A p-adic proof that pi is transcendental

Lindemann

Ferdinand von Lindemann

In my last blog post, I discussed a simple proof of the fact that pi is irrational.  That pi is in fact transcendental was first proved in 1882 by Ferdinand von Lindemann, who showed that if \alpha is a nonzero complex number and e^\alpha is algebraic, then \alpha must be transcendental.  Since e^{i \pi} = -1 is algebraic, this suffices to establish the transcendence of \pi (and setting \alpha = 1 it shows that e is transcendental as well).  Karl Weierstrass proved an important generalization of Lindemann’s theorem in 1885.

The proof by Lindemann that pi is transcendental is one of the crowning achievements of 19th century mathematics.  In this post, I would like to explain a remarkable 20th century proof of the Lindemann-Weierstrass theorem due to Bezivin and Robba [Annals of Mathematics Vol. 129, No. 1 (Jan. 1989), pp. 151-160], which uses p-adic analysis in a key way.  Their original argument was made substantially more elementary by Beukers in this paper; we refer the reader to [American Mathematical Monthly Vol. 97 Issue 3 (Mar. 1990), pp. 193-197] for a lovely exposition of the resulting proof, which rivals any of the usual approaches in its simplicity.  But I’d like to focus here on the original Bezivin-Robba proof, which deserves to be much better known than it is.  In the concluding remarks, we will briefly discuss a 21st century theorem of Bost and Chambert-Loir that situates the Bezivin-Robba approach within a much broader mathematical framework. Continue reading

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.)

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