# Conference on p-adic methods in number theory

After somewhat of a hiatus, I’m back to blogging again.  The purpose of this post is to advertise the conference “p-adic Methods in Number Theory”, which will be held in Berkeley, CA from May 26-30, 2015.  The conference, which I am helping to organize, is in honor of the mathematical legacy of Robert Coleman.  Please spread the word!  Here is the current version of the conference poster, which will be mailed out soon to a math department near you: Many thanks to Janet Ziebell of the Georgia Tech College of Sciences for her help creating this poster, and to Ken McMurdy for designing the conference website.

Here is a memorial article about Robert which I co-authored with Barry Mazur and Ken Ribet.   I encourage you to read it!  It will be published in the new open access journal Research in the Mathematical Sciences, in a special volume dedicated to Robert.

You can find other interesting links related to Robert Coleman’s life and work here, and in this older blog post of mine.

# Newton polygons and Galois groups

A famous result of David Hilbert asserts that there exist irreducible polynomials of every degree $n$ over ${\mathbf Q}$ having the largest possible Galois group $S_n$.  However, Hilbert’s proof, based on his famous irreducibility theorem, is non-constructive.  Issai Schur proved a constructive (and explicit) version of this result: the $n^{\rm th}$ Laguerre polynomial $L_n(x) = \sum_{j=0}^n (-1)^j \binom{n}{j} \frac{x^j}{j!}$ is irreducible and has Galois group $S_n$ over ${\mathbf Q}$.

In this post, we give a simple proof of Schur’s result using the theory of Newton polygons.  The ideas behind this proof are due to Robert Coleman and are taken from his elegant paper On the Galois Groups of the Exponential Taylor Polynomials.  (Thanks to Farshid Hajir for pointing out to me that Coleman’s method applies equally well to the Laguerre polynomials.) Before we begin, here is a quote from Ken Ribet taken from the comments section of this post:

# Effective Chabauty

One of the deepest and most important results in number theory is the Mordell Conjecture, proved by Faltings (and independently by Vojta shortly thereafter). It asserts that if $X / {\mathbf Q}$ is an algebraic curve of genus at least 2, then the set $X({\mathbf Q})$ of rational points on $X$ is finite. At present, we do not know any effective algorithm (in theory or in practice) to compute the finite set $X({\mathbf Q})$. The techniques of Faltings and Vojta lead in principle to an upper bound for the number of rational points on $X$, but the bound obtained is far from sharp and is difficult to write down explicitly. In his influential paper Effective Chabauty, Robert Coleman combined his theory of p-adic integration with an old idea of Chabauty and showed that it led to a simple explicit upper bound for the size of $X({\mathbf Q})$ provided that the Mordell-Weil rank of the Jacobian of $X$ is not too large.  (For a memorial tribute to Coleman, who passed away on March 24, 2014, see this blog post.)

# Robert F. Coleman 1954-2014

I am very sad to report that my Ph.D. advisor, Robert Coleman, died last night in his sleep at the age of 59.  His loving wife Tessa called me this afternoon with the heartbreaking news.  Robert was a startlingly original and creative mathematician who has had a profound influence on modern number theory and arithmetic geometry.  He was an inspiration to me and many others and will be dearly missed.