scholarly journals Ultrametric theta functions and abelian varieties

1978 ◽  
Vol 69 ◽  
pp. 65-96 ◽  
Author(s):  
Horacio Tapia-Recillas
Keyword(s):  
Positive Integer ◽  
Abelian Variety ◽  
Abelian Varieties ◽  
Complete Solution ◽  
Theta Functions ◽  
Partial Solution ◽  
Rational Points ◽  
Graded Ring ◽  
Multiplicative Subgroup ◽  
Natural Way

Let k be a field complete with respect to a non-trivial, non-archimedean valuation and let g be a positive integer. Consider the following question : if Γ is a multiplicative subgroup of Gg = (k*)g satisfying certain “Riemann conditions”, can one construct in a natural way an abelian variety defined over k having Gg/Γ as its set of k-rational points? This problem was first considered by Morikawa [3]. J. Tate provided a complete solution for g = 1 (cf. for example [6]). J. McCabe [2] gave a partial solution for g > 1. He showed how to attach to Γ a graded ring R of theta functions such that A = Proj. R is g-dimensional abelian variety over k.

scholarly journals On the Unicity Conjecture for Generalized Markoff Equation

2018 ◽  
Vol 3 ◽  
pp. 137 ◽  
Author(s):  
Youb Raj Gaire
Keyword(s):  
Positive Integer ◽  
Complete Solution ◽  
Partial Solution ◽  
Famous Conjecture ◽  
Markoff Equation ◽  
College Of Engineering ◽  
Positive Integers

<p>The Markoff equation x<sup>2</sup> + y<sup>2</sup> + z<sup>2</sup> = 3xyz is introduced by A.A. Markoff in 1879. A famous conjecture on the Markoff equation, made by Frobinus in 1913, states that any Markoff triples (x, y, z) with x ≤ y ≤ z is uniquely determined by its largest number z. The complete solution of this equation is still open however the partial solution is given by Barager (1996), Button (2001), Zhang (2007), Srinivasan (2009), Chen and Chen (2013). In 1957, Mordell developed a generalization to the Markoff equation of the form x<sup>2</sup> + y<sup>2</sup> + z<sup>2</sup> = Axyz + B where, A and B are positive integers. In 2015, Donald McGinn take a particular form of above equation with A = 1 and B = A and gave a partial solution to the unicity conjecture to this equation. In this paper, the partial solution to the unicity conjecture to the equation of the form x<sup>2</sup> + y<sup>2</sup> + z<sup>2</sup> =3xyz + A where A is positive integer with A ≤ 4(x<sup>2</sup> –1) is given. </p><p><strong>Journal of Advanced College of Engineering and Management,</strong> Vol. 3, 2017, Page : 137-145</p>

scholarly journals THE MULTILINEAR SUPPORT PROBLEM FOR PRODUCTS OF ABELIAN VARIETIES AND TORI

2012 ◽  
Vol 08 (01) ◽  
pp. 255-264
Author(s):  
ANTONELLA PERUCCA
Keyword(s):  
Abelian Variety ◽  
Number Field ◽  
Abelian Varieties ◽  
Rational Points ◽  
Dependency Relation

Let G be the product of an abelian variety and a torus defined over a number field K. The aim of this paper is detecting the dependence among some given rational points of G by studying their reductions modulo all primes of K. We show that if some simple conditions on the order of the reductions of the points are satisfied then there must be a dependency relation over the ring of K-endomorphisms of G. We generalize Larsen's result on the support problem to several points on products of abelian varieties and tori.

scholarly journals Complete addition laws on abelian varieties

2012 ◽  
Vol 15 ◽  
pp. 308-316 ◽  
Author(s):  
Christophe Arene ◽  
David Kohel ◽  
Christophe Ritzenthaler
Keyword(s):  
Finite Number ◽  
Elliptic Curve ◽  
Galois Group ◽  
Abelian Variety ◽  
Abelian Varieties ◽  
Rational Points ◽  
Absolute Galois Group ◽  
Projective Embedding ◽  
Complete Set

AbstractWe prove that under any projective embedding of an abelian variety A of dimension g, a complete set of addition laws has cardinality at least g+1, generalizing a result of Bosma and Lenstra for the Weierstrass model of an elliptic curve in ℙ2. In contrast, we prove, moreover, that if k is any field with infinite absolute Galois group, then there exists for every abelian variety A/k a projective embedding and an addition law defined for every pair of k-rational points. For an abelian variety of dimension 1 or 2, we show that this embedding can be the classical Weierstrass model or the embedding in ℙ15, respectively, up to a finite number of counterexamples for ∣k∣≤5 .

scholarly journals Computing isogenies between abelian varieties

2012 ◽  
Vol 148 (5) ◽  
pp. 1483-1515 ◽  
Author(s):  
David Lubicz ◽  
Damien Robert
Keyword(s):  
Abelian Variety ◽  
Time Complexity ◽  
Abelian Varieties ◽  
Theta Functions ◽  
Null Point ◽  
Constant Number ◽  
Weil Pairing ◽  
Base Field ◽  
Rational Expression ◽  
Higher Dimensional

AbstractWe describe an efficient algorithm for the computation of separable isogenies between abelian varieties represented in the coordinate system given by algebraic theta functions. Let A be an abelian variety of dimension g defined over a field of odd characteristic. Our algorithm comprises two principal steps. First, given a theta null point for A and a subgroup K isotropic for the Weil pairing, we explain how to compute the theta null point corresponding to the quotient abelian variety A/K. Then, from the knowledge of a theta null point of A/K, we present an algorithm to obtain a rational expression for an isogeny from A to A/K. The algorithm that results from combining these two steps can be viewed as a higher-dimensional analog of the well-known algorithm of Vélu for computing isogenies between elliptic curves. In the case where K is isomorphic to (ℤ/ℓℤ)g for ℓ∈ℕ*, the overall time complexity of this algorithm is equivalent to O(log ℓ) additions in A and a constant number of ℓth root extractions in the base field of A. In order to improve the efficiency of our algorithms, we introduce a compressed representation that allows us to encode a point of level 4ℓ of a g-dimensional abelian variety using only g(g+1)/2⋅4g coordinates. We also give formulas for computing the Weil and commutator pairings given input points in theta coordinates.

Galois Representations Over Fields of Moduli and Rational Points on Shimura Curves

2014 ◽  
Vol 66 (5) ◽  
pp. 1167-1200 ◽  
Author(s):  
Victor Rotger ◽  
Carlos de Vera-Piquero
Keyword(s):  
Abelian Variety ◽  
Moduli Space ◽  
Quadratic Field ◽  
Galois Representations ◽  
Abelian Varieties ◽  
Main Idea ◽  
Galois Representation ◽  
Rational Points ◽  
Shimura Curves ◽  
Fields Of Moduli

AbstractThe purpose of this note is to introduce a method for proving the non-existence of rational points on a coarse moduli space X of abelian varieties over a given number field K in cases where the moduli problem is not fine and points in X(K) may not be represented by an abelian variety (with additional structure) admitting a model over the field K. This is typically the case when the abelian varieties that are being classified have even dimension. The main idea, inspired by the work of Ellenberg and Skinner on the modularity of ℚ-curves, is that one may still attach a Galois representation of Gal(/K) with values in the quotient group GL(Tℓ(A))/ Aut(A) to a point P = [A] ∈ X(K) represented by an abelian variety A/, provided Aut(A) lies in the centre of GL(Tℓ(A)). We exemplify our method in the cases where X is a Shimura curve over an imaginary quadratic field or an Atkin–Lehner quotient over ℚ.

scholarly journals Every positive integer is the order of an ordinary abelian variety over $${{\mathbb {F}}}_2$$

2021 ◽  
Vol 7 (4) ◽  
Author(s):  
Everett W. Howe ◽  
Kiran S. Kedlaya
Keyword(s):  
Positive Integer ◽  
Abelian Variety ◽  
Rational Points

AbstractWe show that for every integer $$m > 0$$ m > 0 , there is an ordinary abelian variety over $${{\mathbb {F}}}_2$$ F 2 that has exactly m rational points.

There are No Transcendental Brauer–Manin Obstructions on Abelian Varieties

2018 ◽  
Vol 2020 (9) ◽  
pp. 2684-2697
Author(s):  
Brendan Creutz
Keyword(s):  
Abelian Variety ◽  
Number Field ◽  
Abelian Varieties ◽  
Rational Points ◽  
Brauer Group

Abstract Suppose X is a torsor under an abelian variety A over a number field. We show that any adelic point of X that is orthogonal to the algebraic Brauer group of X is orthogonal to the whole Brauer group of X. We also show that if there is a Brauer–Manin obstruction to the existence of rational points on X, then there is already an obstruction coming from the locally constant Brauer classes. These results had previously been established under the assumption that A has finite Tate–Shafarevich group. Our results are unconditional.

scholarly journals Level structures on abelian varieties and Vojta’s conjecture

2017 ◽  
Vol 153 (2) ◽  
pp. 373-394 ◽  
Author(s):  
Dan Abramovich ◽  
Anthony Várilly-Alvarado
Keyword(s):  
Positive Integer ◽  
Recent Work ◽  
Abelian Variety ◽  
Number Field ◽  
Abelian Varieties ◽  
Fixed Number ◽  
Vojta's Conjecture

Assuming Vojta’s conjecture, and building on recent work of the authors, we prove that, for a fixed number field $K$ and a positive integer $g$, there is an integer $m_{0}$ such that for any $m>m_{0}$ there is no principally polarized abelian variety $A/K$ of dimension $g$ with full level-$m$ structure. To this end, we develop a version of Vojta’s conjecture for Deligne–Mumford stacks, which we deduce from Vojta’s conjecture for schemes.

Tits polygons

2022 ◽  
Vol 275 (1352) ◽  
Author(s):  
Bernhard Mühlherr ◽  
Richard Weiss ◽  
Holger Petersson
Keyword(s):  
Rational Points ◽  
Jordan Algebras ◽  
Parabolic Subgroups ◽  
Characteristic P ◽  
Arbitrary Group ◽  
Content Type ◽  
Defining Relations ◽  
Moufang Condition ◽  
Unique Vertex ◽  
Natural Way

We introduce the notion of a Tits polygon, a generalization of the notion of a Moufang polygon, and show that Tits polygons arise in a natural way from certain configurations of parabolic subgroups in an arbitrary spherical buildings satisfying the Moufang condition. We establish numerous basic properties of Tits polygons and characterize a large class of Tits hexagons in terms of Jordan algebras. We apply this classification to give a “rank  2 2 ” presentation for the group of F F -rational points of an arbitrary exceptional simple group of F F -rank at least  4 4 and to determine defining relations for the group of F F -rational points of an an arbitrary group of F F -rank  1 1 and absolute type D 4 D_4 , E 6 E_6 , E 7 E_7 or E 8 E_8 associated to the unique vertex of the Dynkin diagram that is not orthogonal to the highest root. All of these results are over a field of arbitrary characteristic.

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