scholarly journals On Elliptic Curves in SL2(ℂ)/Γ.., Schanuel’s Conjecture and Geodesic Lengths

2004 ◽  
Vol 176 ◽  
pp. 159-180 ◽  
Author(s):  
Jörg Winkelmann
Keyword(s):  
Elliptic Curves ◽  
Geodesic Length ◽  
Quotient Manifold ◽  
Cocompact Subgroup ◽  
Discrete Cocompact Subgroup ◽  
Schanuel's Conjecture ◽  
Special Case

AbstractLet Γ be a discrete cocompact subgroup of SL2(ℂ). We conjecture that the quotient manifold X = SL2(ℂ) / Γ contains infinitely many non-isogenous elliptic curves and prove this is indeed the case if Schanuel’s conjecture holds. We also prove it in the special case where Γ ∩ SL2(∝) is cocompact in SL2(ℝ).Furthermore, we deduce some consequences for the geodesic length spectra of real hyperbolic 2- and 3-folds.

scholarly journals On Geometric Flats in the CAT(0) Realization of Coxeter Groups and Tits Buildings

2009 ◽  
Vol 61 (4) ◽  
pp. 740-761 ◽  
Author(s):  
Pierre-Emmanuel Caprace ◽  
Frédéric Haglund
Keyword(s):  
Abelian Group ◽  
Coxeter Group ◽  
Coxeter Groups ◽  
Free Abelian Group ◽  
Gromov Hyperbolic ◽  
Convex Cocompact ◽  
Rank 2 ◽  
Cocompact Subgroup ◽  
Special Case ◽  
Geometric Action

Abstract.Given a complete CAT(0) space X endowed with a geometric action of a group Ⲅ, it is known that if Ⲅ contains a free abelian group of rank n, then X contains a geometric flat of dimension n. We prove the converse of this statement in the special case where X is a convex subcomplex of the CAT(0) realization of a Coxeter group W, and Ⲅ is a subgroup of W. In particular a convex cocompact subgroup of a Coxeter group is Gromov-hyperbolic if and only if it does not contain a free abelian group of rank 2. Our result also provides an explicit control on geometric flats in the CAT(0) realization of arbitrary Tits buildings.

scholarly journals Crystallographic groups with trivial center and outer automorphism group

2017 ◽  
Vol 164 (2) ◽  
pp. 363-368
Author(s):  
RAFAŁ LUTOWSKI ◽  
ANDRZEJ SZCZEPAŃSKI
Keyword(s):  
Automorphism Group ◽  
Outer Automorphism ◽  
Crystallographic Group ◽  
Crystallographic Groups ◽  
Outer Automorphism Group ◽  
Trivial Center ◽  
Cocompact Subgroup ◽  
Discrete Cocompact Subgroup

AbstractLet Γ be a crystallographic group of dimension n, i.e. a discrete, cocompact subgroup of Isom(ℝn) = O(n) ⋉ ℝn. For any n ⩾ 2, we construct a crystallographic group with a trivial center and trivial outer automorphism group.

scholarly journals Abelian surfaces good away from 2

2017 ◽  
Vol 13 (04) ◽  
pp. 991-1001
Author(s):  
Christopher Rasmussen ◽  
Akio Tamagawa
Keyword(s):  
Elliptic Curves ◽  
Number Field ◽  
High Dimension ◽  
Abelian Varieties ◽  
Number Fields ◽  
Sufficient Condition ◽  
Good Reduction ◽  
Abelian Surfaces ◽  
Special Case

Fix a number field [Formula: see text] and a rational prime [Formula: see text]. We consider abelian varieties whose [Formula: see text]-power torsion generates a pro-[Formula: see text] extension of [Formula: see text] which is unramified away from [Formula: see text]. It is a necessary, but not generally sufficient, condition that such varieties have good reduction away from [Formula: see text]. In the special case of [Formula: see text], we demonstrate that for abelian surfaces [Formula: see text], good reduction away from [Formula: see text] does suffice. The result is extended to elliptic curves and abelian surfaces over certain number fields unramified away from [Formula: see text]. An explicit example is constructed to demonstrate that good reduction away from [Formula: see text] is not sufficient, at [Formula: see text], for abelian varieties of sufficiently high dimension.

scholarly journals Variance of sums in arithmetic progressions of divisor functions associated with higher degree L-functions in 𝔽q[t]

2019 ◽  
Vol 16 (05) ◽  
pp. 1013-1030
Author(s):  
Edva Roditty-Gershon ◽  
Chris Hall ◽  
Jonathan P. Keating
Keyword(s):  
Elliptic Curves ◽  
Conjugacy Classes ◽  
Arithmetic Progressions ◽  
Divisor Function ◽  
Legendre Curve ◽  
Divisor Functions ◽  
Matrix Integrals ◽  
Usual Formula ◽  
Special Case

We compute the variances of sums in arithmetic progressions of generalized [Formula: see text]-divisor functions related to certain [Formula: see text]-functions in [Formula: see text], in the limit as [Formula: see text]. This is achieved by making use of recently established equidistribution results for the associated Frobenius conjugacy classes. The variances are thus expressed, when [Formula: see text], in terms of matrix integrals, which may be evaluated. Our results extend those obtained previously in the special case corresponding to the usual [Formula: see text]-divisor function, when the [Formula: see text]-function in question has degree one. They illustrate the role played by the degree of the [Formula: see text]-functions; in particular, we find qualitatively new behavior when the degree exceeds one. Our calculations apply, for example, to elliptic curves defined over [Formula: see text], and we illustrate them by examining in some detail the generalized [Formula: see text]-divisor functions associated with the Legendre curve.

scholarly journals ELLIPTIC CURVES RELATED TO CYCLIC CUBIC EXTENSIONS

2009 ◽  
Vol 05 (04) ◽  
pp. 591-623 ◽  
Author(s):  
RINTARO KOZUMA
Keyword(s):  
Elliptic Curves ◽  
Class Group ◽  
Selmer Group ◽  
Nice Property ◽  
Class Groups ◽  
Cubic Surface ◽  
Unit Class ◽  
Cubic Extension ◽  
Hyperplane Sections ◽  
Special Case

The aim of this paper is to study certain family of elliptic curves [Formula: see text] defined over a number field F arising from hyperplane sections of some cubic surface [Formula: see text] associated to a cyclic cubic extension K/F. We show that each [Formula: see text] admits a 3-isogeny ϕ over F and the dual Selmer group [Formula: see text] is bounded by a kind of unit/class groups attached to K/F. This is proven via certain rational function on the elliptic curve [Formula: see text] with nice property. We also prove that the Shafarevich–Tate group [Formula: see text] coincides with a class group of K as a special case.

Unlikely Intersections in Elliptic Surfaces and Problems of Masser

2012 ◽  
Author(s):  
Umberto Zannier ◽  
David Masser
Keyword(s):  
Elliptic Curves ◽  
Abelian Variety ◽  
Multiplicative Group ◽  
Abelian Varieties ◽  
Elliptic Surfaces ◽  
Group Context ◽  
Special Case ◽  
Relative Case

This chapter turns from the multiplicative-group context to the context of abelian varieties. There are here entirely similar results and conjectures: we have already recalled the Manin–Mumford conjecture, and pointed out that the Zilber conjecture also admits an abelian exact analogue. Actually, abelian varieties have moduli, which introduce new issues with respect to the toric case. The chapter focuses mainly on some new problems, raised by Masser, which represent a relative case of Manin–Mumford–Raynaud, where the relevant abelian variety is no longer fixed but moves in a family. The unlikely intersections of Masser's questions occur in the special case of elliptic surfaces (i.e., families of elliptic curves), and can be dealt with by a method that has been recently introduced.

scholarly journals Galois groups of number fields generated by torsion points of elliptic curves

1986 ◽  
Vol 104 ◽  
pp. 43-53 ◽  
Author(s):  
Kay Wingberg
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Galois Group ◽  
Number Field ◽  
Quadratic Field ◽  
Complex Multiplication ◽  
Number Fields ◽  
Galois Groups ◽  
Torsion Points ◽  
Special Case

Coates and Wiles [1] and B. Perrin-Riou (see [2]) study the arithmetic of an elliptic curve E defined over a number field F with complex multiplication by an imaginary quadratic field K by using p-adic techniques, which combine the classical descent of Mordell and Weil with ideas of Iwasawa’s theory of Zp-extensions of number fields. In a special case they consider a non-cyclotomic Zp-extension F∞ defined via torsion points of E and a certain Iwasawa module attached to E/F, which can be interpreted as an abelian Galois group of an extension of F∞. We are interested in the corresponding non-abelian Galois group and we want to show that the whole situation is quite analogous to the case of the cyclotomic Zp-extension (which is generated by torsion points of Gm).

scholarly journals Bagger–Witten line bundles on moduli spaces of elliptic curves

2016 ◽  
Vol 31 (35) ◽  
pp. 1650188 ◽  
Author(s):  
Wei Gu ◽  
Eric Sharpe
Keyword(s):  
Line Bundle ◽  
Elliptic Curves ◽  
Moduli Spaces ◽  
Line Bundles ◽  
Deformation Parameters ◽  
Special Cases ◽  
Moduli Stack ◽  
Upper Half Plane ◽  
Ordinary Line ◽  
Special Case

In this paper, we discuss Bagger–Witten line bundles over moduli spaces of SCFTs. We review how in general they are “fractional” line bundles, not honest line bundles, twisted on triple overlaps. We discuss the special case of moduli spaces of elliptic curves in detail. There, the Bagger–Witten line bundle does not exist as an ordinary line bundle, but rather is necessarily fractional. As a fractional line bundle, it is nontrivial (though torsion) over the uncompactified moduli stack, and its restriction to the interior, excising corners with enhanced stabilizers, is also fractional. It becomes an honest line bundle on a moduli stack defined by a quotient of the upper half plane by a metaplectic group, rather than [Formula: see text]. We review and compare to results of recent work arguing that well-definedness of the worldsheet metric implies that the Bagger–Witten line bundle admits a flat connection (which includes torsion bundles as special cases), and gives general arguments on the existence of universal structures on moduli spaces of SCFTs, in which superconformal deformation parameters are promoted to nondynamical fields ranging over the SCFT moduli space.

Valuations of Somos 4 sequences and canonical local heights on elliptic curves

2011 ◽  
Vol 150 (3) ◽  
pp. 385-397 ◽  
Author(s):  
YUKIHIRO UCHIDA
Keyword(s):  
Asymptotic Behaviour ◽  
Elliptic Curves ◽  
Recurrence Relation ◽  
Special Case

AbstractSomos 4 sequences are sequences of numbers defined by a bilinear recurrence relation of order 4 and include elliptic divisibility sequences as a special case. In this paper, we describe valuations of Somos 4 sequences in terms of canonical local heights on the associated elliptic curves. We consider both Archimedean and non-Archimedean valuations. As applications, we study the asymptotic behaviour of valuations of Somos 4 sequences and obtain another proof of the integrality of certain Somos 4 sequences.

scholarly journals Critically separable rational maps in families

2012 ◽  
Vol 148 (6) ◽  
pp. 1880-1896 ◽  
Author(s):  
Clayton Petsche
Keyword(s):  
Fixed Point ◽  
Elliptic Curves ◽  
Number Field ◽  
Rational Maps ◽  
Finiteness Theorem ◽  
Good Reduction ◽  
Rational Map ◽  
Arithmetic Complexity ◽  
Special Case

AbstractGiven a number field K, we consider families of critically separable rational maps of degree d over K possessing a certain fixed-point and multiplier structure. With suitable notions of isomorphism and good reduction between rational maps in these families, we prove a finiteness theorem which is analogous to Shafarevich’s theorem for elliptic curves. We also define the minimal critical discriminant, a global object which can be viewed as a measure of arithmetic complexity of a rational map. We formulate a conjectural bound on the minimal critical discriminant, which is analogous to Szpiro’s conjecture for elliptic curves, and we prove that a special case of our conjecture implies Szpiro’s conjecture in the semistable case.

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