Related papers: Explicit Coleman Integration in Larger Characteris…
The Coleman integral is a $p$-adic line integral that plays a key role in computing several important invariants in arithmetic geometry. We give an algorithm for explicit Coleman integration on curves, using the algorithms of the second…
Coleman's theory of p-adic integration figures prominently in several number-theoretic applications, such as finding torsion and rational points on curves, and computing p-adic regulators in K-theory (including p-adic heights on elliptic…
Vologodsky's theory of $p$-adic integration plays a central role in computing several interesting invariants in arithmetic geometry. In contrast with the theory developed by Coleman, it has the advantage of being insensitive to the…
In this paper, we introduce an algorithm for computing p-adic integrals on bad reduction hyperelliptic curves. For bad reduction curves, there are two notions of p-adic integration: Berkovich-Coleman integrals which can be performed…
We describe an algorithm to compute the local component at p of the Coleman-Gross p-adic height pairing on divisors on hyperelliptic curves. As the height pairing is given in terms of a Coleman integral, we also provide new techniques to…
In this paper, we develop an algorithm for computing Coleman--Gross (and hence Nekov\'a\v{r}) $p$-adic heights on hyperelliptic curves over number fields with arbitrary reduction type above $p$. This height is defined as a sum of local…
We describe an algorithm to compute the local Coleman-Gross p-adic height at p on a hyperelliptic curve. Previously, this was only possible using an algorithm due to Balakrishnan and Besser, which was limited to odd degree. While we follow…
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 characteristic polynomial of Frobenius.…
The purpose of this paper is to prove integrality for certain $p$-adic iterated Coleman integrals. As underlying geometry we will take the complement of a divisor $D\subset X$ with good reduction, where $X$ is the projective line or an…
Coleman integrals is a major tool in the explicit arithmetic of algebraic varieties, notably in the study of rational points on curves. One of the inputs to compute Coleman integrals is the availability of an affine model. We develop a…
We present a new quadratic Chabauty method to compute the integral points on certain even degree hyperelliptic curves. Our approach relies on a nontrivial degree zero divisor supported at the two points at infinity to restrict the $p$-adic…
Let $p$ be an odd prime number. We propose an algorithm for computing rational representations of isogenies between Jacobians of hyperelliptic curves via-adic differential equations with a sharp analysis of the loss of precision.…
In this paper, we give an explicit description of the de Rham and p-adic polylogarithms for elliptic curves using the Kronecker theta function. We prove in particular that when the elliptic curve has complex multiplication and good…
In this paper we present a p-adic algorithm to compute the zeta function of a nondegenerate curve over a finite field using Monsky-Washnitzer cohomology. The paper vastly generalizes previous work since all known cases, e.g. hyperelliptic,…
Consider an elliptic curve defined over an imaginary quadratic field $K$ with good reduction at the primes above $p\geq 5$ and has complex multiplication by the full ring of integers $\mathcal{O}_K$ of $K$. In this paper, we construct…
The aim of this paper is to propose an ``elementary" approach to Coleman's theory of p-adic abelian integrals. Our main tool is a theory of commutative p-adic Lie groups (the logarithm map); we use neither dagger analysis nor…
Let g >= 1 and let Q be a monic, squarefree polynomial of degree 2g + 1 in Z[x]. For an odd prime p not dividing the discriminant of Q, let Z_p(T) denote the zeta function of the hyperelliptic curve of genus g over the finite field F_p…
We extend the result of a previous work to the case of curves with semi-stable reduction. In this case, one can replace Coleman integration by Vologodsky integration to extend the Coleman-Gross definition of a $p$-adic height pairing. we…
We return to some past studies of hyperkloosterman sums ([9,10]) via $p$-adic cohomology with an aim to improve earlier results. In particular, we work here with Dwork's $\theta_\infty$-splitting function and a better choice of basis for…
We describe an algorithm to compute the zeta function of any non-hyperelliptic genus 3 plane curve $C$ over a finite field with automorphism group $G = \mathbb{Z} / 2 \mathbb{Z}$. This algorithm computes in the Monsky-Washnitzer cohomology…