Jinbi Jin | Searcharxiv

arXiv2017

Explicit computation of the first étale cohomology on curves

In this paper, we describe an algorithm that, for a smooth connected curve $X$ over a field $k$ with normal completion having arithmetic genus $p_a(X)$, a finite locally constant sheaf $\mathcal A$ on $X_{et}$ of abelian groups of torsion invertible in $k$, represented by a smooth curve with normal completion having arithmetic genus $p_a(\mathcal A)$ and degree $n$ over $X$, computes the first étale cohomology $H^1(X_{k^{sep},et},\mathcal A)$ and the first étale cohomology with proper support $H^1_c(X_{k^{sep},et},\mathcal A)$ as sets of torsors, in arithmetic complexity exponential in $n^{\log n}$, $p_a(X)$, and $p_a(\mathcal A)$. This is done via the computation of a groupoid scheme classifying the relevant torsors (with extra rigidifying data).

math.AG↗

arXiv2015

Homogeneous division polynomials for Weierstrass elliptic curves

Starting from the classical division polynomials we construct homogeneous polynomials $α_n$, $β_n$, $γ_n$ such that for $P = (x:y:z)$ on an elliptic curve in Weierstrass form over an arbitrary ring we have $nP = \bigl(α_n(P):β_n(P):γ_n(P)\bigr)$. To show that $α_n,β_n,γ_n$ indeed have this property we use the a priori existence of such polynomials, which we deduce from the Theorem of the Cube. We then use this result to show that the equations defining the modular curve $Y_1(n)_{\mathbb C}$ computed for example by Baaziz, in fact are equations of $Y_1(n)$ over $\mathbb Z[1/n]$.

math.AG↗