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Number theory, arithmetic geometry
My research is in number theory and more specifically in arithmetic geometry. I like to study the family of compatible Galois representations associated to various arithmetic objects (eg. abelian varieties, modular forms, Drinfeld modules); these representations encode much, if not all, of the arithmetic of the originally object. Galois representations can also be used to study the absolute the Galois group of the rational numbers, and I have recently been playing with some applications to the Inverse Galois Problem.
- Explicit class field theory for global function fields, J. Number Theory 133 (2013), no.3, 1062–1078.
- Splitting fields of characteristic polynomials of random elements in arithmetic groups (with F. Jouve, E. Kowalski) Israel J. Math. 193 (2013) no.1, 263–307.
- A refinement of Koblitz’s conjecture, Int. J. Number Theory 7 (2011), no.3, 739–769.
- Elliptic curves with maximal Galois action on their torsion points, Bull. Lond. Math. Soc. 42 (2010), no. 5, 811–826.