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The general idea is that for an arithmetic variety $X$, different choices of the pair $(H^\bullet,\mathcal{C})$, where $H^\bullet$ is a cohomology theory and $\mathcal{C}$ is a complex, yield differen …

This was alredy answered in the comments, it is the $p$-adic completion of $\mathbb{Q}_p(p^{1/p^\infty})$ that is a perfectoid field. But here is a reference for completeness Matthias Wulkau, Revie …

Apparently this issue was worked out shortly after, during Go Yamashita's stay at the IHES in 2006, and after some discussion with Deligne. For any Tate motive $M$ unramified at $v$, its crystalline …

This is completely worked out in Walsh's paper: P. G. Walsh, Integer Solutions to the Equation $y^2=x(x^2\pm p^k)$ (2008) In particular, for your elliptic curve $y^2=x^3+px$, there are at most 2 p …

I think that not much has changed since 2012, in terms of general consensus within the mathematical community. There's some very interesting opinions and notes on the topic (see for example the one b …

I accidentally found the answer to this question in the book "The Abel Prize. 2003-2007 The First Five Years", page 74: "A first lecture on zeta and L-functions in the setting of the theory of s …

Vojta's general statement (the "conjectured refinement of Roth's theorem", analogous to the second main theorem in Nevanlinna theory) is: Let $X$ be a smooth complete variety over $k$, let $D$ be a n …

Most research on the Hasse-Weil zeta function focuses on some particular type of algebraic variety, and general surveys usually deal mostly with the better understood elliptic curve case. I am lookin …

If the rank of elliptic curves over $\mathbb{Q}$ is unbounded, there is no known reason for this to be the case. It could very well be that the set of ranks has density $0$ on the naturals, for all we …

As a concrete exampe of how much the answer depends on what particular results you are interested, take the problem of understanding the rational points on a non-singular algebraic curve over $\mathbb …

This is covered at the end of Serre's book "Abelian l-adic Representations and Elliptic Curves". Basically, since $\mathrm{Gal}(\bar{K}/K)$ acts semi-simply on $V_p(E)$ (you might have to go to Lang' …

The proof of this result is worked out in detail in page 250 (theorem 7.5) of the book Eberhard Freitag, Reinhardt Kiehl "Etale Cohomology and the Weil Conjecture" You can preview that page in par …

The way to reconstruct the differential sheaf from the fundamental group is, of course, $$\pi_1(X)\curvearrowright X\curvearrowright \omega_X$$ But I don't think you can get to the differential shea …

You might want to also consider the geometric ("symplectic") version of the conjecture, since Mochizuki alredy has a paper outlining the relationship between Bogomolov's proof and his own IUT theory. …

By Mordell-Weil, for any number field $K$ we have $$C(K)=\mathbb{Z}^r \times E(K)_{\mathrm{tors}}$$ As you mention, Mazur showed all the possible options for $E(\mathbb{Q})_{\mathrm{tors}}$ in his f …