Christopher Skinner | |
---|---|
Born | Little Rock, Arkansas | 4 June 1972
Alma mater | University of Michigan, Princeton University |
Known for | Main conjecture of Iwasawa theory for modular curves |
Scientific career | |
Fields | Mathematics |
Institutions | Princeton University |
Thesis | Deformations of Galois Representations (1997) |
Doctoral advisor | Andrew Wiles |
Doctoral students | Ellen Eischen |
Christopher McLean Skinner (born June 4, 1972) is an American mathematician and professor at Princeton University. He works in algebraic number theory and arithmetic aspects of the Langlands program.
Skinner was born on June 4, 1972 in Little Rock, Arkansas. [1] Skinner graduated with a B.A. from the University of Michigan in 1993. [1] He received a Ph.D. from Princeton University in 1997 under the supervision of Andrew Wiles. [1]
Skinner was a member of the Institute for Advanced Study from 1997 to 2000. [1] He was then an associate professor of mathematics at the University of Michigan from 2000 to 2004, and then a full professor from 2004 to 2006. [1] He became a professor of mathematics at Princeton University in 2006. [1]
Skinner and Wiles proved modularity results for residually reducible Galois representations in joint work. [2]
Skinner and Eric Urban proved many cases of Iwasawa–Greenberg main conjectures for a large class of modular forms. [3] As a consequence, for a modular elliptic curve over the rational numbers, they prove that the vanishing of the Hasse–Weil L-function L(E, s) of E at s = 1 implies that the p-adic Selmer group of E is infinite. Combined with theorems of Gross–Zagier and Kolyvagin, this gave a conditional proof (on the Tate–Shafarevich conjecture) of the conjecture that E has infinitely many rational points if and only if L(E, 1) = 0, a (weak) form of the Birch–Swinnerton-Dyer conjecture. These results were used by Manjul Bhargava, Skinner, and Wei Zhang to prove that a positive proportion of elliptic curves satisfy the Birch–Swinnerton-Dyer conjecture. [4] [5]
Skinner was a Packard Foundation Fellow from 2001 to 2006 [1] [6] and a Sloan Research Fellow from 2001 to 2002. [1] He was named an inaugural Fellow of the American Mathematical Society in 2013. [7] In 2015, he was named a Simons Investigator in Mathematics. [8] [9]
He was an invited speaker at the International Congress of Mathematicians in Madrid in 2006. [10]
Sir Andrew John Wiles is an English mathematician and a Royal Society Research Professor at the University of Oxford, specialising in number theory. He is best known for proving Fermat's Last Theorem, for which he was awarded the 2016 Abel Prize and the 2017 Copley Medal and for which he was appointed a Knight Commander of the Order of the British Empire in 2000. In 2018, Wiles was appointed the first Regius Professor of Mathematics at Oxford. Wiles is also a 1997 MacArthur Fellow.
John Henry Coates was an Australian mathematician who was the Sadleirian Professor of Pure Mathematics at the University of Cambridge in the United Kingdom from 1986 to 2012.
In mathematics, the Birch and Swinnerton-Dyer conjecture describes the set of rational solutions to equations defining an elliptic curve. It is an open problem in the field of number theory and is widely recognized as one of the most challenging mathematical problems. It is named after mathematicians Bryan John Birch and Peter Swinnerton-Dyer, who developed the conjecture during the first half of the 1960s with the help of machine computation. Only special cases of the conjecture have been proven.
In mathematics, arithmetic geometry is roughly the application of techniques from algebraic geometry to problems in number theory. Arithmetic geometry is centered around Diophantine geometry, the study of rational points of algebraic varieties.
This is a glossary of arithmetic and diophantine geometry in mathematics, areas growing out of the traditional study of Diophantine equations to encompass large parts of number theory and algebraic geometry. Much of the theory is in the form of proposed conjectures, which can be related at various levels of generality.
Victor Alexandrovich Kolyvagin is a Russian mathematician who wrote a series of papers on Euler systems, leading to breakthroughs on the Birch and Swinnerton-Dyer conjecture, and Iwasawa's conjecture for cyclotomic fields. His work also influenced Andrew Wiles's work on Fermat's Last Theorem.
A modular elliptic curve is an elliptic curve E that admits a parametrization X0(N) → E by a modular curve. This is not the same as a modular curve that happens to be an elliptic curve, something that could be called an elliptic modular curve. The modularity theorem, also known as the Taniyama–Shimura conjecture, asserts that every elliptic curve defined over the rational numbers is modular.
Bryan John Birch FRS is a British mathematician. His name has been given to the Birch and Swinnerton-Dyer conjecture.
Manjul Bhargava is a Canadian-American mathematician. He is the Brandon Fradd, Class of 1983, Professor of Mathematics at Princeton University, the Stieltjes Professor of Number Theory at Leiden University, and also holds Adjunct Professorships at the Tata Institute of Fundamental Research, the Indian Institute of Technology Bombay, and the University of Hyderabad. He is known primarily for his contributions to number theory.
In mathematics, a Heegner point is a point on a modular curve that is the image of a quadratic imaginary point of the upper half-plane. They were defined by Bryan Birch and named after Kurt Heegner, who used similar ideas to prove Gauss's conjecture on imaginary quadratic fields of class number one.
In mathematics, elliptic units are certain units of abelian extensions of imaginary quadratic fields constructed using singular values of modular functions, or division values of elliptic functions. They were introduced by Gilles Robert in 1973, and were used by John Coates and Andrew Wiles in their work on the Birch and Swinnerton-Dyer conjecture. Elliptic units are an analogue for imaginary quadratic fields of cyclotomic units. They form an example of an Euler system.
Wiles's proof of Fermat's Last Theorem is a proof by British mathematician Sir Andrew Wiles of a special case of the modularity theorem for elliptic curves. Together with Ribet's theorem, it provides a proof for Fermat's Last Theorem. Both Fermat's Last Theorem and the modularity theorem were believed to be impossible to prove using previous knowledge by almost all living mathematicians at the time.
In mathematics, the main conjecture of Iwasawa theory is a deep relationship between p-adic L-functions and ideal class groups of cyclotomic fields, proved by Kenkichi Iwasawa for primes satisfying the Kummer–Vandiver conjecture and proved for all primes by Mazur and Wiles. The Herbrand–Ribet theorem and the Gras conjecture are both easy consequences of the main conjecture. There are several generalizations of the main conjecture, to totally real fields, CM fields, elliptic curves, and so on.
Barry Charles Mazur is an American mathematician and the Gerhard Gade University Professor at Harvard University. His contributions to mathematics include his contributions to Wiles's proof of Fermat's Last Theorem in number theory, Mazur's torsion theorem in arithmetic geometry, the Mazur swindle in geometric topology, and the Mazur manifold in differential topology.
In arithmetic geometry, the Tate–Shafarevich groupШ(A/K) of an abelian variety A (or more generally a group scheme) defined over a number field K consists of the elements of the Weil–Châtelet group , where is the absolute Galois group of K, that become trivial in all of the completions of K (i.e., the real and complex completions as well as the p-adic fields obtained from K by completing with respect to all its Archimedean and non Archimedean valuations v). Thus, in terms of Galois cohomology, Ш(A/K) can be defined as
James S. Milne is a New Zealand mathematician working in arithmetic geometry.
Wei Zhang is a Chinese mathematician specializing in number theory. He is currently a Professor of Mathematics at the Massachusetts Institute of Technology.
Arul Shankar is an Indian mathematician at the University of Toronto specializing in number theory, particularly arithmetic statistics.
Sarah Livia Zerbes is a German algebraic number theorist at ETH Zurich. Her research interests include L-functions, modular forms, p-adic Hodge theory, and Iwasawa theory, and her work has led to new insights towards the Birch and Swinnerton-Dyer conjecture, which predicts the number of rational points on an elliptic curve by the behavior of an associated L-function.
Eric Jean-Paul Urban is a professor of mathematics at Columbia University working in number theory and automorphic forms, particularly Iwasawa theory.