On the Existence of Absolutely Simple Abelian Varieties of a Given Dimension over an Arbitrary Field

Howe, Everett W.; Zhu, Hui June

doi:10.1006/jnth.2001.2697

Cited by 44 publications

(41 citation statements)

References 14 publications

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“…Since an ordinary abelian variety over F q is simple if and only if its Weil polynomial is irreducible, we can look for the existence of simple ordinary jacobian varieties of every dimension. However, Howe and Zhu in [3] proved that for any finite prime fields there exist absolutely simple ordinary abelian varieties of every dimension. But the question of whether there exist absolutely simple jacobians of every dimension over a given finite field is still open (whereas the answer is yes if the field is the algebraic closure of a finite field, if it has characteristic p but is not algebraic over F p or if it has characteristic zero), see [3].…”

Section: Conjecturementioning

confidence: 97%

On some questions related to the Gauss conjecture for function fields

Aubry

Blache

2008

Journal of Number Theory

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International audienceWe show that, for any finite field Fq , there exist infinitely many real quadratic function fields over Fq such that the numerator of their zeta function is a separable polynomial. As pointed out by Anglès, this is a necessary condition for the existence, for any finite field Fq, of infinitely many real function fields over Fq with ideal class number one (the so-called Gauss conjecture for function fields). We also show conditionally the existence of infinitely many real quadratic function fields over Fq such that the numerator of their zeta function is an irreducible polynomial

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Section: Conjecturementioning

confidence: 97%

On some questions related to the Gauss conjecture for function fields

Aubry

Blache

2008

Journal of Number Theory

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“…Proof. By [15,Proposition 3], in order to prove that A is absolutely simple it is sufficient to prove that: (a) there is no d > 1 such that the characteristic polynomial of the Frobenius is in Z[T d ], and (b) there is no d > 1 and no primitive d-th root of unity ζ such that Q(π d ) is a proper sub-field of Q(π) and Q(π) = Q(π d , ζ). If (b) does not hold and Q(π) = Q(π d , ζ), then the degree ϕ(d) of the extension Q(ζ)/Q must divide deg Q(π) = 2 dim A.…”

Section: Cubic Threefoldsmentioning

confidence: 99%

Irrationality of generic cubic threefold via Weil's conjectures

Markushevich

Roulleau

2018

Commun. Contemp. Math.

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An arithmetic method of proving the irrationality of smooth projective 3-folds is described, using reduction modulo p. It is illustrated by an application to a cubic threefold, for which the hypothesis that its intermediate Jacobian is isomorphic to the Jacobian of a curve is contradicted by reducing modulo 3 and counting points over appropriate extensions of F3. As a spin-off, it is shown that the 5dimensional Prym varieties arising as intermediate Jacobians of certain cubic 3-folds have the maximal number of points over Fq which attains Perret's and Weil's upper bounds.

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“…Heuristically, over large finite fields, randomly sampled abelian varieties are vanilla with overwhelming probability. Indeed, being vanilla is invariant in isogeny classes, and Howe and Zhu have shown in [14,Theorem 2] that the fraction of isogeny classes of g-dimensional abelian varieties over F q that are ordinary and absolutely simple tends to 1 as q → ∞. All absolutely simple ordinary abelian varieties are vanilla, except those whose endomorphism algebras contain roots of unity; but the number of such isogeny classes for fixed g is asymptotically negligible.…”

Section: Vanilla Abelian Varietiesmentioning

confidence: 99%

Isogenies for Point Counting on Genus Two Hyperelliptic Curves with Maximal Real Multiplication

Ballentine

Guillevic

García

et al. 2017

Association for Women in Mathematics Series

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Schoof's classic algorithm allows point-counting for elliptic curves over finite fields in polynomial time. This algorithm was subsequently improved by Atkin, using factorizations of modular polynomials, and by Elkies, using a theory of explicit isogenies. Moving to Jacobians of genus-2 curves, the current state of the art for point counting is a generalization of Schoof's algorithm. While we are currently missing the tools we need to generalize Elkies' methods to genus 2, recently Martindale and Milio have computed analogues of modular polynomials for genus-2 curves whose Jacobians have real multiplication by maximal orders of small discriminant. In this article, we prove Atkin-style results for genus-2 Jacobians with real multiplication by maximal orders, with a view to using these new modular polynomials to improve the practicality of point-counting algorithms for these curves. IntroductionEfficiently computing the number of points on the Jacobian of a genus 2 curve over a finite field is an important problem in experimental number theory and numbertheoretic cryptography. When the characteristic of the finite field is small, Kedlaya's algorithm and its descendants provide an efficient solution (see [18], [13], and [12]), while in extremely small characteristic we have extremely fast AGM-style algorithms (see for example [25], [26], and [3]). However, the running times of these algorithms are exponential in the size of the field characteristic; the hardest case, therefore (and also the most important case for contemporary cryptographic applications) is where the characteristic is large, or even where the field is a prime field.So let q be a power of a large prime p, and let C be a genus-2 curve over F q . Our fundamental problem is to compute the number of F q -rational points on the Jacobian J C of C .

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On the Existence of Absolutely Simple Abelian Varieties of a Given Dimension over an Arbitrary Field

Cited by 44 publications

References 14 publications

On some questions related to the Gauss conjecture for function fields

On some questions related to the Gauss conjecture for function fields

Irrationality of generic cubic threefold via Weil's conjectures

Isogenies for Point Counting on Genus Two Hyperelliptic Curves with Maximal Real Multiplication

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