Kedlaya's Algorithm in Larger Characteristic

@article{Harvey2006KedlayasAI,
  title={Kedlaya's Algorithm in Larger Characteristic},
  author={David Harvey},
  journal={arXiv: Number Theory},
  year={2006}
}
  • David Harvey
  • Published 31 October 2006
  • Computer Science
  • arXiv: Number Theory
We show that the linear dependence on $p$ of the running time of Kedlaya's point-counting algorithm in characteristic $p$ may be reduced to $p^{1/2}$. 

Figures and Tables from this paper

Efficient computation of p-adic heights
We analyse and drastically improve the running time of the algorithm of Mazur, Stein and Tate for computing the canonical cyclotomic p-adic height of a point on an elliptic curve E/Q, where E has
Computing Hasse-Witt matrices of hyperelliptic curves in average polynomial time
We present an algorithm that computes the Hasse-Witt matrix of given hyperelliptic curve over Q at all primes of good reduction up to a given bound N. It is simpler and faster than the previous
Counting points on curves using a map to P1
  • J. Tuitman
  • Mathematics, Computer Science
    Math. Comput.
  • 2016
TLDR
A new algorithm to compute the zeta function of a curve over a finite field using a map to the projective line is introduced and all the necessary bounds are developed.
Computing zeta functions of cyclic covers in large characteristic
We describe an algorithm to compute the zeta function of a cyclic cover of the projective line over a finite field of characteristic $p$ that runs in time $p^{1/2 + o(1)}$. We confirm its
Computing Zeta Functions of Superelliptic Curves in Larger Characteristic
  • M. Minzlaff
  • Computer Science, Mathematics
    Math. Comput. Sci.
  • 2010
TLDR
Harvey’s optimisation for large characteristic p to the superelliptic case is introduced and the most general algorithm to compute zeta functions that runs soft linear in p1/2 is state.
Counting points on curves using a map to P1, II
  • J. Tuitman
  • Mathematics, Computer Science
    Finite Fields Their Appl.
  • 2017
Computing L-Series of Hyperelliptic Curves
We discuss the computation of coefficients of the L-seriesassociated to a hyperelliptic curve over Q of genus at most 3, using pointcounting, generic group algorithms, and p-adic methods.
Explicit Coleman integration in larger characteristic
  • Alex J Best
  • Mathematics, Computer Science
    The Open Book Series
  • 2019
TLDR
A more efficient algorithm to compute p-adic Coleman integrals on odd degree hyperelliptic curves for large primes p by using fast linear recurrence techniques when reducing differentials in Monsky-Washnitzer cohomology.
A Point Counting Algorithm for Cyclic Covers of the Projective Line
TLDR
A Kedlaya-style point counting algorithm for cyclic covers over a finite field with refined bounds on the p-adic precision, and an alternative pseudo-basis for the Monsky-Washnitzer cohomology which leads to an integral matrix when $p \geq 2r$.
Variation of Néron–Severi Ranks of Reductions of K3 Surfaces
TLDR
These ranks for reductions of smooth quartic surfaces modulo all primes p < 216 in several representative examples are computed and the resulting statistics are investigated.
...
...

References

SHOWING 1-10 OF 21 REFERENCES
Linear Recurrences with Polynomial Coefficients and Computation of the Cartier-Manin Operator on Hyperelliptic Curves
We improve an algorithm originally due to Chudnovsky and Chudnovsky for computing one selected term in a linear recurrent sequence with polynomial coefficients. Using baby-steps / giant-steps
Computing Zeta Functions via p-Adic Cohomology
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
Counting Points in Medium Characteristic Using Kedlaya's Algorithm
TLDR
This work investigates more precisely this dependence on the characteristic, and shows that after a few modifications using fast algorithms for radix-conversion, Kedlaya's algorithm works in time almost linear in p, and can be applied to medium values of p.
Counting Points on Hyperelliptic Curves using Monsky-Washnitzer Cohomology
We describe an algorithm for counting points on an arbitrary hyperelliptic curve over a finite field of odd characteristic, using Monsky-Washnitzer cohomology to compute a p-adic approximation to the
The Middle Product Algorithm I
TLDR
A new algorithm is presented – MiddleProduct or, for short, MP – computing the n middle coefficients of a (2n−1)×n full product in the same number of multiplications as a full n×n product.
Computation of p-Adic Heights and Log Convergence
TLDR
A new notion of log convergence is introduced and it is proved that E2, the p-adic modular form associated to the elliptic curve, is log convergent.
Point counting after Kedlaya, EIDMA-Stieltjes Graduate course, Leiden, September 22-26, 2003
TLDR
This series of three lectures of one hour each was preceded by two introductory lectures by Henk van Tilborg about applications of discrete logarithms (in multiplicative groups as well as elliptic curves) to cryptography to serve as motivation for the coming three lectures.
SAGE: system for algebra and geometry experimentation
SAGE is a framework for number theory, algebra, and geometry computation that is initially being designed for computing with elliptic curves and modular forms. The current implementation is primarily
Point Counting in Families of Hyperelliptic Curves
TLDR
A deterministic algorithm for computing the zeta function of the curve EΓ by using Dwork deformation in rigid cohomology is presented, which turns out to be quite efficient for n big enough.
Gaussian elimination is not optimal
t. Below we will give an algorithm which computes the coefficients of the product of two square matrices A and B of order n from the coefficients of A and B with tess than 4 . 7 n l°g7 arithmetical
...
...