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The method explained in Husemöller's book on elliptic curves is as follows: Take a general quartic $v^2=f_4(u)=a_ou^4+a_1u^3+a_2u^2+a_3u+a_4$, and let $$u=\frac{ax+b}{cx+d}\qquad v=\frac{ad-bc}{(cx+d) …

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 …

Probably too late, but... (1) Not quite. The Hodge index theorem only works for $\mathbb{C}$. To extend it to all characteristic $0$ ground fields, you need the Lefschetz principle and the comparison …

Perhaps this helps: Kyungho Oh, Vanishing Theorems for Singular Varieties, (Proceedings of the American Mathematical Society).

I'm not sure if this is what you are looking for, but consider the Tate curve $E_q$ on your local field $k$. It sits on the exact sequence of $G_k$ modules. $$0 \to \mu_n \to E_q[n] \to \mathbb{Z}/n\ …

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 …

Some of this stuff is explained in much detail in this paper: Charles Weibel, History of Homological Algebra (1999) In particular, after reviewing the work of Riemman and Betti, it says: Insp …

The anabelian conjectures for small dimensions have been known for quite some time. In full generality the results are: Dimension 0. Finitely generated fields are anabelian (Pop) Dimension 1. Hyperb …

It seems that Grothendieck's familly has given permission for the distribution of his unpublished works, so I hope it is ok to ask this. Is there any way to obtain a copy (online or not) of "La lo …

All the manuscripts have been finally made avaible by Montpellier university. You can find all of them here, avaible in pdf. The items related to “La longue Marche" are: La "Longue Marche" à traver …

For future references. Feel free to edit to include new cases, or any improvements. As for 2015, the standard conjectures on algebraic cycles is unconditionally (at lest) known for $X$: Lefschetz st …

Partially inspired by an unpublished work of Wojtkoviak for $\zeta (3)$, all of this follows from this result by Deligne: Theorem. $\pi_1(\mathbb{P}^1-\{0,1,\infty\})$ is a smooth mixed Tate motive o …

I think the motivation for the Hirzebruch-Jung algorithm is not the algorithm itself, but the fact the it yields directly a very interesting continued fraction expansion. It is those, now called Hirze …

Elliptic curves over real quadratic fields were proven to be modular very recently by Freitas, Le Hung and Siksek: Nuno Freitas, Bao Le Hung, Samir Siksek, Elliptic Curves over Real Quadratic Fields …

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 …