Wednesday, 10-12 and 14-15, ESI, Schrödinger lecture hall
starting on March 1, 2006

 
Heights arose naturally in connexion with Weil's proof in 1928 of the finite generation of the group of rational points on an abelian variety, and shortly afterwards in 1929 Siegel used heights in his proof of the finiteness of the set of integral points on a curve of positive genus. Somewhat later they were recognized as a very efficient tool for measuring algebraic numbers in the context of standard constructions in transcendence theory. Nowadays it is impossible to do much diophantine geometry without them.

Our course will start without diophantine geometry or even algebraic number theory. We will give basic definitions and properties, and then prove, at least for the harmless curve x+y=1, a special case of a famous 1992 result of Zhang about lower bounds. Using elementary algebraic number theory we will then establish some additional properties of heights and give a shorter second proof of Zhang's result.

In 1997 Bilu with his Equidistribution Theorem provided a considerable strengthening of Zhang's result. We will give a proof based on an analytic inequality of Erdös-Turàn together with a standard transcendence construction. This opens the way for a treatment of the more classical 1979 result of Dobrowolski, also about lower bounds, which remains one of the most useful for applications.

More recently there has been interest in upper bounds for heights. We will give a simple proof that the height is bounded above for points on the curve x+y=1 whose coordinates x,y are multiplicatively dependent; this means that the points lie in the union of all proper subgroups of the algebraic group G_m². We will also give an introduction to the theory in higher dimensions. Zhang proved his lower bound result for arbitrary subvarieties X in any G_mⁿ; now it is necessary to exclude points lying on subgroups contained in X. By contrast, the upper bound result is known essentially only for curves in G_mⁿ. There has been some recent work on planes, but still hardly anything is known for arbitrary surfaces. The general case is covered only by conjectures. The upper bounds can be applied to give new finiteness statements. Thus if C is a typical curve in G_mⁿ, then there are only finitely many points on C for which there are two independent multiplicative relations on the coordinates. The proof uses a deep generalization of Dobrowolski's Theorem established in 1999 by Amoroso and David. If time permits, we will treat in detail the case n=3, where it suffices to use another generalization due to Amoroso and Zannier in 2000. Just as for upper bounds, the corresponding finiteness statements for arbitrary varieties are covered only by conjectures. Interestingly enough these have recently turned up in other contexts: Zhang himself (unpublished), Zilber 2002 in connexion with Schanuel's Conjecture in transcendence theory, and Pink 2005 who treats the most general case of mixed Shimura varieties.