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16
Bounding Picard numbers of surfaces using padic cohomology, arXiv:math/0601508v2
 Arithmetic, Geometry and Coding Theory (AGCT 2005), Societé Mathématique de
, 2007
"... Motivated by an application to LDPC (low density parity check) algebraic geometry codes described by Voloch and Zarzar, we describe a computational procedure for establishing an upper bound on the arithmetic or geometric Picard number of a smooth projective surface over a finite field, by computing ..."
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Motivated by an application to LDPC (low density parity check) algebraic geometry codes described by Voloch and Zarzar, we describe a computational procedure for establishing an upper bound on the arithmetic or geometric Picard number of a smooth projective surface over a finite field, by computing the Frobenius action on padic cohomology to a small degree of padic accuracy. We have implemented this procedure in Magma; using this implementation, we exhibit several examples, such as smooth quartics over F2 and F3 with arithmetic Picard number 1, and a smooth quintic over F2 with geometric Picard number 1. We also produce some examples of smooth quartics with geometric Picard number 2, which by a construction of van Luijk also have trivial geometric automorphism group.
POINT COUNTING IN FAMILIES OF HYPERELLIPTIC CURVES IN CHARACTERISTIC 2
"... Let ĒΓ be a family of hyperelliptic curves over F2 alg cl with general Weierstrass equation given over a very small field F. We describe in this paper an algorithm for computing the zeta function of Ē¯γ, with ¯γ in a degree n extension field of F, which has as time complexity Õ(n3) bit operations ..."
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Cited by 12 (5 self)
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Let ĒΓ be a family of hyperelliptic curves over F2 alg cl with general Weierstrass equation given over a very small field F. We describe in this paper an algorithm for computing the zeta function of Ē¯γ, with ¯γ in a degree n extension field of F, which has as time complexity Õ(n3) bit operations and memory requirements O(n2) bits. With a slightly different algorithm we can get time O(n2.667) and memory O(n2.5), and the computation for n curves of the family can be done in time Õ(n3.376). All of these algorithms are polynomialtime in the genus.
Explicit Coleman Integration for Hyperelliptic Curves
"... Abstract. Coleman’s theory of padic integration figures prominently in several numbertheoretic applications, such as finding torsion and rational points on curves, and computing padic regulators in Ktheory (including padic heights on elliptic curves). We describe an algorithm for computing Cole ..."
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Abstract. Coleman’s theory of padic integration figures prominently in several numbertheoretic applications, such as finding torsion and rational points on curves, and computing padic regulators in Ktheory (including padic heights on elliptic curves). We describe an algorithm for computing Coleman integrals on hyperelliptic curves, and its implementation in Sage. 1
Class number approximation in cubic function fields
 Contributions to Discrete Mathematics
"... Abstract. We develop explicitly computable bounds for the order of the Jacobian of a cubic function field. We use approximations via truncated Euler products and thus derive effective methods of computing the order of the Jacobian of a cubic function field. Also, a detailed discussion of the zeta fu ..."
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Abstract. We develop explicitly computable bounds for the order of the Jacobian of a cubic function field. We use approximations via truncated Euler products and thus derive effective methods of computing the order of the Jacobian of a cubic function field. Also, a detailed discussion of the zeta function of a cubic function field extension is included. 1.
Computing zeta functions in families of Ca,b curves using deformation
"... Abstract. We apply deformation theory to compute zeta functions in a family of Ca,b curves over a finite field of small characteristic. The method combines Denef and Vercauteren’s extension of Kedlaya’s algorithm to Ca,b curves with Hubrechts ’ recent work on point counting on hyperelliptic curves u ..."
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Abstract. We apply deformation theory to compute zeta functions in a family of Ca,b curves over a finite field of small characteristic. The method combines Denef and Vercauteren’s extension of Kedlaya’s algorithm to Ca,b curves with Hubrechts ’ recent work on point counting on hyperelliptic curves using deformation. As a result, it is now possible to generate Ca,b curves suitable for use in cryptography in a matter of minutes. 1
Approximating Euler products and class number computation in algebraic function fields
"... Abstract. We provide a number of results that can be used to derive approximations for the Euler product representation of the zeta function of an arbitrary algebraic function field. Three such approximations are given here. Our results have two main applications. They lead to a computationally suit ..."
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Abstract. We provide a number of results that can be used to derive approximations for the Euler product representation of the zeta function of an arbitrary algebraic function field. Three such approximations are given here. Our results have two main applications. They lead to a computationally suitable algorithm for computing the class number of an arbitrary function field. The ideas underlying the class number algorithms in turn can be used to analyze the distribution of the zeros of its zeta function. 1.
On nondegeneracy of curves
 Algebra & Number Theory
, 2009
"... Abstract. We study the conditions under which an algebraic curve can be modelled by a Laurent polynomial that is nondegenerate with respect to its Newton polytope. We prove that every curve of genus g ≤ 4 over an algebraically closed field is nondegenerate in the above sense. More generally, let be ..."
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Abstract. We study the conditions under which an algebraic curve can be modelled by a Laurent polynomial that is nondegenerate with respect to its Newton polytope. We prove that every curve of genus g ≤ 4 over an algebraically closed field is nondegenerate in the above sense. More generally, let be the locus of nondegenerate curves inside the moduli space of curves of genus g ≥ 2. Then we show that dim Mnd g = min(2g + 1,3g − 3), except for g = 7 where dim Mnd 7 = 16; thus, a generic curve of genus g is nondegenerate M nd g if and only if g ≤ 4.
Algebraic Independence in Positive Characteristic – A padic Calculus
 Trans. Amer. Math. Soc
, 2014
"... Abstract. A set of multivariate polynomials, over a field of zero or large characteristic, can be tested for algebraic independence by the wellknown Jacobian criterion. For fields of other characteristic p> 0, no analogous characterization is known. In this paper we give the first such criterio ..."
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Abstract. A set of multivariate polynomials, over a field of zero or large characteristic, can be tested for algebraic independence by the wellknown Jacobian criterion. For fields of other characteristic p> 0, no analogous characterization is known. In this paper we give the first such criterion. Essentially, it boils down to a nondegeneracy condition on a lift of the Jacobian polynomial over (an unramified extension of) the ring of padic integers. Our proof builds on the functorial de RhamWitt complex, which was invented by Illusie (1979) for crystalline cohomology computations, and we deduce a natural explicit generalization of the Jacobian. This new avatar we call the WittJacobian. In essence, we show how to faithfully differentiate polynomials over Fp (i.e., somehow avoid ∂xp/∂x = 0) and thus capture algebraic independence.
Computing zeta functions of nondegenerate hypersurfaces with few monomials
 SUBMITTED EXCLUSIVELY TO THE LONDON MATHEMATICAL SOCIETY
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Effective padic cohomology for cyclic cubic threefolds
, 2008
"... These are the lecture notes from a series of six lectures given at a summer school on padic cohomology held in Mainz in the fall of 2008. (They may be viewed as a sequel to the author’s notes from the Arizona Winter School in 2007 [49].) The goal of the notes is to describe how to use padic cohomo ..."
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These are the lecture notes from a series of six lectures given at a summer school on padic cohomology held in Mainz in the fall of 2008. (They may be viewed as a sequel to the author’s notes from the Arizona Winter School in 2007 [49].) The goal of the notes is to describe how to use padic cohomology to make effective, provably correct numerical computations of zeta functions. More specifically, we discuss three techniques in detail: • use of the Hodge filtration to infer the zeta function from point counts; • the “direct cohomological method ” of computing the Frobenius action on the padic cohomology of a single variety; • the “deformation method ” of computing the Frobenius structure on the padic cohomologies of a oneparameter family of varieties, using the associated PicardFuchs differential equation. We make these methods explicit for cyclic cubic threefolds (cubic threefolds in P 4 admitting an automorphism of order 3), and demonstrate with a numerical example over the field F7. To demonstrate these calculations, we use the opensource computer algebra system Sage [77]; however, some of the calculations depend on the nonfree system Magma [63], which may