# Applications of étale cohomology

It is well-known that étale cohomology is used in the proof of Weil conjectures and that SGA 4.5 is devoted to it. Also it seems(from a brief perusal of Milne's notes) that it is a kind of Galois Cohomology/Kummer theory for arbitrary algebraic varieties.

However I have heard a lot of people praising it, and this leads me to suspect that it must have applications beyond proving the Weil conjectures. I would be grateful if some of these can be given. I am sorry if this is a stupid question. The wikipedia page, Milne's article, etc., did not give any extra applications and so I hope asking people is more sensible. Please provide references also if available.

• Let me ask a related question. Do you know any applications of cohomology?? Any cohomology theory at all? Isn't it nice when the objects we're studying can be studied using cohomology, because cohomology linearises everything and turns stuff abelian, it's tangent spaces and things like that but it's oh so general. And now, if the things I happen to like studying are algebraic varieties, isn't it great that there's a cohomology theory! I now have access to all the wonderful linearisation tricks that it offers. Isn't that reason enough? – Kevin Buzzard Feb 23 '10 at 22:51
• @Regenbogen Sheaf cohomology for varieties in char p give vector spaces in char p. If you want to study, say, algebraic cycles, you need a characteristic zero cohomology. It's not just for the Weil conjectures, it's for everything. – Felipe Voloch Feb 23 '10 at 23:21
• Fantastic answers to the same question here: mathoverflow.net/questions/6070/etale-cohomology-why-study-it – Hunter Brooks Feb 23 '10 at 23:34
• "Why is it the case ... ?" Because of the particular aspects of geometry that you have studied. If you studied other aspects of algebraic geometry over the complex numbers, you would see many applications of singular cohomology. If you studied arithmetic geometry, you would see many application of etale cohomology (which plays the same role, roughly, as singular cohomology does in the complex setting). – Emerton Feb 23 '10 at 23:55
• Regenbogen- etale cohomology IS sheaf cohomology. Most often people use that name to refer to the cohomology of the constant sheaf (an analogue of topological cohomology, which is the cohomology of the constant sheaf in a usual topology), but the point of the etale cohomology is this: while taking cohomology in the Zariski topology works with coherent sheaves, for some other types of sheaves, it gives the wrong answer, so you need to take a different sort of topology on an algebraic variety, the etale topology (which isn't really a topology, but a generalization of one). – Ben Webster Feb 24 '10 at 2:08

One of the most important applications of etale cohomology is to Deligne-Lusztig theory, and the large subsequent body of work approaching the representation of finite groups of Lie type using $\ell$-adic cohomology. For me, this is the most important application beyond the Weil conjectures.

In addition to the original paper of Deligne and Lusztig "Representations of reductive groups over finite fields" in Ann. of Math 1976, you might be interested in the book "Weil Conjectures, Perverse Sheaves and l'adic Fourier Transform", by Weissauer and Kiehl.

If $X$ is a variety over $k$, the $\ell$-adic cohomology groups $H^i(X\otimes_k \overline{k},\mathbb{Q}_\ell)$ carry an action of $Gal(\overline{k}/k)$. This makes etale cohomology a very efficient (and the only?) tool to produce interesting Galois representations.

A typical example is $H^1(E_{\overline{Q}},Q_\ell)$ for an elliptic curve $E$ over $\mathbb{Q}$. This is dual to $(\varprojlim E[\ell^n]) \otimes_{Z_\ell} Q_\ell$ where $E[\ell^n] \subset E(\overline{\mathbb{Q}})$ is the set of $\ell^n$-torsion points of the curve. The action of $Gal(\overline{\mathbb{Q}}/\mathbb{Q})$ on $H^1(E_{\overline{\mathbb{Q}}},\mathbb{Q}_\ell)$ corresponds to its action on $E(\overline{Q})$. So for each elliptic curve over $\mathbb{Q}$, one gets a 2 dimensional representation of $Gal(\overline{\mathbb{Q}}/\mathbb{Q})$ (actually one for each $\ell$). These are central objects in the arithmetic of elliptic curves.

The proof of the Langlands correspondance for $GL_n$ by Drinfeld and Lafforgue uses the same principle. To associate a Galois representation to an automorphic representation they realize it as some $\ell$-adic cohomology group of a space.

• My only nitpick with this answer: It's not quite accurate to say that representations of Gal(Q) are central to the arithmetic of elliptic curves; I'd say that they're right near the center of number theory! – Harrison Brown Feb 24 '10 at 0:12

It can be used to give a short proof of the weak Mordell-Weil theorem, see my answer in Proofs of Mordell-Weil theorem.

Another application: Proving the ($p'$-part of the) conjecture of Birch and Swinnerton-Dyer for Abelian schemes over $1$- or higher dimensional bases over finite fields under the assumption that the algebraic rank equals the analytic rank or one $\ell$-primary subgroup of the Tate-Shafarevich group is finite.