Eric Urban
Eric Jean-Paul Urban is a professor of mathematics at Columbia University working in number theory and automorphic forms, particularly Iwasawa theory.
Eric Urban | |
|---|---|
Urban at the Mathematical Research Institute of Oberwolfach in 2018 | |
| Alma mater | Paris-Sud University |
| Awards | Guggenheim Fellowship (2007) |
| Scientific career | |
| Fields | Mathematics |
| Institutions | Columbia University |
| Thesis | Arithmétique des formes automorphes pour GL(2) sur un corps imaginaire quadratique (1994) |
| Doctoral advisor | Jacques Tilouine |
Career
Urban received his PhD in mathematics from Paris-Sud University in 1994 under the supervision of Jacques Tilouine.[1] He is a professor of mathematics at Columbia University.[2]
Research
Together with Christopher Skinner, 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 (in joint work with Manjul Bhargava and Wei Zhang) to prove that a positive proportion of elliptic curves satisfy the Birch–Swinnerton-Dyer conjecture.[4][5]
Awards
Urban was awarded a Guggenheim Fellowship in 2007.[6]
Selected publications
- Urban, Eric (2011). "Eigenvarieties for reductive groups". Annals of Mathematics. Second Series. 174 (3): 1685–1784. doi:10.4007/annals.2011.174.3.7. ISSN 0003-486X.
- Skinner, Christopher; Urban, Eric (2014). "The Iwasawa Main Conjectures for GL2". Inventiones Mathematicae. 195 (1): 1–277. Bibcode:2014InMat.195....1S. doi:10.1007/s00222-013-0448-1. ISSN 0020-9910. S2CID 120848645.
References
- ^ Eric Urban at the Mathematics Genealogy Project
- ^ "Eric Jean-Paul Urban » Department Directory". Columbia University. Retrieved 3 March 2020.
- ^ Skinner, Christopher; Urban, Eric (2014). "The Iwasawa Main Conjectures for GL2". Inventiones Mathematicae. 195 (1): 1–277. Bibcode:2014InMat.195....1S. doi:10.1007/s00222-013-0448-1. ISSN 0020-9910. S2CID 120848645.
- ^ Bhargava, Manjul; Skinner, Christopher; Zhang, Wei (2014-07-07). "A majority of elliptic curves over $\mathbb Q$ satisfy the Birch and Swinnerton-Dyer conjecture". arXiv:1407.1826 [math.NT].
- ^ Baker, Matt (2014-03-10). "The BSD conjecture is true for most elliptic curves". Matt Baker's Math Blog. Retrieved 2019-02-24.
- ^ "Eric Urban". John Simon Guggenheim Memorial Foundation. Retrieved 9 March 2021.
External links
- Eric Urban at the Mathematics Genealogy Project