New paper: genus $1$ curves and Brauer groups
Asher Auel and I have posted our paper [AA] on splitting Brauer classes with genus $1$ curves. This paper was eluded to in the post on jonality. In the language of that post, our main theorem gives a bunch of new cases where the jonality of a Severi–Brauer variety is the lowest possible, i.e., $1$. Equivalently, we give new cases where the following question has a postive answer.
Question. Let $k$ be a field and let $\alpha\in\Br(k)$ be a Brauer class. Is there a genus $1$ curve $C$ over $k$ such that $\alpha$ pulls back to zero in $\Br(k(C))$?
Two previous papers deal directly with this topic. The first is a paper [dJH] of Johan de Jong and Wei Ho. They prove that if $k$ is a field and $\alpha\in\Br(k)$ is a Brauer class of degree $2,3,4,5$, then $\alpha$ is split by a genus $1$ curve. The arguments in their paper are all geometric. For example, if $D$ is a degree $3$ division algebra with Severi–Brauer $P$, then a general anticanonical section is a genus $1$ curve $X$ splitting inside $P$ which necessarily splits $\alpha$. The case of $d=2$ is similar and those of $d=4,5$ are similar, but more complicated.
The second paper is Saltman’s [S], who exhaustively analyzed the degree $3$ case to answer for example whether it is always possible to split with a genus $1$ curve of any given $j$-invariant (no).
A closely related paper [HL] of Wei Ho and Max Lieblich establishes that every Brauer class is split by a torsor for an abelian variety which may be taken to be either a Jacobian of a high genus curve or a product of such a Jacobian with an elliptic curve.
Our contribution is to consider the problem from a cohomological perspective which is especially suitable for use in the context of global fields. In some sense, our work is an inverse to the work of Ciperiani and Krashen who actually compute the Brauer classes split by a given genus $1$ curve.
Splitting $\mu_N$-gerbes
Our strongest results are for classes of smallish degree over global fields. We tackle these by answering a harder question in many cases.
Question. Let $k$ be a field and let $\beta\in\H^2(\Spec k,\mu_N)$ be a $\mu_N$-cohomology class (for example lifting $\alpha\in\Br(k)$). Is there a genus $1$ curve $C$ defined over $k$ such that $\beta$ pulls back to zero in $\H^2(C,\mu_N)$.
The conclusion is strictly stronger than asking for the vanishing of the class in the function field of $C$. This turns out to be a subtle phenomenon. In general, the curve $C$ might split $\alpha$ but not $\beta$, or it might be that it splits a $2$-torsion class of $\H^2(\Spec k,\mu_4)$ but not its unique lift to $\H^2(\Spec k,\mu_2)$.
Note that the Severi-Brauer variety of a central simple algebra $D$ of class $\alpha$ never splits $\beta$ in the sense above. Using a Leray-Serre spectral sequence, one sees that in order to split $\beta$ there must be rational $N$-torsion in the Picard scheme, which rules out many interesting classes of varieties. It also means, by Mazur’s theorem, that there is no hope of a positive answer to the question for non-zero classes of $\H^2(\Spec\bQ,\mu_p)$ when $p\geq 11$ is prime.
Here is one of our main theorems.
Theorem A. Let $k$ be a field and let $\beta\in\H^2(\Spec k,\mu_N)$. If $\beta$ is cyclic, then $\beta$ is split by a genus $1$ curve in the following cases:
- $N=2,3,4,5$,
- $N=6,7,10$ and $k$ is global,
- $N=8$, $k$ is global, and, if the characteristic of $k$ is not $2$, then $k$ contains $\zeta_8$,
- $N=9$, $k$ is global, and, if the characteristic of $k$ is not $3$, then $k$ contains $\zeta_9+\zeta_9^{-1}$, and
- $N=12$, $k$ is global, and, if the characteristic of $k$ is not $2$, then $k$ contains $\zeta_4$.
Note that every class $\beta$ is cyclic when $k$ is global.
We were very excited about this result because it’s the first positive result in this direction involving $p=7$. Of course, the requirement that $k$ be global and that means that this theorem does not have the same applicability as the earlier result of de Jong and Ho.
The main idea in the proof is rather simple, although implementing it required a key idea of Saltman. The idea is to look at $\mu_N$-isogenies of elliptic curves
\[0\rightarrow\mu_N\rightarrow E\rightarrow E'\rightarrow 0,\]which guarantee that $E’$ has an exact order $N$-point $P\in E’(k)$. There is then an obstruction class $\delta(P)\in\H^1(\Spec k,\mu_N)$ to lifting $P$ to a rational point of $E$. Then, the boundary map
\[\H^1(\Spec k,\bZ/N)\rightarrow\H^2(\Spec k,\mu_N)\]induced from the $N$-torsion groupscheme $E[N]$ is of the form
\[\chi\mapsto[\chi,\delta(P)],\]the cyclic class corresponding cupping a character $\chi$ with the class $\delta(P)$. It is easy to see that if $X_\chi$ is the $E$-torsor corresponding to $\chi$ under the map $\H^1(\Spec k,\bZ/N)\rightarrow\H^1(\Spec k,E’)$, then $X_\chi$ splits $[\chi,\delta(P)]$. Thus, one wants to find ways of generating lots of possible $\delta(P)$s. For $N=2,3,4,5,$ one can find a $\mu_N$-isogeny as above where $\delta(P)$ is any given element of $k^\times/(k^\times)^N$, which is enough to prove Theorem A.
For larger $N$, this seems to be impossible. Instead, in the global field case, one can find an isogeny where at least the extension $k(\delta(P)^{1/N})$ splits the cyclic class $\alpha=[\chi,u]$. Then, theorems of Albert, Vishne, and Mináč-Wadsworth, imply that you can pick a different character $\chi’$ such that $\alpha=[\chi,u]=[\chi’,\delta(P)]$ under the assumption on roots of unity in Theorem A.
The population of the argument with lots of $\delta(P)$s uses that that the
modular curves $X_1(N)$ are rational and have lots of rational points, even
over $\bQ$, when $N=2,3,4,5,6,7,8,9,10,12.$ Then, an explicit calculation in
MAGMA, explained to us by Tom Fischer, produces formulas for the $\delta(P)$
in terms of a parameter on these modular curves.
For example, when $N=7$ we use the elliptic curve $E’$
\[y^2+(1+\lambda-\lambda^2)xy+\lambda(1-\lambda)^2 y=x^3+\lambda(1-\lambda)^2x^2,\]which has an exact order $7$ point at $(0,0)$. Fischer had already computed $\delta(P)=\lambda^6(\lambda-1)^3$ up to $7$th powers appears in an early paper.
Splitting Brauer classes with full torsion
Cathy O’Neil’s thesis was about an obstruction theory for when the period of a genus $1$ curve is equal to its index. This work was also taken up by Pete Clark, and we further extend it to prove the following theorem.
Theorem C. Suppose that $E$ is an elliptic curve over a field $k$. If $E$ admits a full level $N$ structure $E[N]\cong\bZ/N\times\mu_N$, then every cyclic class of $\Br(k)[N]$ is split by an $E$-torsor.
Using Theorem C and the Merkurjev-Suslin theorem, one can prove for instance that if $k$ contains $\bQ$, then every Brauer class is split by a product of genus $1$ curves and one can choose the Jacobians of those curves to have any given $j$-invariants in $\bQ$. Or, if $k$ contains $\overline{\bF}_p$, then using the theorem above as well as as a result of Albert, then every class of $\Br(k)[p^\infty]$ is split by a genus $1$ curve.
Beyond cyclicity
It seems very difficult for our cohomological methods to extend beyond the cyclic algebra case. However, when $k$ contains a primitive $N$th root of unity and $E[N]$ has full level $N$ structure, then the methodology of Theorem A says that certain $E$-torsors simultaneously split two cyclic algebras. At present, we do not know if this is enough to split for example even all biconic class of $\H^2(\Spec k,\mu_2)$.
References
[AA] Antieau, Auel, Explicit descent on elliptic curves and splitting Brauer classes, arXiv:2106.04291.
[dJH] de Jong, Ho, Genus one curves and Brauer-Severi varieties, Math. Res. Lett. 19 (2012) no. 6, 1357–1359. arXiv:1207.4810.
[HL] Ho, Lieblich, Splitting Brauer classes using the universal Albanese. arXiv:1805.12566.
[S] Saltman, Genus one curves in Severi-Brauer surfaces, arXiv:2105.09986.