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I don't think this will be true in general. Say $K=\mathbf{Q}$ and $K'=\mathbf{Q}(\sqrt{2})$, and let $X_0$ be $Spec(R')$. Then $X_0$ is regular of dimension 1 and the map down to $S$ is projective a …

Let $K$ be a finite extension of $\mathbf{Q}_p$, with integer ring $R$ and residue field $k$. Say $G/R$ is a finite flat (commutative) group scheme of order $p^2$, killed by $p$. Say the special fibre …

In general one can say very little. There are some positive results (as indicated in the comments) in special cases, but the below example kills any hope that one can say something in general. NB "the …

The right way to do this sort of question is to apply Saito's local-global theorem, which says that the (semisimplification of the) Weil-Deligne representation built from $D_{pst}(\rho_{f,p})$ by forg …

[I took the time to chase this up so may as well post it as an answer.] There is a (cuspidal) modular (eigen)form of level $\Gamma_0(512)$ and weight 2, which if I remember correctly was shown to me …

As you've spotted, there are two ways to do $p$-adic modular forms. The point is that at some point you take a limit of classical modular forms, and there are a whole host of modular forms which may h …

Here is a completely different kind of answer to this question. A perfectoid space is a term of type PerfectoidSpace in the Lean theorem prover. Here's a quote from the source code: structure perfe …

Here are the answers to some of your questions. If $P$ is a (non-zero) prime ideal of the integers of $F$ then $B$ will either be split or ramified at $P$, depending on whether $B\otimes_F F_P$ is iso …

If $E/\mathbf{Q}$ is an elliptic curve and we put it into minimal Weierstrass form, we can count how many integral points it has. A theorem of Siegel tells us that this number $n(E)$ is finite, and th …

I have never understood this proof either. What in my mind makes it very odd is that a very similar argument can sometimes be used to prove in some sense the exact opposite---that certain equations ha …

Surprisingly, the case of modular curves is misleading! General theory of correspondences, plus the theory of the mod $p$ reduction of curves like $X_0(Np)$ ($p$ doesn't divide $N$) give a relationshi …

One example I always found useful was that if you consider an elliptic curve like (the projective model of) y^2=x^3+1, then this equation gives an elliptic curve not only over the complex numbers but …

The Eichler-Shimura relation doesn't just prove the Hasse-Weil conjecture for modular curves. It e.g. attaches Galois representations to modular forms of weight 2. More delicate arguments (using etale …

Let me highlight some issues that Emerton doesn't: 1) you seem to hint that you don't know that modular forms can be viewed as a product of a bunch of local terms. So there is an adelic story, where …