p-adic Integration on Bad Reduction Hyperelliptic Curves

p-ADIC HEIGHT PAIRINGS AND INTEGRAL POINTS ON HYPERELLIPTIC CURVES

We give a formula for the component at p of the p-adic height pairing of a divisor of degree 0 on a hyperelliptic curve. We use this to give a Chabauty-like method for finding p-adic approximations to p-integral points on such curves when the Mordell-Weil rank of the Jacobian equals the genus. In this case we get an explicit bound for the number of such p-integral points, and we are able to use...

Explicit p-adic methods for elliptic and hyperelliptic curves

We give an overview of some p-adic algorithms for computing with elliptic and hyperelliptic curves, starting with Kedlaya’s algorithm. While the original purpose of Kedlaya’s algorithm was to compute the zeta function of a hyperelliptic curve over a finite field, it has since been used in a number of applications. In particular, we describe how to use Kedlaya’s algorithm to compute Coleman inte...

Some properties of $τ$-adic expansions on hyperelliptic Koblitz curves

This paper explores two techniques on a family of hyperelliptic curves that have been proposed to accelerate computation of scalar multiplication for hyperelliptic curve cryptosystems. In elliptic curve cryptosystems, it is known that Koblitz curves admit fast scalar multiplication, namely, the τ -adic non-adjacent form (τ -NAF). It is shown that the τ -NAF has the three properties: (1) existen...

ON p-ADIC FUNCTIONAL INTEGRATION

p-Adic generalization of the Feynman path integrals in quantum mechanics is considered. The probability amplitude Kv(x, t′′; x′, t′) (v = ∞, 2, 3, · · · , p, · · ·) for a particle in a constant field is calculated. Path integrals over Qp have the same form as those over R.

SYNTOMIC REGULATORS AND p-ADIC INTEGRATION II: K2 OF CURVES