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Suppose $K$ is a number field, $r=r_1+r_2-1$ (the rank of the unit group) and $u_1,\dots, u_r$ are a basis of fundamental units. Suppose this basis minimizes the max over $i=1,\dots, r$ of the largest …

I'm glad to have stumbled on your question -- I'm actually working on something along the lines of what you're asking about now with Eric Larson. If what we think we've proven is true (and don't quote …

It's known that if $A$ is an abelian variety of totally multiplicative reduction over a p-adic field K, then, after taking a finite field extension, it becomes isomorphic, as a rigid analytic group, t …

For $E$ an elliptic curve, let $LS(E)$ be the group of line bundles on $E$ with a flat connection. This is an $\mathbb{A}^1$-torsor over $E^\vee$. By Riemann-Hilbert (since the gauge group acts trivia …

Over an algebraically closed field of characteristic zero, e.g. $\overline{\mathbb{Q}}$, the Deligne-Mumford stack $\overline{\mathcal{M}}_{g,n}$ represents the functor $$\overline{\mathcal{M}}_{g,n}( …

There is an idea (I think originally due to Tate) that class field theory is fundamentally a consequence of Pontrjagin duality and Hilbert Theorem 90. I'm curious whether this can phrased using modern …

Thanks to modern techniques (such as the pro-etale site), we can now understand etale (co)homology of varieties and motives as "genuinely" derived (e.g. DG) Galois-equivariant objects. I'm looking for …

If you do not impose an algebraically closed condition, no two are equivalent. This basically follows from your (3). Namely, observe that An extension is finite if and only if it has finitely many …

Suppose $R$ is a perfectoid ring in mixed characteristic, and $R'$ its characteristic-$p$ tilt. Scholze's results on tilting say that the étale theories over $R$ and $R'$ are equivalent in an almost s …