[Paper Review] Logarithmic good reduction and the index

Kentaro Mitsui, Arne Smeets|arXiv (Cornell University)|Nov 30, 2017

Algebraic Geometry and Number Theory16 references3 citations

TL;DR

This paper investigates the index of smooth, proper varieties over discretely valued fields with logarithmic good reduction. It proves that the index is prime to the residue characteristic $p$ when the $ε$-adic Euler characteristic is non-zero, but shows this fails in general for genus 1 curves, fully characterizing such cases via Galois action and cohomological flatness conditions.

ABSTRACT

Let $K$ be the fraction field of a complete discrete valuation ring, with algebraically closed residue field of characteristic $p > 0$. This paper studies the index of a smooth, proper $K$-variety $X$ with logarithmic good reduction. We prove that it is prime to $p$ in `most' cases, for example if the Euler number of $X$ does not vanish, but (perhaps surprisingly) not always. We also fully characterise curves of genus $1$ with logarithmic good reduction, thereby completing classical results of T. Saito and Stix valid for curves of genus at least $2$.

Motivation & Objective

• To determine whether the index of a smooth, proper $K$-variety with logarithmic good reduction is always prime to the residue characteristic $p$.
• To extend classical results on good reduction for curves of genus $\geq 2$ to the case of genus 1 curves.
• To characterize precisely which genus 1 curves over $K$ admit logarithmic good reduction, especially when the index is divisible by $p$.
• To resolve the tension between expectations from Hensel’s lemma and the actual behavior in positive characteristic, particularly in the absence of rational points.
• To provide a complete criterion for logarithmic good reduction of genus 1 curves using Galois representation tameness and cohomological flatness.

Proposed method

• Uses the $\ell$-adic Euler characteristic $\chi(X) = \sum_{i\geq 0}(-1)^i \dim_{\mathbb{Q}_\ell} H^i(X_{K^s}, \mathbb{Q}_\ell)$ as a key invariant to detect the existence of $K^t$-rational points.
• Applies results from logarithmic geometry and log regular models to analyze the structure of the minimal regular model $\mathcal{C}$ of a genus 1 curve $C$ over the valuation ring $S$.
• Employs the theory of Néron models and the Jacobian $J$ of $C$ to relate the period (equal to the index) to the reduction type and Galois action.
• Analyzes the tameness of the Galois action on $H^1(C_{K^s}, \mathbb{Q}_\ell)$ as a necessary condition for logarithmic good reduction.
• Introduces cohomological flatness of $\mathcal{C}$ over $S$ as a key technical condition when $p$ divides the period $m$.
• Relies on known constructions of elliptic surfaces with wild fibers in positive characteristic (e.g., Katsura–Ueno, Harbourne–Lang) to produce counterexamples.

Experimental results

Research questions

• RQ1 Is the index of a smooth, proper $K$-variety with logarithmic good reduction always prime to the residue characteristic $p$?
• RQ2 What conditions on the Galois action and cohomological structure are necessary and sufficient for a genus 1 curve to have logarithmic good reduction?
• RQ3 Can the index be divisible by $p$ even when the curve has logarithmic good reduction, and if so, under what conditions?
• RQ4 How does the interplay between the period, the Jacobian’s reduction type, and the cohomological flatness of the minimal model affect the existence of $K^t$-rational points?
• RQ5 To what extent do classical results on good reduction for curves of genus $\geq 2$ extend to genus 1 curves without rational points?

Key findings

• If the $\ell$-adic Euler characteristic $\chi(X)$ of a smooth, proper $K$-variety $X$ with logarithmic good reduction is non-zero, then $X(K^t) \neq \emptyset$, so $\iota(X)$ is prime to $p$.
• For curves of genus 1, logarithmic good reduction holds if and only if (a) the Galois action on $H^1(C_{K^s}, \mathbb{Q}_\ell)$ is tamely ramified, and (b) if $p$ divides the period $m$, then the Jacobian has good reduction and the minimal model $\mathcal{C}$ is cohomologically flat over $S$.
• There exist genus 1 curves with logarithmic good reduction for which $C(K^t) = \emptyset$, i.e., $p$ divides the index, showing that the index need not be prime to $p$ in general.
• The period of a genus 1 curve over $K$ equals its index due to the vanishing of the Brauer group of $K$, a result of Lichtenbaum.
• The counterexamples arise from elliptic surfaces with wild fibers in characteristic $p$, as constructed by Katsura–Ueno and others.
• The result completes the classification of logarithmic good reduction for curves, unifying the genus 1 case with the previously known results for genus $\geq 2$.