sous Galois des invariants modulaires associés à des courbes elliptiques complexes sans multiplication complexe variant dans une même classe d'isogénie s'équidistribuent dans la courbe modulaire vers la probabilité hyperbolique. La démonstration repose sur des arguments de théorie ergodique, notamment le théorème de Ratner (cf. [A. Eskin et H. Oh, Ergodic theoretic proof of equidistribution of Hecke points, Ergodic Theory Dynam. Systems26(1) (2006) 163–167]), ainsi que sur le théorème de l'image ouverte de Serre [J.-P. Serre, Abelian l-Adic Representations and Elliptic Curves (W. A. Benjamin, New York, 1968); Propriétés Galoisiennes des points d'ordre fini des courbes elliptiques, Invent. Math.15(4) (1972) 259–331] dans le cas où les invariants modulaires considérés sont algébriques sur Q, et des résultats de G. Shimura dans le cas transcendant [Introduction to the Arithmetic Theory of Automorphic Functions, Publications of the Mathematical Society of Japan (Princeton University Press, Princeton, NJ, 1994)]. In this article, it is shown that Galois orbits of invariants associated with non-CM and pairwise isogeneous complex elliptic curves equidistribute in the classical modular curve towards the hyperbolic probability measure. The proof is based on arguments from ergodic theory, especially Ratner's theorem on unipotent flows (cf. [A. Eskin and H. Oh, Ergodic theoretic proof of equidistribution of Hecke points, Ergodic Theory Dynam. Systems26(1) (2006) 163–167]), as well as on Serre's open image theorem [J.-P. Serre, Abelian l-Adic Representations and Elliptic Curves (W. A. Benjamin, New York, 1968); Propriétés Galoisiennes des points d'ordre fini des courbes elliptiques, Invent. Math.15(4) (1972) 259–331] in case of algebraic invariants, and on G. Shimura's work in the transcendant case [Introduction to the Arithmetic Theory of Automorphic Functions, Publications of the Mathematical Society of Japan (Princeton University Press, Princeton, NJ, 1994)].

scholarly journals Galois orbits of TQFTs: symmetries and unitarity

2022 ◽  
Vol 2022 (1) ◽  
Author(s):  
Matthew Buican ◽  
Rajath Radhakrishnan
Keyword(s):  
Fixed Point ◽  
Sufficient Conditions ◽  
Field Theories ◽  
Quantum Field Theories ◽  
Quantum Field ◽  
Fermion Model ◽  
Topological Quantum ◽  
Galois Orbits ◽  
Galois Actions ◽  
Toric Code

Abstract We study Galois actions on 2+1D topological quantum field theories (TQFTs), characterizing their interplay with theory factorization, gauging, the structure of gapped boundaries and dualities, 0-form symmetries, 1-form symmetries, and 2-groups. In order to gain a better physical understanding of Galois actions, we prove sufficient conditions for the preservation of unitarity. We then map out the Galois orbits of various classes of unitary TQFTs. The simplest such orbits are trivial (e.g., as in various theories of physical interest like the Toric Code, Double Semion, and 3-Fermion Model), and we refer to such theories as unitary “Galois fixed point TQFTs”. Starting from these fixed point theories, we study conditions for preservation of Galois invariance under gauging 0-form and 1-form symmetries (as well as under more general anyon condensation). Assuming a conjecture in the literature, we prove that all unitary Galois fixed point TQFTs can be engineered by gauging 0-form symmetries of theories built from Deligne products of certain abelian TQFTs.

scholarly journals The distribution of Galois orbits of points of small height in toric varieties

2019 ◽  
Vol 141 (2) ◽  
pp. 309-381
Author(s):  
José Ignacio Burgos Gil ◽  
Patrice Philippon ◽  
Juan Rivera-Letelier ◽  
Martín Sombra
Keyword(s):  
Toric Varieties ◽  
Galois Orbits ◽  
Small Height

scholarly journals GENERALISED FERMAT HYPERMAPS AND GALOIS ORBITS

2009 ◽  
Vol 51 (2) ◽  
pp. 289-299 ◽  
Author(s):  
ANTOINE D. COSTE ◽  
GARETH A. JONES ◽  
MANFRED STREIT ◽  
JÜRGEN WOLFART
Keyword(s):  
Galois Group ◽  
Riemann Surfaces ◽  
Prime Power ◽  
Algebraic Curves ◽  
Complete Bipartite Graph ◽  
Number Fields ◽  
Absolute Galois Group ◽  
Graph Theoretic ◽  
Galois Orbits ◽  
Fermat Curves

AbstractWe consider families of quasiplatonic Riemann surfaces characterised by the fact that – as in the case of Fermat curves of exponent n – their underlying regular (Walsh) hypermap is an embedding of the complete bipartite graph Kn,n, where n is an odd prime power. We show that these surfaces, regarded as algebraic curves, are all defined over abelian number fields. We determine their orbits under the action of the absolute Galois group, their minimal fields of definition and in some easier cases their defining equations. The paper relies on group – and graph – theoretic results by G. A. Jones, R. Nedela and M. Škoviera about regular embeddings of the graphs Kn,n [7] and generalises the analogous results for maps obtained in [9], partly using different methods.

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