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I think this question is based on a misconception. “Being an eigenvariety” isn’t a rigorously defined property of a space (or map of spaces) which you could prove to hold or to not hold. It’s more lik …

Let $X$ be a smooth algebraic variety (over some field of char 0), $Z$ a smooth closed subvariety of codimension 1, $i : Z \hookrightarrow X$ the inclusion, and $j : U \hookrightarrow X$ the complemen …

Some conjectures – like this one and also the BSD conjecture – are hard to find in precise form in a single place, because the community's understanding of statement of the conjecture changed over tim …

Are you absolutely sure you want to compute $p$-adic etale cohomology for a smooth proper $\mathbb{Z}[1/S]$-scheme with $p \notin S$, so $p$ is not invertible on $X$? This will be painful, and I stron …

I don't think this statement is equivalent to the Hodge conjecture, but it does follow from a certain natural generalisation of the Hodge conjecture. See this blog post for further discussion: https:/ …

This question already has multiple nice answers, but I am going to add one more thing which isn't quite covered by the existing posts. One distinctive advantage of the $\Gamma_0(N)$ and $\Gamma_1(N)$ …

The correct viewpoint is not "$\Lambda$ is like a disc", but "$\Lambda$ is like the functions on a disc". To see this, ask yourself: given an element $f \in \mathbb{Z}_p[[T]]$, what values can we plug …

This is a subtle issue (which has come up before on this site several times, see e.g. is the modular curve X(N) defined over Q? for a related question). Your $S(N)$ is naturally a scheme over $\mathbb …

The answer to your example question is "No". Let $G$ be isomorphic to $\widehat{\mathbf{Z}}$, and $\phi$ the topological generator of $G$ (corresponding to $1 \in \widehat{\mathbf{Z}}$). Then one comp …

I am sure I've already written an expository account of this somewhere, but I looked over the lecture and seminar notes on my webpage and couldn't find it, so I'll write one here instead. Suppose we s …

There is a rather concrete construction in Landsburg's 1991 paper "Relative Chow groups" which gives an explicit isomorphism from $CH^k(X, 1)$ to the degree 1 homology of the Gersten complex, which is …

The curve $X_1(Np^n)$ is connected, but the ordinary locus in this curve is not: if you remove the residue discs of the supersingular points, what's left "falls apart" into a disjoint union of several …

Since this question has come alive again, let me point out that the Hecke operators cannot give a splitting of $h^1(E)$ into two pieces over $F$, since the Hecke correspondences on a modular curve are …

Doesn't this kind of prove itself? Pick some elements $\alpha_1, \dots, \alpha_n \in \mathfrak{m}$ which represent $\mathfrak{m} / (p, \mathfrak{m}^2)$. Clearly sending $t_i$ to $\alpha_i$ defines a m …

Let $R$ be a complete local $\mathbf{Z}_p$-algebra, for some prime $p$. In the 1970 paper Group schemes of prime order by Oort and Tate, they write down an explicit finite flat group scheme $G_R(a, b) …