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Disclaimer: I know absolutely nothing about p-adic cohomology, so it is possible that even the premises of this question are incorrect. But it turns out that I need to apply the theory of rigid cohomo …

Note that the defining equation for $X$ can be rewritten as $$(x+y)^3+(x-y)^3+(z+w)^3+(z-w)^3=-2.$$ As the linear transformation $(x,y,z,w)\mapsto(x+y,x-y,z+w,z-w)$ is invertible over any field of cha …

For upper bounds, there is the paper https://arxiv.org/abs/1801.05397 of Schreieder, which shows (over any uncountable field of characteristic not equal to two) that for $N>2$, a very general $N$-dime …

As pointed out in the comments, this is Theorem 6.3.1.2 of Hilburn-Raskin. (It certainly was known much earlier, but I'm not sure what to give as a reference.) Their proof is stated quite elegantly, i …

The Beilinson-Bernstein localization theorem states roughly that the category of $D$-modules on the flag variety $G/B$ is equivalent to the category of modules over the universal enveloping algebra $U …

They do not exist in general. The simplest example is maybe to take $X\rightarrow Y$ to be the closed embedding of the origin inside $\mathbb{A}^1.$ Then $f_*$ sends a vector space $V$ to the $\mathca …

The automorphisms $a_1,a_2,a_3$ are part of a $\mathbb{Z}/p\mathbb{Z}\times\mathbb{Z}/p\mathbb{Z}$ action on $F_k$, which in turn induces an action on the space $H^0(\Omega^1_{F_k})$ of holomorphic di …

Maybe let me try to synthesize my comments into an answer. All of this is contained in Raskin's beautiful paper arxiv.org/abs/1611.04937 on Whittaker categories. Convention: We work here in the derive …

(In characteristic zero) Flatness implies the other two definitions; integrability and formal lifting are very weak conditions (in fact if I haven't made a mistake I think this notion of integrability …

Here is one way to construct an object $X_{mdR}$ for $X$ say, a complex variety. Recall that $X_{dR}$ can be constructed as follows. Let $\hat{X}$ be the formal completion of the diagonal inside $X\ti …

Here is one way to see it, via classifying $G$-invariant radical ideals. (This has the bonus that it implicitly describes the boundary.) Lemma: $G$-invariant ideals $I$ of $\mathbb{C}[G/U]$ are in bij …

Here is the worst possible proof of all but one case, namely $m=r=3$ (but this is not included in your question as stated, though I think it is claimed in the paper.) Let $r$ denote the dimension of …

Denote $X\backslash\text{Sing}(X)$ by $X_0$ and its preimage in $Y$ as $Y_0$. Note that $Y_0$ is the Galois cover of $X_0$ corresponding to the normal subgroup $\mathbb{Z}[i]\times\mathbb{Z}[i]$ insid …

A cubic threefold is a smooth degree $3$ hypersurface in $\mathbb{P}^4$. Is there a cubic threefold $X$ over any field $k$ (possibly of positive characteristic) and an odd degree rational map $\mathbb …

The answers to your questions can be found in this article: https://arxiv.org/abs/1108.5351. I highly recommend reading it before trying to understand Arinkin-Gaitsgory. Let me try to resolve your di …