Computing Zeta Functions via p-Adic Cohomology

Kedlaya, Kiran S.

doi:10.1007/978-3-540-24847-7_1

Cited by 29 publications

(19 citation statements)

References 37 publications

(55 reference statements)

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“…However, it is a remarkably useful algorithm for the case dim(X) = 1, where it has been extensively studied and implemented; see [26]. Problem 2.4 was solved for smooth projective hypersurfaces using relative rigid cohomology in [30].…”

Section: Theorem 22 (Dwork) the Zeta Function Is A Rational Functiomentioning

confidence: 99%

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A Recursive Method for Computing Zeta Functions of Varieties

Lauder¹

2006

LMS J. Comput. Math.

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This paper is dedicated to Richard P. Brent on the occasion of his sixtieth birthday. AbstractWe present an algorithm that reduces the problem of calculating a numerical approximation to the action of absolute Frobenius on the middle-dimensional rigid cohomology of a smooth projective variety over a finite field, to that of performing the same calculation for a smooth hyperplane section. When combined with standard geometric techniques, this yields a method for computing zeta functions which proceeds 'by induction on the dimension'. The 'inductive step' combines previous work of the author on the deformation of Frobenius with a higher rank generalisation of Kedlaya's algorithm. The analysis of the loss of precision during the algorithm uses a deep theorem of Christol and Dwork on p-adic solutions to differential systems at regular singular points. We apply our algorithm to compute the zeta functions of compactifications of certain surfaces which are double covers of the affine plane.

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Section: Theorem 22 (Dwork) the Zeta Function Is A Rational Functiomentioning

confidence: 99%

“…Let D( ) be an 'approximate solution' computed 'modulo p N ' to the differential system (23) modulo N . LetD( ) be an 'approximate solution' computed 'modulo p N ' to the dual system (26) …”

Section: Proof First Observe That We Have the Local Factorisation Armentioning

confidence: 99%

A Recursive Method for Computing Zeta Functions of Varieties

Lauder¹

2006

LMS J. Comput. Math.

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“…Thus any sign that this problem might be "easy" has some relevance; the main result of this note (originally written as an addendum to [13]) is one such sign, if only an indirect one.…”

Section: Introductionmentioning

confidence: 93%

Quantum computation of zeta functions of curves

Kedlaya

2006

comput. complex.

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We exhibit a quantum algorithm for determining the zeta function of a genus g curve over a finite field F q , which is polynomial in g and log(q). This amounts to giving an algorithm to produce provably random elements of the class group of a curve, plus a recipe for recovering a Weil polynomial from enough of its cyclic resultants. The latter effectivizes a result of Fried in a restricted setting.

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“…However the above result is incomparable with our work since they assume dense input format and also that n is fixed. There has been considerable work on this topic in computational number theory, see [10,11,27] and the references therein.…”

Section: Problem History and Motivationmentioning

confidence: 99%

Algorithms for Modular Counting of Roots of Multivariate Polynomials

2007

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Given a multivariate polynomial P (X 1 , . . . , X n ) over a finite field F q , let N(P ) denote the number of roots over F n q . The modular root counting problem is given a modulus r, to determine N r (P ) = N(P ) mod r. We study the complexity of computing N r (P ), when the polynomial is given as a sum of monomials. We give an efficient algorithm to compute N r (P ) when the modulus r is a power of the characteristic of the field. We show that for all other moduli, the problem of computing N r (P ) is NP-hard. We present some hardness results which imply that our algorithm is essentially optimal for prime fields. We show an equivalence between maximum-likelihood decoding for Reed-Solomon codes and a root-finding problem for symmetric polynomials.

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Computing Zeta Functions via p-Adic Cohomology

Abstract: Abstract. We survey some recent applications of p-adic cohomology to machine computation of zeta functions of algebraic varieties over finite fields of small characteristic, and suggest some new avenues for further exploration.

Cited by 29 publications

References 37 publications

A Recursive Method for Computing Zeta Functions of Varieties

A Recursive Method for Computing Zeta Functions of Varieties

Quantum computation of zeta functions of curves

Algorithms for Modular Counting of Roots of Multivariate Polynomials

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