# Dwork's use of p-adic analysis in algebraic geometry

Using p-adic analysis, Dwork was the first to prove the rationality of the zeta function of a variety over a finite field. From what I have seen, in algebraic geometry, this method is not used much and Grothendieck's methods are used instead. Is this because it is felt that Dwork's method is not general or powerful enough; for example, Deligne proved the Riemann Hypothesis for these zeta functions with Grothendieck's methods, is it felt that Dwork's method can't do this?

The premise of the question is not correct. Dwork's methods (and modern descendants of them) are a major part of modern arithmetic geometry over $p$-adic fields, and of $p$-adic analysis. You could look at the two volume collection Geometric aspects of Dwork theory to get some sense of these developments.

Just to say a little bit here: Dwork's approach led to Monsky--Washnitzer cohomology, which in turn was combined with the theory of crystalline cohomology by Bertheolt to develop the modern theory of rigid cohomology. The $p$-adic analysis of Frobenius actions is also at the heart of the $p$-adic theory of modular and automorphic forms, and of much of the machinery underlying $p$-adic Hodge theory. The theory of $F$-isocrystals (the $p$-adic analogue of variations of Hodge structure) also grew (at least in part) out of Dworks ideas.

To get a sense for some of Dwork's techniques, you can look at the Bourbaki report Travaux de Dwork, by Nick Katz, and also at Dwork's difficult masterpiece $p$-adic cycles, which has been a source of insipiration for many arithmetic geometers.

In some sense the $p$-adic theory is more difficult than the $\ell$-adic theory, which is why it took longer to develop. (It is like Hodge theory as compared to singular cohomology. The latter is already a magnificent theory, but the former is more difficult in the sense that it has more elaborate foundations and a more elaborate formalism, and (related to this) has ties to analysis (elliptic operators, differential equations) that are not present in the same way in the latter.) [For the experts: I am alluding to $p$-adic Hodge theory, syntomic cohomology, $p$-adic regulators, Serre--Tate theory, and the like.]

• Nice answer! As for the bunny's specific question, you could perhaps add that in the last few years a purely p-adic proof of the Weil conjectures has been completed using rigid cohomology (by Kiran Kedlaya, building on work of others). Mar 22, 2010 at 0:30
• "Dwork's difficult masterpiece p-adic cycles" I wonder since some time if I should read that? A Zentralblatt review indicates that it may be outdated, an article by Katz on parts of it is described as unreadable. At this occasion an other question: Do you have a text of your talk: math.uchicago.edu/~emerton/AlgGeom/AlgGeomAbstracts/p-adic.html ? Mar 22, 2010 at 11:05

I agree with Emerton's answer (and had similar thoughts, but since he is a leading expert in this field, it's better if the answer comes from him). I would say that if anything the opposite is true: nowadays people (especially those of an arithmetic bent) are more interested in $p$-adic cohomology than $\ell$-adic cohomology and the former is viewed as richer and more difficult than the latter. Thus the importance of Dwork's work could scarcely be higher.

Flipping to the other side of the Weil conjectures, I think it is also not quite fair to say that Deligne proved the Riemann hypothesis "with Grothendieck's methods". I know you mean that he used Grothendieck's methods ($\ell$-adic cohomology) rather than Dwork's methods ($p$-adic analysis) but it doesn't do justice to the range of new ideas that Deligne brought to the table (as well as the ideas that were left on Grothendieck's table, to Grothendieck's eternal consternation).

Flipping back again, note that the Weil conjectures (and even parts of Deligne's significant generalization "Weil II") have since been proven completely by $p$-adic methods, c.f.:

Kedlaya, Kiran S.(1-MIT) Fourier transforms and $p$-adic `Weil II'. (English summary) Compos. Math. 142 (2006), no. 6, 1426--1450.

http://arxiv.org/abs/math/0210149

This is another sign that the "Dwork school" is going strong in contemporary research.

If you want to actually compute zeta functions of varieties over finite fields, then the p-adic methods are often more efficient. See e.g. the work of Kedlaya.

• I'm upvoting this answer, because I think that the work of Kedlaya is most relevant to the question. But I also think this answer should be expanded to include more history on Monsky-Washnitzer cohomology, etc.. My impression is that Kedlaya's work is strong enough to prove the Weil conjectures with purely p-adic methods, but I'm not expert enough to say who else contributed to this before Kedlaya. Also, I do not know whether p-adic methods are strong enough to prove everything in Weil II. Perhaps an expert can expand on this answer? Mar 22, 2010 at 0:35
• Kedlaya built on the foundational work of Berthelot and others, and was also very much inspired by Laumon's proof in the $\ell$-adic context via the Fourier transform. Of course, Kedlaya added a lot himself, and here it is also worth mentioning that his underlying techniques (the $p$-adic analysis of Frobenius) are squarely in the tradition of Dwork. Mar 22, 2010 at 0:41
• @Marty: Good question. In my answer (which was written independently of Felipe's answer and Bjorn's comment) I also mentioned Kedlaya's p-adic work. But I implied that his methods give not only a p-adic proof of Weil conjectures but also "Weil II". (This has since been fixed.) Upon a closer look, it seems that he proves some but not all of p-adic Weil II. I would indeed like to hear an expert weigh in on what remains to be done. Mar 22, 2010 at 0:43
• Would Satoh's algorithm for computing zeta functions, that relies on Dwork's work on deformations, be related to p-adic analysis? It's different than Kedlaya's work, but still uses p-adic geometry stuff that I don't understand. Mar 22, 2010 at 9:09
• Satoh's algorithm is indeed related although, for elliptic curves, the relevant p-adic theory was known before Dwork. Mar 22, 2010 at 11:19