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Suppose that $X$ is a (projective) K3 surface over a field $k$. A polarization of $X$ is an element $\lambda\in Pic_X(k)$ that is represented over an algebraic closure $\overline{k}$ by an ample line bundle $L$ over $X_{\overline{k}}$. Given such a $\lambda$, we can consider its self-intersection number $(\lambda,\lambda)\in 2\mathbb{Z}$. Suppose that $k$ has finite characteristic $p>2$. Is it possible to find a finite extension $k'/k$ and a polarization $\lambda$ of $X_{k'}$ such that $(\lambda,\lambda)$ is co-prime to $p$?

This question has a negative answer for polarizations of abelian varieties (see BCnrd's answer herehere), so I'm not optimistic that things are any better for K3 surfaces. On the other hand, one has an affirmative answer to the analogue for polarizations of abelian varieties, if one is allowed to modify the abelian variety up to isogeny.

So here's my backup question: Is there an analogous notion of an 'isogeny' of K3 surfaces that might help here? At the least, two K3 surfaces that are `isogenous' should have isometric Neron-Severi lattices (up to tensoring with $\mathbb{Q}$) and also isomorphic $l$-adic realizations, compatible with the cycle class map from the Neron-Severi lattices.

Suppose that $X$ is a (projective) K3 surface over a field $k$. A polarization of $X$ is an element $\lambda\in Pic_X(k)$ that is represented over an algebraic closure $\overline{k}$ by an ample line bundle $L$ over $X_{\overline{k}}$. Given such a $\lambda$, we can consider its self-intersection number $(\lambda,\lambda)\in 2\mathbb{Z}$. Suppose that $k$ has finite characteristic $p>2$. Is it possible to find a finite extension $k'/k$ and a polarization $\lambda$ of $X_{k'}$ such that $(\lambda,\lambda)$ is co-prime to $p$?

This question has a negative answer for polarizations of abelian varieties (see BCnrd's answer here), so I'm not optimistic that things are any better for K3 surfaces. On the other hand, one has an affirmative answer to the analogue for polarizations of abelian varieties, if one is allowed to modify the abelian variety up to isogeny.

So here's my backup question: Is there an analogous notion of an 'isogeny' of K3 surfaces that might help here? At the least, two K3 surfaces that are `isogenous' should have isometric Neron-Severi lattices (up to tensoring with $\mathbb{Q}$) and also isomorphic $l$-adic realizations, compatible with the cycle class map from the Neron-Severi lattices.

Suppose that $X$ is a (projective) K3 surface over a field $k$. A polarization of $X$ is an element $\lambda\in Pic_X(k)$ that is represented over an algebraic closure $\overline{k}$ by an ample line bundle $L$ over $X_{\overline{k}}$. Given such a $\lambda$, we can consider its self-intersection number $(\lambda,\lambda)\in 2\mathbb{Z}$. Suppose that $k$ has finite characteristic $p>2$. Is it possible to find a finite extension $k'/k$ and a polarization $\lambda$ of $X_{k'}$ such that $(\lambda,\lambda)$ is co-prime to $p$?

This question has a negative answer for polarizations of abelian varieties (see BCnrd's answer here), so I'm not optimistic that things are any better for K3 surfaces. On the other hand, one has an affirmative answer to the analogue for polarizations of abelian varieties, if one is allowed to modify the abelian variety up to isogeny.

So here's my backup question: Is there an analogous notion of an 'isogeny' of K3 surfaces that might help here? At the least, two K3 surfaces that are `isogenous' should have isometric Neron-Severi lattices (up to tensoring with $\mathbb{Q}$) and also isomorphic $l$-adic realizations, compatible with the cycle class map from the Neron-Severi lattices.

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Keerthi Madapusi
Keerthi Madapusi
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Polarizations of K3 surfaces over finite fields

Suppose that $X$ is a (projective) K3 surface over a field $k$. A polarization of $X$ is an element $\lambda\in Pic_X(k)$ that is represented over an algebraic closure $\overline{k}$ by an ample line bundle $L$ over $X_{\overline{k}}$. Given such a $\lambda$, we can consider its self-intersection number $(\lambda,\lambda)\in 2\mathbb{Z}$. Suppose that $k$ has finite characteristic $p>2$. Is it possible to find a finite extension $k'/k$ and a polarization $\lambda$ of $X_{k'}$ such that $(\lambda,\lambda)$ is co-prime to $p$?

This question has a negative answer for polarizations of abelian varieties (see BCnrd's answer here), so I'm not optimistic that things are any better for K3 surfaces. On the other hand, one has an affirmative answer to the analogue for polarizations of abelian varieties, if one is allowed to modify the abelian variety up to isogeny.

So here's my backup question: Is there an analogous notion of an 'isogeny' of K3 surfaces that might help here? At the least, two K3 surfaces that are `isogenous' should have isometric Neron-Severi lattices (up to tensoring with $\mathbb{Q}$) and also isomorphic $l$-adic realizations, compatible with the cycle class map from the Neron-Severi lattices.