# AMS Colloquium Lectures: Benedict Gross and “Complex Multiplication: Past, Present, Future”

Benedict Gross, this year’s AMS Colloquium series lecturer.

Benedict Gross kicked off his series of talks in the AMS Colloquium Lectures on Tuesday by speaking about the past, with a plan to reach the future of Number Theory by Friday.  Gross, former MacArthur Fellow and winner of the Cole Prize in Number Theory is the George Vasmer Leverett Professor of Mathematics, Department of Mathematics, Harvard University. The series, entitled “Complex Multiplication: Past, Present, Future,” considers the interplay between imaginary quadratic fields and the theory of elliptic curves. The area “has a long and twisted history,” according to Gross. The first talk covered the two hundred years from 1751 to 1951, beginning with Euler reviewing Fagnano’s work on the lemniscate, and beginning his investigations of “elliptic integrals”  of the form

$\int\frac{dx}{\sqrt{ax^3+bx^2+cx+d}},$

which lead to elliptic curves. Legendre and Gauss studied positive definite binary forms up to equivalence under the special linear group SL_2(Z).  The number of equivalence classes of forms with a given discriminant is called the class number of the discriminant.  The connection between these class numbers (and their modern variants) and elliptic curves becomes the story of complex multiplication.

Benedict Gross delivering his lecture.

Gross’ talk followed this story through his first meeting with Don Zagier and their famous Gross-Zagier formula, which connects the L-functions of elliptic curves to the heights of Heegner points. Gross expressed his great admiration for Zagier, who was 24 when they first met and already a professor. In Gross’ telling, Zagier graduated high school at 13 and applied to Oxford, where he was not accepted because there were rules against undergraduates under 16.  So, Zagier instead went to MIT and finished in three years, and was then accepted as a graduate student at Oxford. Wow. That’s one path in mathematics! Ahem. Gross and Zagier’s formula is an enormous result in the field with many applications, including implying cases of the Birch and Swinnerton-Dyer conjecture.

Bjorn Poonen, Claude Shannon Professor of Mathematics at the Massachusetts Institute of Technology, at the talk.

The first installment of the series ended in 1951 with the work of Kurt Heegner, a German radio engineer and mathematical scholar who proved that there are only nine quadratic imaginary fields with class number one. Gross explained that Heegner phrased all of this in terms of Diophantine equations, and “nobody really noticed or believed his arguments” at the time.  In 1966, when Harold Stark and Bryan Birch reproved the result, it was realized that Heegner’s argument was in fact correct!  Apparently Alan Baker’s 1967 theorem bounding linear forms in logarithms also implies this result, by very different methods.

The second installment of the talks (in progress now!) covers the next fifty years, from 1952 to 2002, and the final installment, at 1 PM on Friday in Ballrooms I and II, will cover the latest developments in the area.

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