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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 $(a_n)_{n \ge 1}$ be a sequence of complex numbers, and suppose that the sequence has a well-defined "average", in the sense that $$ \lim_{N \to \infty} \frac{1}{N}\sum_{i = 1}^N a_i = R$$ for som …

The set $\mathcal{L}_{\rho}$ is either empty, or a singleton finite. This follows from Faltings' isogeny theorem, which states that for any two elliptic curves (or, more generally, abelian varieties) …

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 …

You might want to study the work of Darmon--Lauder--Rotger, notably this paper: https://web.mat.upc.edu/victor.rotger/docs/DLR1.pdf They study cases of the equivariant BSD conjecture where $\rho$ is a …

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 …

Are you asking for a proof of existence, or an explicit construction? These are very different things! It is immediate from the definition that there exists a finite family $(W_i, W_i')_{i \in I}$ wit …

I'm adding an answer to this very old question of mine, since someone just contacted me about it. The problem is rather comprehensively solved in this 2019 preprint of Taeko Okazaki: Takeo Okazaki, L …

The slogan "number theorists aim to understand $\operatorname{Gal}(\overline{\mathbb{Q}} / \mathbb{Q})$" is one that gets used a lot, but it's perhaps a tiny little bit misleading. Understanding the s …

(1) Borel's article in the Corvallis proceedings does this slightly differently: he chooses a specific Frobenius element $\sigma$, and then looks at the subset $\widehat{G} \times \{\sigma\}$ of ${}^L …

Yes, this is true. It works with arbitrary algebraic varieties, no need to be specific to Shimura varieties. Let $X \to^{\pi} Spec(K)$ be an algebraic variety, $\mathcal{F}$ an etale sheaf on $X$, and …

I found the answer myself with the help of a very useful hint from user "nobody" in the comments, so I'm going to post a community-wiki answer in case anyone else finds it useful. My question is answe …

$\DeclareMathOperator\SL{SL}\DeclareMathOperator\GL{GL}\DeclareMathOperator\W{W}$Let $k$ be a finite field, and let $\W_n(k)$ be the degree $n$ Witt vectors over $k$ (so $\W_1(k) = k$). Does there ex …

I don't really understand the setup of this question: what is $R\langle x_1/r_1, \dots \rangle$ supposed to mean if $r_i$ is a real number? That said, if $R = \mathbb{Z}_p$, then $\mathbb{Z}_p[[x_1, \ …

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:/ …