scholarly journals Explicit endomorphism of the Jacobian of a hyperelliptic function field of genus 2 using base field operations

2015 ◽  
Vol 52 (2) ◽  
pp. 265-276
Author(s):  
Eduardo Ruiz Duarte ◽  
Octavio Páez Osuna
Keyword(s):  
Function Field ◽  
Base Field ◽  
Disjoint Support ◽  
Genus 2 ◽  
Explicit Formulae ◽  
Hyperelliptic Function

We present an efficient endomorphism for the Jacobian of a curve C of genus 2 for divisors having a Non disjoint support. This extends the work of Costello and Lauter in [12] who calculated explicit formulæ for divisor doubling and addition of divisors with disjoint support in JF(C) using only base field operations. Explicit formulæ is presented for this third case and a different approach for divisor doubling.

scholarly journals Genus-2 curves and Jacobians with a given number of points

2015 ◽  
Vol 18 (1) ◽  
pp. 170-197 ◽  
Author(s):  
Reinier Bröker ◽  
Everett W. Howe ◽  
Kristin E. Lauter ◽  
Peter Stevenhagen
Keyword(s):  
Finite Field ◽  
Elliptic Curves ◽  
Polynomial Time ◽  
Rational Points ◽  
The Other ◽  
Base Field ◽  
Arbitrary Base ◽  
The Family ◽  
Genus 2 ◽  
Genus 2 Curves

AbstractWe study the problem of efficiently constructing a curve $C$ of genus $2$ over a finite field $\mathbb{F}$ for which either the curve $C$ itself or its Jacobian has a prescribed number $N$ of $\mathbb{F}$-rational points.In the case of the Jacobian, we show that any ‘CM-construction’ to produce the required genus-$2$ curves necessarily takes time exponential in the size of its input.On the other hand, we provide an algorithm for producing a genus-$2$ curve with a given number of points that, heuristically, takes polynomial time for most input values. We illustrate the practical applicability of this algorithm by constructing a genus-$2$ curve having exactly $10^{2014}+9703$ (prime) points, and two genus-$2$ curves each having exactly $10^{2013}$ points.In an appendix we provide a complete parametrization, over an arbitrary base field $k$ of characteristic neither two nor three, of the family of genus-$2$ curves over $k$ that have $k$-rational degree-$3$ maps to elliptic curves, including formulas for the genus-$2$ curves, the associated elliptic curves, and the degree-$3$ maps.Supplementary materials are available with this article.

scholarly journals DUALITY FOR COHOMOLOGY OF CURVES WITH COEFFICIENTS IN ABELIAN VARIETIES

2018 ◽  
Vol 240 ◽  
pp. 42-149 ◽  
Author(s):  
TAKASHI SUZUKI
Keyword(s):  
Function Field ◽  
Positive Characteristic ◽  
Abelian Varieties ◽  
Base Field ◽  
Tate Pairing ◽  
Global Function ◽  
Crystalline Cohomology ◽  
Local Duality ◽  
Height Pairing ◽  
Finite Base

In this paper, we formulate and prove a duality for cohomology of curves over perfect fields of positive characteristic with coefficients in Néron models of abelian varieties. This is a global function field version of the author’s previous work on local duality and Grothendieck’s duality conjecture. It generalizes the perfectness of the Cassels–Tate pairing in the finite base field case. The proof uses the local duality mentioned above, Artin–Milne’s global finite flat duality, the nondegeneracy of the height pairing and finiteness of crystalline cohomology. All these ingredients are organized under the formalism of the rational étale site developed earlier.

The Zeta Function of a Pair of Quadratic Forms

2001 ◽  
Vol 44 (2) ◽  
pp. 242-256
Author(s):  
Laura Mann Schueller
Keyword(s):  
Finite Field ◽  
Zeta Function ◽  
Function Field ◽  
Quadratic Forms ◽  
Arbitrary Characteristic ◽  
Number Of Zeros ◽  
Even Order ◽  
Hyperelliptic Function

AbstractThe zeta function of a nonsingular pair of quadratic forms defined over a finite field, k, of arbitrary characteristic is calculated. A. Weil made this computation when char k ≠ 2. When the pair has even order, a relationship between the number of zeros of the pair and the number of places of degree one in an appropriate hyperelliptic function field is established.

Variations on a theme of rationality of cycles

2013 ◽  
Vol 11 (6) ◽  
Author(s):  
Nikita Karpenko
Keyword(s):  
Function Field ◽  
Algebraic Cycles ◽  
Base Field ◽  
Arbitrary Characteristic ◽  
Steenrod Operations ◽  
Cobordism Theory ◽  
Characteristic 2 ◽  
Algebraic Cobordism ◽  
Variations On A Theme

AbstractWe prove certain weak versions of some celebrated results due to Alexander Vishik comparing rationality of algebraic cycles over the function field of a quadric and over the base field. The original proofs use Vishik’s symmetric operations in the algebraic cobordism theory and work only in characteristic 0. Our proofs use the modulo 2 Steenrod operations in the Chow theory and work in any characteristic ≠ 2. Our weak versions are still sufficient for existing applications. In particular, Vishik’s construction of fields of u-invariant 2r + 1, for r ≥ 3, is extended to arbitrary characteristic ≠ 2.

scholarly journals Pseudo-elliptic integrals, units, and torsion

2005 ◽  
Vol 79 (3) ◽  
pp. 335-347 ◽  
Author(s):  
Francesco Pappalardi ◽  
Alfred J. Van Der Poorten
Keyword(s):  
Function Field ◽  
Function Fields ◽  
Quadratic Function ◽  
Elliptic Integrals ◽  
Square Root ◽  
Base Field ◽  
Quadratic Function Field

AbstractWe remark on pseudo-elliptic integrals and on exceptional function fields, namely function fields defined over an infinite base field but nonetheless containing non-trivial units. Our emphasis is on some elementary criteria that must be satisfied by a squarefree polynomialD(x)whose square root generates a quadratic function field with non-trivial unit. We detail the genus I case.

scholarly journals Around rationality of cycles

2013 ◽  
Vol 11 (6) ◽  
Author(s):  
Raphaël Fino
Keyword(s):  
Function Field ◽  
Algebraic Cycles ◽  
Base Field ◽  
Steenrod Operations ◽  
Cobordism Theory ◽  
Characteristic 2 ◽  
Algebraic Cobordism

AbstractWe prove certain results comparing rationality of algebraic cycles over the function field of a quadric and over the base field. These results have already been obtained by Alexander Vishik in the case of characteristic 0, which allowed him to work with algebraic cobordism theory. Our proofs use the modulo 2 Steenrod operations in the Chow theory and work in any characteristic ≠ 2.

l-PARTS OF DIVISOR CLASS GROUPS OF CYCLIC FUNCTION FIELDS OF DEGREE l

2007 ◽  
Vol 03 (02) ◽  
pp. 171-190 ◽  
Author(s):  
CHRISTIAN WITTMANN
Keyword(s):  
Field Theory ◽  
Function Field ◽  
Hyperelliptic Curve ◽  
Galois Module ◽  
Class Groups ◽  
Divisor Class ◽  
Deterministic Algorithms ◽  
Finite Set ◽  
Cyclic Function ◽  
Hyperelliptic Function

Let l be a prime number and K be a cyclic extension of degree l of the rational function field 𝔽q(T) over a finite field of characteristic ≠ = l. Using class field theory we investigate the l-part of Pic 0(K), the group of divisor classes of degree 0 of K, considered as a Galois module. In particular we give deterministic algorithms that allow the computation of the so-called (σ - 1)-rank and the (σ - 1)2-rank of Pic 0(K), where σ denotes a generator of the Galois group of K/𝔽q(T). In the case l = 2 this yields the exact structure of the 2-torsion and the 4-torsion of Pic 0(K) for a hyperelliptic function field K (and hence of the 𝔽q-rational points on the Jacobian of the corresponding hyperelliptic curve over 𝔽q). In addition we develop similar results for l-parts of S-class groups, where S is a finite set of places of K. In many cases we are able to prove that our algorithms run in polynomial time.

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