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One expects that the majority of algebraic curves over number fields having genus $> 1$ should not be modular in this sense. For instance, take a sufficiently general genus 2 curve $C$ over $\mathbf{Q …

There are two questions here: which version of the theory is easiest to get off the ground axiomatically, and which version is more convenient to work with in applications? For the first question, it' …

The correct setting for this construction turns out to be projective varieties, so let me suppose we have a smooth variety $X$ inside $\mathbf{P}^N$, for some $N \ge 1$, defined by the vanishing of so …

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 $\Gamma_1(N)$ …

One major application of research in integral $p$-adic Hodge theory is in proving modularity results, e.g. for elliptic curves. Here one wants to understand liftings of global mod p Galois representat …

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 …

Suppose $\mathfrak{X}$ is a smooth projective scheme of finite type over $\mathbb{Z}_p$, with generic fibre $X$. Then there are comparison theorems relating de Rham and crystalline cohomology, $H^i_{\ …

For the first question: it can happen that $\pm \Gamma$ is congruence but $\Gamma$ is not; there is a beautiful paper on this phenomenon, with lots of examples, by Kiming, Schütt and Verril here. Fo …

This question is somehow a "characteristic 0" question, so let me treat $Y = Y_1(N)$ and $X = X_1(N)$ as $\mathbf{Q}$-varieties rather than doing anything complicated with integral models. There's an …

Let me explain a bit more what that footnote was supposed to mean. As I'm sure you know, an Euler system for a Galois representation $V$ over a number field $K$ consists of a bunch of classes in $H^1 …

Your belief is correct. A $\mathbb{Z}$-point has to reduce to an $\mathbb{F}_p$-point for all $p$, which kills examples with gcd > 1. If you want to make this precise, try writing down an explicit d …

(Synthesis of answers from comments, posted as community-wiki answer for convenience.) If $k = \mathbb{C}$ then algebraic de Rham cohomology, defined a la Hartshorne using the completion of $X$ alon …

For simplicity, let's take $K = \mathbf{Q}_p$. One of the few things about $\mathbf{B}_{\mathrm{cris}}$ that one can straightforwardly prove directly from its definition is that it contains $\widehat …

Here's an example, if I'm not mistaken. Let $E / K$ be an elliptic curve and $x \in E$ a non-torsion $K$-point. Then the image of the divisor $\{x\} - \{\infty\}$ under the etale cycle class map is a …

The statement you want follows fairly straightforwardly from Bass' conjecture -- sufficiently straightforwardly that it may well not have a separate name of its own. If $\Sigma$ is a sufficiently lar …