# Comptes Rendus
Mathématique

Brian Lawrence and Shizhang Li

A counterexample to an optimistic guess about étale local systems

Volume 359, issue 8 (2021), p. 923-924

Published online: 6 October 2021

https://doi.org/10.5802/crmath.229

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Number theory / Théorie des nombres

# A counterexample to an optimistic guess about étale local systems

Brian Lawrence*, a and Shizhang Li b

a University of California, Los Angeles, USA.

b University of Michigan, USA.

E-mails: brianrl@math.ucla.edu (B. Lawrence), shizhang@umich.edu (S. Li)

Manuscript received 16th June 2020, accepted 1st June 2021.

Relative p-adic Hodge theory aims at extending known results in p-adic Hodge theory to a p-adic local system on a rigid variety. Let X be a geometrically connected, quasi-compact rigid analytic variety over a p-adic field K and let $\mathbb{E}$ be a $\mathbb{Q}_p$-local system on the étale site $X_{\acute{e}t}$. Liu and Zhu [3] showed that if at one point $\bar{x} \in X(\bar{K})$ the stalk $\mathbb{E}_{\bar{x}}$ is de Rham as a p-adic Galois representation, then $\mathbb{E}$ is a de Rham local system; in particular, the stalk of $\mathbb{E}_{\bar{y}}$ at any point $\bar{y} \in X(\bar{K})$ would be de Rham as well. They noted that the similar statements replacing “de Rham” by either “crystalline” or “semistable” are both wrong. However, inspired by potential semistability of de Rham representations [1], they ask [3, Remark 1.4] if a de Rham local system $\mathbb{E}$ on X would become semistable<fnref n="1" /> after pulling the system back to a finite étale cover of X, or even after enlarging the ground field K by a finite extension. While the former guess may well be true, in this paper we construct an example illustrating the failure of the latter.

**Theorem 1.** There exist a projective variety X over a p-adic field K, and an étale local system $\mathbb{E}$ on X, such that $\mathbb{E}$ is de Rham, but for every finite extension $K'/K$, the restriction of $\mathbb{E}$ to $X_{K'}$ is not semistable.

More precisely, for every finite extension $K'$ of K, there exist a further extension $L/K'$ and a point $x \in X(L)$, such that $\mathbb{E}_x$ is not semistable as a representation of $\text{Gal}_L$.

**Proof.** Let $p$ be an odd prime, and let X be the elliptic curve over $\mathbb{Q}_p$ with Weierstrass equation

$$y^2 = x(x - p)(x + 1).$$

Let $X'$ be (the normalization of) the double cover of X given by

$$y^2 = x(x - p)(x + 1), z^2 = x;$$

<footnote>* Corresponding author.</footnote>

<footnote n="1">¹Here we make an ad hoc definition for a local system $\mathbb{E}$ on $X_{\acute{e}t}$ to be semistable: its stalk $\mathbb{E}_{\bar{y}}$ at any point $\bar{y} \in X(\bar{K})$ is semistable as a p-adic Galois representation. This is the weakest requirement, any reasonable definition should imply our condition.</footnote>
it is again an elliptic curve, Galois over $X$ with deck transformation group $Z/2Z$ cyclic of order 2.

Let $\pi: X' \to X$ be the covering map. Let $\mathbb{E}$ be the rank-1 direct summand of the rank-2 étale sheaf $\pi_*\mathbb{Q}_p$ on $X$, corresponding to the nontrivial character of the deck transformation group $Z/2Z$. This $\mathbb{E}$ is de Rham because it comes from geometry.

On the other hand, for any $L$ of absolute ramification degree at least 3, we can find a point $(x, y)$ above which $\mathbb{E}$ is not semistable, as follows. Let $x$ be any uniformizer for $L$; then $x(x-p)(x+1)$ is a square in $L$, so we can find $y \in L$ such that $(x, y) \in X(L)$.

Now $\mathbb{E}_{(x,y)}$ is the rank-one representation of $\mathrm{Gal}_L$

$$ \rho_{L'/L}: \mathrm{Gal}_L \to \{\pm 1\} \subseteq \mathrm{GL}_1(\mathbb{Q}_p) $$

having kernel $\mathrm{Gal}_{L'}$, where $L' = L[\sqrt{x}]$ is a ramified extension. By [2, Proposition 7.17], $\rho_{L'/L}$ is not semistable. $\square$

## Acknowledgements

We would like to thank Bhargav Bhatt, Zarathustra Brady, Johan de Jong, and Xinwen Zhu for interesting discussions. B. L. would like to thank the University of Michigan for its hospitality, and to acknowledge support from the National Science Foundation.

## References

[1] L. Berger, "Représentations $p$-adiques et équations différentielles", Invent. Math. 148 (2002), no. 2, p. 219-284.

[2] J.-M. Fontaine, Y. Ouyang, "Theory of $p$-adic Galois Representations", 2008, preprint, available at https://www.math.u-psud.fr/~fontaine/galoisrep.pdf.

[3] R. Liu, X. Zhu, "Rigidity and a Riemann-Hilbert correspondence for $p$-adic local systems", Invent. Math. 207 (2017), no. 1, p. 291-343.