For any integer $d \geq 3$ and any algebraically closed field $k$ of characteristic $0$ and transcendence degree at least $d-3$, or of characteristic $p>2$ and transcendence degree at least $d-2$, we construct a $d$-dimensional variety over $k$ with a Brauer class of period $2$ and index $2^d$, violating the period-index conjecture; in particular, the period-index conjecture fails in dimension $3$ over $\overline{\mathbf{Q}}$ and $\overline{\mathbf{F}_p(t)}$, and in all dimensions $d \geq 3$ over $\mathbf{C}$ and $\overline{\mathbf{F}_p((t))}$. To bound the index of Brauer classes from below, we employ an obstruction of motivic nature, which requires proving the nonexistence of integral Hodge or Tate classes satisfying a certain equation modulo $2$. Motivated by these counterexamples, we propose a period-index conjecture with corrections at small primes and, as evidence, prove its Hodge-theoretic counterpart.
Let $k$ be algebraically closed of characteristic $p$. Reichstein and Vistoli showed that every $\Z/p^n$-torsor becomes at most one-dimensional over a field extension of degree prime to $p$, and Ledet conjectured $\ed_k(\Z/p^n)=n$. Using the finite-degree profile $\edd d_k(\tau)$ and its jump degrees introduced in \cit...
Oort's conjecture asserts that for any $g \geq 2$ and any prime $p$, every geometric generic member in the supersingular locus $\mathcal{S}_g$ has automorphism group $\{ \pm 1 \}$. This has been proved very recently by Viehmann in full generality, with previously known counterexamples for $p=2$ and $g=2,3$. We construc...
Let $K$ be a Henselian discretely valued field with excellent ring of integers $\mathcal{O}_K$ and algebraically closed residue field $k$. Let $X\to\mathbb{P}^1_K$ be a cyclic cover of degree $n$ prime to the characteristic of $k$. In joint work with Obus and Srinivasan (arXiv:2609.20553), we define an integer called t...
Let $X$ be a curve of genus $g\ge2$ over an algebraically closed field of odd characteristic $p$, and let $G=\mathrm{Aut}(X)$. A theorem of Giulietti and Korchmaros shows that if $G$ fixes no point and a $p$-subgroup $S\le G_P$ satisfies $|S|>\frac{p}{p-1}g$, then $X$ is one of the classical hyperelliptic, Hermitian, o...
We prove conductor-discriminant inequalities for all $\mathbb{Z}/n$-covers of $\mathbb{P}^1$ defined over discretely valued fields $K$ with excellent valuation ring $\mathcal{O}_K$ and perfect residue field of characteristic not dividing $n$, modulo some calculations appearing in work of the third author (arXiv:2609.20...
Andrew Obus, P. Srinivasan, Connor Stewart· 0 citations
For a smooth projective curve $C/\mathbb{Q}$ of genus $\geq 2$ and $L/\mathbb{Q}$ an extension, we write $C(L)_{\text{new}}=\{P\in C(L):\mathbb{Q}(P)=L\}$. Recent work of Khawaja and Siksek conjectures that this set is empty for $100\%$ of degree $n$ number fields $L$, when ordered by absolute discriminant. Moreover, t...
D. Mocanu, G. C. Turcas· 0 citations
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