Harold Stark

This article is about the mathematician Harold Stark. On the admiral, see Harold R. Stark .

Harold Mead Stark (born August 6, 1939 in Los Angeles ) is an American number theorist .

Life

Stark received his PhD in 1964 from the University of California, Berkeley with Derrick Henry Lehmer with the dissertation On the Tenth Complex Quadratic Field with Class Number One . In 1968 he became a Sloan Research Fellow . In 1970/71 he was at the Institute for Advanced Study in Princeton, New Jersey . After being a professor at the Massachusetts Institute of Technology (MIT), he is currently at the University of California, San Diego .

Stark is known for solving the Gaussian class number problem, the proof that there are exactly nine imaginary square number fields with the class number 1. As early as the 1930s it was known from Deuring , Heilbronn and Linfoot that there is at most one other with class number 1 in addition to the nine bodies already suspected by Carl Friedrich Gauß . Stark proved his theorem with methods of analytical number theory, he examined the behavior of the functions of binary quadratic forms at that point . It was later found that an earlier proof of Kurt Heegner's Stark theorem was also essentially correct. Alan Baker also gave a proof of this theorem at about the same time (1966) as Stark, using completely different methods. ${\displaystyle L}$ ${\displaystyle s=1}$

The Stark conjectures come from him, which provide a connection between analytical and algebraic quantities in finite Galois extensions of algebraic number fields . More precisely, the first non-vanishing coefficient of the Taylor expansion of the Artin function (a kind of Dirichlet function, which is formed with the help of representations of the Galois group ) is given at the point according to the conjecture of the regulator of the Stark units of . Stark proved these conjectures for -functions of Abelian extensions over the rational numbers and over imaginary quadratic number fields. The Stark conjectures provide (by determining these units) algorithms for partial solutions to Hilbert's 12th problem of the explicit construction of class fields and are a current research area in algebraic number theory. ${\displaystyle K}$ ${\displaystyle k}$ ${\displaystyle L}$ ${\displaystyle L}$ ${\displaystyle G}$ ${\displaystyle s=0}$ ${\displaystyle K}$ ${\displaystyle L}$

In 1970 he was invited speaker at the International Congress of Mathematicians in Nice ( Class number problems in quadratic fields ). He was inducted into the American Academy of Arts and Sciences in 1983 and the National Academy of Sciences in 2007. His PhD students include M. Ram Murty , Andrew Odlyzko , Kenneth Rosen, Jeffrey Lagarias , Jeffrey Hoffstein.

Fonts

• Class-number problems in quadratic fields. International Congress of Mathematicians 1970, p. 511.
• Class numbers of complex quadratic fields (= Lecture Notes in Mathematics. Volume 320). 1973, pp. 153-174.
• Class fields and modular forms of weight one. In: Modular Functions of one Variable ${\displaystyle V}$ (= Lectures Notes in Mathematics. Volume 601). 1976, p. 277.
• An Introduction to Number Theory. MIT Press, 1978, ISBN 0-262-69060-8 .
• Galois theory, algebraic number theory, and zeta functions. In Michel Waldschmidt , Claude Itzykson , Jean-Marc Luck, Pierre Moussa (Eds.): Number Theory and Physics. Les Houches 1989, Springer, 1992.

literature

• John T. Tate : Le Conjectures de Stark sur les Fonctions d'Artin en . ${\displaystyle L}$ ${\displaystyle s=0}$Birkhäuser, 1984.

Web links

• Harold Stark in the Mathematics Genealogy Project (English)
• Homepage at the University of California at San Diego
• Curt Meyer: Comments on Heegner-Stark's theorem about the imaginary-quadratic number fields with the class number one. Journal of Pure and Applied Mathematics, Volume 242, 1970
• Siegel: To prove Stark's theorem. Inv. Math. 1968
• Hayes lectures on the Stark Conjectures
• Xavier Francois-Roblot: Stark's conjectures and Hilbert's twelfth problem. Experimental Mathematics Volume 9, 2000

swell

1. ↑ Stark: There is no tenth complex quadratic field with class-number one , Proceedings National Academy of Sciences, Volume 57, 1967, p. 216, online , A complete determination of the complex quadratic fields of class-number one , Michigan Mathematical Journal Volume 14, 1967, pp. 1-27
2. ↑ Stark: On the “gap” in a theorem of Heegner , Journal of Number Theory, Volume 1, 1969, pp. 16–27, Max Deuring: Imaginary quadratic number fields with the class number one ( Memento of the original from November 30, 2015 in Internet Archive ) Info: The archive link was inserted automatically and has not yet been checked. Please check the original and archive link according to the instructions and then remove this notice. , Inventiones Mathematicae Volume 5, 1968, p. 169, Carl Ludwig Siegel : To the proofs of Stark's theorem , Inventiones Mathematicae Volume 5, 1968, p. 180 
3. ↑ Baker Mathematika Volume 12, 1966, p. 204
4. ↑ Stark: -functions at , part 1-4, Advances in Mathematics, Vol. 7, 1971, p. 301, Vol. 17, 1975, p. 60, Vol. 22, 1976, p. 64, Vol. 35, 1980, p. 197. The values ​​at are linked to those at via a functional equation. ${\displaystyle L}$ ${\displaystyle s=1}$ ${\displaystyle s=1}$ ${\displaystyle s=0}$