# Frobenius on Monsky-Washnitzer cohomology of a hyperelliptic curve over a largish prime finite field¶

This is a wrapper for the matrix() function in hypellfrob.cpp.

AUTHOR:

• David Harvey (2007-05)
• David Harvey (2007-12): rewrote for hypellfrob version 2.0
sage.schemes.hyperelliptic_curves.hypellfrob.hypellfrob(p, N, Q)

Compute the matrix of Frobenius acting on the Monsky-Washnitzer cohomology of a hyperelliptic curve $$y^2 = Q(x)$$, with respect to the basis $$x^i dx/y$$, $$0 \leq i < 2g$$.

INPUT:

• p – a prime
• Q – a monic polynomial in $$\ZZ[x]$$ of odd degree. Must have no multiple roots mod p.
• N – precision parameter; the output matrix will be correct modulo $$p^N$$.

PRECONDITIONS:

Must have $$p > (2g+1)(2N-1)$$, where $$g = (\deg(Q)-1)/2$$ is the genus of the curve.

ALGORITHM:

Described in “Kedlaya’s algorithm in larger characteristic” by David Harvey. Running time is theoretically soft-$$O(p^{1/2} N^{5/2} g^3)$$.

Todo

Remove the restriction on $$p$$. Probably by merging in Robert’s code, which eventually needs a fast C++/NTL implementation.

EXAMPLES:

sage: from sage.schemes.hyperelliptic_curves.hypellfrob import hypellfrob
sage: R.<x> = PolynomialRing(ZZ)
sage: f = x^5 + 2*x^2 + x + 1; p = 101
sage: M = hypellfrob(p, 4, f); M
[ 91844754 + O(101^4)  38295665 + O(101^4)  44498269 + O(101^4)  11854028 + O(101^4)]
[ 93514789 + O(101^4)  48987424 + O(101^4)  53287857 + O(101^4)  61431148 + O(101^4)]
[ 77916046 + O(101^4)  60656459 + O(101^4) 101244586 + O(101^4)  56237448 + O(101^4)]
[ 58643832 + O(101^4)  81727988 + O(101^4)  85294589 + O(101^4)  70104432 + O(101^4)]
sage: -M.trace()
7 + O(101^4)
sage: sum([legendre_symbol(f(i), p) for i in range(p)])
7
sage: ZZ(M.det())
10201
sage: M = hypellfrob(p, 1, f); M
[     O(101)      O(101) 93 + O(101) 62 + O(101)]
[     O(101)      O(101) 55 + O(101) 19 + O(101)]
[     O(101)      O(101) 65 + O(101) 42 + O(101)]
[     O(101)      O(101) 89 + O(101) 29 + O(101)]


AUTHORS:

• David Harvey (2007-05)
• David Harvey (2007-12): updated for hypellfrob version 2.0