Eric Jean-Paul Urban is a professor of mathematics at

Columbia University Columbia University in the City of New York, commonly referred to as Columbia University, is a Private university, private Ivy League research university in New York City. Established in 1754 as King's College on the grounds of Trinity Churc ...

working in

number theory Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example ...

and

automorphic forms In harmonic analysis and number theory, an automorphic form is a well-behaved function from a topological group ''G'' to the complex numbers (or complex vector space) which is invariant under the group action (mathematics), action of a discrete s ...

, particularly

Iwasawa theory In number theory, Iwasawa theory is the study of objects of arithmetic interest over infinite Tower of fields, towers of number fields. It began as a Galois module theory of ideal class groups, initiated by (), as part of the theory of cyclotomic ...

Career

Urban received his PhD in mathematics from

Paris-Sud University Paris-Sud University (), also known as the University of Paris — XI (or as the Orsay Faculty of Sciences, University of Paris before 1971), was a French research university distributed among several campuses in the southern suburbs of Paris, ...

in 1994 under the supervision of Jacques Tilouine. He is a professor of mathematics at Columbia University.

Research

Together with Christopher Skinner, Urban proved many cases of Iwasawa–Greenberg main conjectures for a large class of

modular form In mathematics, a modular form is a holomorphic function on the complex upper half-plane, \mathcal, that roughly satisfies a functional equation with respect to the group action of the modular group and a growth condition. The theory of modul ...

s. As a consequence, for a

modular elliptic curve A modular elliptic curve is an elliptic curve ''E'' that admits a parametrization ''X''0(''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 a ...

over the

rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (for example, The set of all ...

s, 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 (in joint work with

Manjul Bhargava Manjul Bhargava (born 8 August 1974) 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 A ...

and Wei Zhang) to prove that a positive proportion of elliptic curves satisfy the Birch–Swinnerton-Dyer conjecture.

Awards

Urban was awarded a

Guggenheim Fellowship Guggenheim Fellowships are Grant (money), grants that have been awarded annually since by the John Simon Guggenheim Memorial Foundation, endowed by the late Simon Guggenheim, Simon and Olga Hirsh Guggenheim. These awards are bestowed upon indiv ...

in 2007.

Selected publications

* *

References

External links

* {{DEFAULTSORT:Urban, Eric 20th-century French mathematicians 21st-century French mathematicians Number theorists Living people Date of birth missing (living people) Place of birth missing (living people) Columbia University faculty University of Paris alumni Year of birth missing (living people)