**Section:**001

*Abelian Varieties*

**Term:**FA 2017

**Subject:**Mathematics (MATH)

**Department:**LSA Mathematics

Let X be an algebraic variety over the field of rational numbers. Can the Betti numbers dimHi(X(C),Q) be defined purely algebraically? Étale cohomology provides an affirmative answer to this question by realising the related vector spaces Hi(X(C),Q) ' Hi(X(C),Q) Q' purely algebraically for any prime number . It follows that the spaces Hi(X(C),Q') come equipped with an action of the group G = Aut(C) of automorphisms of the field C. This interaction between the topology of X(C) and the representation theory of G leads to a fascinating story that has been at the heart of numerous fundamental advances in twentieth century mathematics. The goal of the present class is to develop enough étale cohomology theory to appreciate some of these applications, most notably to arithmetic algebraic geometry.

**
References**

My favourite reference is [Del77, Arcata], which I would follow for the first part of course; this book is available online from the Springer website. Other sources covering similar content that will be useful references are:

- “Étale cohomology” by Milne (see [Mil80]).
- “Étale cohomology and the Weil conjecture” by Freitag and Kiehl (see [FK88]).
- Brian Conrad’s secret notes.
- Johan de Jong’s notes on étale cohomology available as part of the “Stacks project” (see [Sta]).

- [Del74] Pierre Deligne. La conjecture de Weil. I. Inst. Hautes Études Sci. Publ. Math., (43):273–307, 1974.
- [Del77] P. Deligne. Cohomologie ´etale. Springer-Verlag, Berlin, 1977. Séminaire de Géométrie Algébrique du Bois-Marie SGA4 12 , Avec la collaboration de J. F. Boutot, A. Grothendieck, L. Illusie et J. L. Verdier, Lecture Notes in Mathematics, Vol.569.
- [DL76] P. Deligne and G. Lusztig. Representations of reductive groups over finite fields. Ann. of Math. (2), 103(1):103–161, 1976.
- [FK88] Eberhard Freitag and Reinhardt Kiehl. Étale cohomology and the Weil conjecture, volume 13 of Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)]. Springer-Verlag, Berlin, 1988. Translated from the German by Betty S. Waterhouse and William C. Waterhouse, With an historical introduction by J. A. Dieudonné.
- [Har77] Robin Hartshorne. Algebraic geometry. Springer-Verlag, New York, 1977. Graduate Texts in Mathematics, No. 52.
- [Mil80] James S. Milne. Étale cohomology, volume 33 of Princeton Mathematical Series. Princeton University Press, Princeton, N.J., 1980.
- [Qui68] Daniel G. Quillen. Some remarks on étale homotopy theory and a conjecture of Adams. Topology, 7:111–116, 1968.
- [Sta] The Stacks Project. Available at http://math.columbia.edu/algebraic geometry/stacks-git/.

**Course Requirements:**

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**Intended Audience:**

Students in arithmetic algebraic geometry, and more generally, students in any field that benefits from contact with étale cohomology such as algebraic geometry, commutative algebra, number theory, representation theory and algebraic topology.

**Class Format:**

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