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
Alma materParis-Sud University
Scientific career
FieldsMathematics
InstitutionsColumbia University
ThesisArithmétique des formes automorphes pour GL(2) sur un corps imaginaire quadratique (1994)
Doctoral advisorJacques 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]

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
  1. Eric Urban at the Mathematics Genealogy Project
  2. "Eric Jean-Paul Urban » Department Directory". Columbia University. Retrieved 3 March 2020.
  3. 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.
  4. 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].
  5. Baker, Matt (2014-03-10). "The BSD conjecture is true for most elliptic curves". Matt Baker's Math Blog. Retrieved 2019-02-24.
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