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Eric Urban

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Eric Urban
Urban at the Mathematical Research Institute of Oberwolfach in 2018
Alma materParis-Sud University
Scientific career
FieldsMathematics
InstitutionsColumbia University
Thesis Arithmétique des formes automorphes pour GL(2) sur un corps imaginaire quadratique  (1994)
Doctoral advisorJacques Tilouine

Eric Jean-Paul Urban is a professor of mathematics at Columbia University working in number theory and automorphic forms, particularly Iwasawa theory.

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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