Kummer for Genus One over Prime Order Fields

2017 ◽  
pp. 3-32 ◽  
Author(s):  
Sabyasachi Karati ◽  
Palash Sarkar
Keyword(s):  
Prime Order ◽  
Genus One

Kummer for Genus One Over Prime-Order Fields

2019 ◽  
Vol 33 (1) ◽  
pp. 92-129
Author(s):  
Sabyasachi Karati ◽  
Palash Sarkar
Keyword(s):  
Prime Order ◽  
Genus One

ACTION OF A GROTHENDIECK–TEICHMÜLLER GROUP ON TORSION ELEMENTS OF FULL TEICHMÜLLER MODULAR GROUPS OF GENUS ONE

2012 ◽  
Vol 08 (03) ◽  
pp. 763-787
Author(s):  
BENJAMIN COLLAS
Keyword(s):  
Finite Order ◽  
Prime Order ◽  
Mapping Class Groups ◽  
Mapping Class ◽  
Generalized Equations ◽  
Class Groups ◽  
Galois Action ◽  
Modular Groups ◽  
Genus One ◽  
Invent Math

In this paper we define a new version of Grothendieck–Teichmüller group [Formula: see text] defined by three generalized equations coming from finite-order diffeomorphisms, and we prove that it is isomorphic to the known version IΓ of the Grothendieck–Teichmüller defined in [H. Nakamura and L. Schneps, On a subgroup of the Grothendieck–Teichmüller group acting on the tower of profinite Teichmüller modular groups, Invent. Math.141 (2000) 503–560]. We show that [Formula: see text] acts on the full mapping class groups [Formula: see text] for 2g - 2 + n > 0. We then prove that the conjugacy classes of prime-order torsion of [Formula: see text] are exactly the discrete prime-order ones of [Formula: see text]. Using this we prove that [Formula: see text] acts on prime-order torsion elements of [Formula: see text] in a particular way called λ-conjugacy, analogous to the Galois action on inertia.

scholarly journals Sequence of a Coxiella endosymbiont of the tick Amblyomma nuttalli suggests a pattern of convergent genome reduction in the Coxiella genus

2020 ◽  
Author(s):  
Tiago Nardi ◽  
Emanuela Olivieri ◽  
Edward Kariuki ◽  
Davide Sassera ◽  
Michele Castelli
Keyword(s):  
Coxiella Burnetii ◽  
Phylogenetic Analyses ◽  
Genome Reduction ◽  
Metabolic Profiles ◽  
The Novel ◽  
Bacterial Symbionts ◽  
Human Pathogen ◽  
Host Genus ◽  
Supportive Role ◽  
Genus One

Abstract Ticks require bacterial symbionts for the provision of necessary compounds that are absent in their hematophagous diet. Such symbionts are frequently vertically transmitted and, most commonly, belong to the Coxiella genus, which also includes the human pathogen Coxiella burnetii. This genus can be divided in four main clades, presenting partial but incomplete co-cladogenesis with the tick hosts. Here we report the genome sequence of a novel Coxiella, endosymbiont of the African tick Amblyomma nuttalli, and the ensuing comparative analyses. Its size (~1 Mb) is intermediate between symbionts of Rhipicephalus species and other Amblyomma species. Phylogenetic analyses show that the novel sequence is the first genome of the B clade, the only one for which no genomes were previously available. Accordingly, it allows to draw an enhanced scenario of the evolution of the genus, one of parallel genome reduction of different endosymbiont lineages, which are now at different stages of reduction from a more versatile ancestor. Gene content comparison allows to infer that the ancestor could be reminiscent of Coxiella burnetii. Interestingly, the convergent loss of mismatch repair could have been a major driver of such reductive evolution. Predicted metabolic profiles are rather homogenous among Coxiella endosymbionts, in particular vitamin biosynthesis, consistently with a host-supportive role. Concurrently, similarities among Coxiella endosymbionts according to host genus and despite phylogenetic unrelatedness hint at possible host-dependent effects.

scholarly journals Mazur’s rational torsion result for pointless genus one curves: examples

2021 ◽  
Vol 7 (1) ◽  
Author(s):  
Arjan Dwarshuis ◽  
Majken Roelfszema ◽  
Jaap Top
Keyword(s):  
Algebraic Geometry ◽  
Elliptic Curves ◽  
Rational Point ◽  
Torsion Points ◽  
Arbitrary Genus ◽  
Mazur’S Theorem ◽  
Genus One

AbstractThis note reformulates Mazur’s result on the possible orders of rational torsion points on elliptic curves over $$\mathbb {Q}$$ Q in a way that makes sense for arbitrary genus one curves, regardless whether or not the curve contains a rational point. The main result is that explicit examples are provided of ‘pointless’ genus one curves over $$\mathbb {Q}$$ Q corresponding to the torsion orders 7, 8, 9, 10, 12 (and hence, all possibilities) occurring in Mazur’s theorem. In fact three distinct methods are proposed for constructing such examples, each involving different in our opinion quite nice ideas from the arithmetic of elliptic curves or from algebraic geometry.

scholarly journals Two-loop superstring five-point amplitudes. Part II. Low energy expansion and S-duality

2021 ◽  
Vol 2021 (2) ◽  
Author(s):  
Eric D’Hoker ◽  
Carlos R. Mafra ◽  
Boris Pioline ◽  
Oliver Schlotterer
Keyword(s):  
Moduli Space ◽  
Effective Interaction ◽  
Type Ii ◽  
Leading Order ◽  
Effective Interactions ◽  
Heterotic String Theory ◽  
Genus Two ◽  
Genus One ◽  
Graph Function ◽  
Point Amplitude

Abstract In an earlier paper, we constructed the genus-two amplitudes for five external massless states in Type II and Heterotic string theory, and showed that the α′ expansion of the Type II amplitude reproduces the corresponding supergravity amplitude to leading order. In this paper, we analyze the effective interactions induced by Type IIB superstrings beyond supergravity, both for U(1)R-preserving amplitudes such as for five gravitons, and for U(1)R-violating amplitudes such as for one dilaton and four gravitons. At each order in α′, the coefficients of the effective interactions are given by integrals over moduli space of genus-two modular graph functions, generalizing those already encountered for four external massless states. To leading and sub-leading orders, the coefficients of the effective interactions D2ℛ5 and D4ℛ5 are found to match those of D4ℛ4 and D6ℛ4, respectively, as required by non-linear supersymmetry. To the next order, a D6ℛ5 effective interaction arises, which is independent of the supersymmetric completion of D8ℛ4, and already arose at genus one. A novel identity on genus-two modular graph functions, which we prove, ensures that up to order D6ℛ5, the five-point amplitudes require only a single new modular graph function in addition to those needed for the four-point amplitude. We check that the supergravity limit of U(1)R-violating amplitudes is free of UV divergences to this order, consistently with the known structure of divergences in Type IIB supergravity. Our results give strong consistency tests on the full five-point amplitude, and pave the way for understanding S-duality beyond the BPS-protected sector.

Functional encryption for computational hiding in prime order groups via pair encodings

2017 ◽  
Vol 86 (1) ◽  
pp. 97-120 ◽  
Author(s):  
Jongkil Kim ◽  
Willy Susilo ◽  
Fuchun Guo ◽  
Man Ho Au
Keyword(s):  
Prime Order ◽  
Functional Encryption

scholarly journals There are genus one curves of every index over every infinite, finitely generated field

2019 ◽  
Vol 2019 (749) ◽  
pp. 65-86
Author(s):  
Pete L. Clark ◽  
Allan Lacy
Keyword(s):  
Elliptic Curve ◽  
Abelian Variety ◽  
Finitely Generated ◽  
Weil Theorem ◽  
Genus One

Abstract We show that a nontrivial abelian variety over a Hilbertian field in which the weak Mordell–Weil theorem holds admits infinitely many torsors with period any given n>1 that is not divisible by the characteristic. The corresponding statement with “period” replaced by “index” is plausible but open, and it seems much more challenging. We show that for every infinite, finitely generated field K, there is an elliptic curve E_{/K} which admits infinitely many torsors with index any given n>1 .

scholarly journals On the normalizers of elements of prime order in simple groups

1974 ◽  
Vol 29 (3) ◽  
pp. 387-400 ◽  
Author(s):  
J.A Cohn
Keyword(s):  
Prime Order ◽  
Simple Groups

scholarly journals The eta invariant and equivariant bordism of flat manifolds with cyclic holonomy group of odd prime order

2009 ◽  
Vol 37 (3) ◽  
pp. 275-306 ◽  
Author(s):  
Peter B. Gilkey ◽  
Roberto J. Miatello ◽  
Ricardo A. Podestá
Keyword(s):  
Prime Order ◽  
Holonomy Group ◽  
Eta Invariant ◽  
Flat Manifolds ◽  
Equivariant Bordism
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