Tuesday, February 7, 2017

Connes 70

I am happy to report that to  celebrate Alain Connes' 70th birthday, 3 conferences on noncommutative geometry and its interactions with different fields are planned to take place in Shanghai, China. Students, postdocs, young faculty and all those interested in the subject are encouraged to participate. Please check the  Conference webpage for more details.



Tuesday, January 3, 2017

Gamma functions and nonarchimedean analysis

Happy New Year!

I view blog writing as a great opportunity to reach out to members of the mathematics community and especially the younger members; so in this sense blog writing is, for me, very similar to writing for Math Reviews. I have enjoyed doing both for many years (and many many years for MR!). Recently I wrote a review for MR on the paper ``Twisted characteristic p zeta functions'' written
by Bruno Angles, Tuan Ngo Dac and Floric Ribeiro Tavares (``MR Number: MR3515815''). I am attaching the review here with the permission of Math Reviews. You can find it, in preprint form, here with the original (with live hyperlinks to papers) on the MR site.

The paper being reviewed makes some demands of the reader. But the devoted reader will be rewarded with an early view of a beautiful new world. Those readers familiar with Drinfeld modules know that they exist in incredible profusion: One starts with a smooth projective, geometrically connected curve X over the finite field F_q with q elements. Then one chooses a fixed closed point \infty of X and defines the algebra A to be the Dedekind domain of functions regular away from \infty; so plays the role of the integers Z in the Drinfeld theory. One instance of such an A is, of course, the ring F_q[\theta] which is, like Z, Euclidean, and indeed most of the work done so far
is concentrated on this particular A as it is both easy to work with and very similar to classical arithmetic. However, ultimately, the theory should work for general A just as the theory of Drinfeld modules (and generalizations) does. As general A is very far from factorial, one can imagine that many interesting issues arise (and the paper being reviewed discusses them from an axiomatic viewpoint).

Of course, the theory of the zeta function is intimately connected with the theory of the Gamma function and so one should also expect analogs of Gamma functions to appear in the characteristic p theory with the correct one being given decades ago by Greg Anderson and Dinesh Thakur in the polynomial case. Their function appears firstly as an element of the Tate algebra of functions inconverging on the closed unit disc. One fascinating aspect of the paper being reviewed is that this Tate algebra is replaced by Tate algebras created out of the general rings A (and so lie inside curves of
higher genus as opposed to the affine line). This is the beautiful new world I mentioned above…..



Sunday, July 24, 2016

A motivic product formula

The classical product formula for number fields is a fundamental tool in arithmetic. In 1993, Pierre Colmez published a truly inspired generalization of this to the case of Grothendieck's motives. In turn, this spring Urs Hartl and Rajneesh Kumar Singh put an equally inspired manuscript on the arXiv devoted to translating Colmez into the theory of Drinfeld modules and the like. Underneath the mountains of terminology there is a fantastic similarity between these two beautiful papers and I have created a blog to bring this to the attention of the community. Please see:
https://drive.google.com/open?id=0BwCbLZazAtweamZYckpaTy15cFU

Tuesday, June 14, 2016

What is a functional equation?

Like all number theorists I am fascinated (to say the least) with the functional equation of 
classical  L-series. Years ago, I came up with a simple characterization of functional equations basically using only complex conjugation. This point being that, via a canonical change of variables (going back to Riemann), such L-series are, up to a nonzero scalar, given by real power series with the expectation that the zeroes are also real. In characteristic p the best one can hope is also that the zeroes will be as rational as the coefficients (though this statement needs to be modified to take care of standard factorizations as well as the great generality of Drinfeld's base rings A).

For those interested, a two page pdf can be found at the following link: https://drive.google.com/open?id=0BwCbLZazAtweTmNIa1ZSc0h2UEE

Saturday, March 26, 2016

An indirect consequence of the famous Lucas congruence...

So, in the course of function field arithmetic, one runs into the binomial coefficients (like one does most everywhere in mathematics); or rather the coefficients modulo a prime p. The primary result about binomial coefficients modulo p is of course the congruence of Lucas. In function field arithmetic
one seems to be unable to avoid the group obtained by permuting p-adic (or q-adic) coefficients of a number. I recently discovered a congruence using these permutations and the product of two binomial coefficients that I decided to blog about. The proof is an indirect consequence of Lucas and is perhaps more interesting than the result itself. One is then led to look for something related with the Carlitz polynomials, which are the function field analog of the binomial coefficients.

I put all of this into a three page pdf which, if you are interested, you can find here:
https://drive.google.com/file/d/0BwCbLZazAtweN293bkxwYUZEYVk/view?usp=sharing

Sunday, November 22, 2015

Review of "Arithmetic of characteristic p special L-values" by B. Anglès and L. Taelman

Bruno Anglès and Lenny Taelman have published a profound study of finite characteristic special values as listed in the title. This appeared in Proc. London Math. Soc. (3) 110 (2015) 1000-1032. The analogy with classical theory is clear throughout the paper, as well as some potentially interesting differences; these results bode very well for the future of the subject. I have written a long review of this paper as well as pointing to a very new one on the web. This is been submitted to Math Reviews/MathSciNet and appears here with their explicit permission.

(Added 12-11-2015: A revised version of the paper by Anglès and Tavares Rebeiro cited in the review (along with a third author Tuan Ngo Dac) has been put on the arXiv as arXiv:1511.06209v2. In this new version the result is established in complete generality for all q.)

 arXiv:1511.06209v2

 arXiv:1511.06209v

A url for a pdf of this review is

https://drive.google.com/file/d/0BwCbLZazAtweQUJrNF8zeXNPMUk/view?usp=sharing






Friday, October 30, 2015

C. Armana's Formula for the coefficients of $h$

(full pdf is at https://drive.google.com/file/d/0BwCbLZazAtweOC01dFgyQjBmUlU/view?usp=sharing)