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I am really not an expert in the field, so I apologize in advance for omissions or mistakes - I would indeed be glad to get corrections. But let me try, anyhow... You are asking for a motivation for r …

Let $k$ be a finite field, say with $q=p^a$ elements. Honda-Tate theory states that there is a bijection between isogeny classes of simple abelian varieties over $k$ and $\mathrm{Gal}(\overline{\mathb …

You finish your question by insisting that the isomorphism be such but "not in the sense of derived categories", which I do not understand completely, since both objects you have at hands naturally li …

I think a good reference is Schneider's book Nonarchimedean Funtional Analysis, but my answer is simply an expanded version of GTA's comments. In Chapter IV, §14, A and B you find a description of tw …

[EDIT]: After getting a very nice answer by Kevin Buzzard I realize that my question was a little bit too vague and I try to restate it more precisely. As the title says, I would like to understand a …

As Keith Conrad observed in his comment, what you are asking about is Washington's theorem which is either in his book, referenced as Theorem 16.12, or in his original paper in Invent. Math. He proved …

I post this as an auto-answer mainly not to leave the question open. After googling a bit better, I discovered two recent works by M. Lapidus and L. Hung (both available on Lapidus' webpage ) “Non …

This is not really about the embedding, rather about the identification of points on the Jacobian with degree-0-divisors. If you are given a cuspidal form $f\in S_2(\Gamma_0(N),\mathbb{Q})$, by defini …

As Daniel Litt says, the choice of $C$ is actually an untilt. The "classical" approach to period rings, which you might have in mind, was to start with a certain complete, algebraically closed field $ …

Well, although there is a typo (Wiles forgot to close his parenthesis, and wrote $H^2(G,\mu_{p^r}\to H^2(G,\mu_{p^s})$ in his proof), his claim is correct. Let, as ibid. $F$ be the finite extension of …

[EDIT]: Following Qiaochu Yuan's comment, it is better to clarify that I do not know what the right definition of a fractal in the following question should be. But a nice answer might contain such a …

If $\lambda\in\overline{\mathbb{Q}}$, the elliptic curve $$ E_\lambda\colon y^2=x(x-1)(x-\lambda) $$ has $(\lambda,0)$ as $2$-torsion point and is defined over (a subfield of) $L=\mathbb{Q}(\lambda)$ …

I don't think we know how to prove this directly. Indeed, recent works by Wake and Wake–Erickson show that this cyclicity is equivalent to a conjectured improvement of Mazur–Wiles' result to the effec …