## Ernst Kani

Title: |
Professor |

Office: |
Jeffery Hall, Rm. 211 |

Phone: |
(613) 533-2435 |

Email: |
kani@queensu.ca |

Website: |
mast.queensu.ca/~kani |

Research: |
Diophantine geometry, algebraic geometry, number theory, cryptography, modular forms |

** Degrees & Accolades**

Dr. rer. Nat. (University Heidelberg, Germany)

M.Sc. (University of Toronto)

B.Sc. (University of Toronto)

** Research Profile**

My main research area is Arithmetic Geometry, which is a synthesis of Number Theory and Algebraic Geometry, and which includes the realm of Diophantine questions such as Fermat's Last Theorem. In addition, I am interested in applications of Arithmetic Geometry to Public Key Cryptography.

More precisely, my research focuses on Galois representations, particularly those attached to (modular) elliptic curves; these played a crucial role in Wiles' proof of Fermat's Last Theorem. The question of when such representations can be isomorphic is of special interest to me, for a precise understanding of this question would have important Diophantine consequences, as G. Frey (Essen) has observed. This leads to four inter-connected yet separate lines of investigations:

1) Modular Diagonal Quotient Surfaces, which classify such isomophisms;

2) Curves of genus 2 with elliptic differentials, which are obtained by fusing together two elliptic curves with isomorphic Galois representations (in part joint work with G. Frey).

3) Hurwitz schemes arising from genus 2 covers over elliptic curves.

4) Representations of fundamental groups (in part joint work with G. Frey and H. Volklein).

The above topics, particularly the modular diagonal quotient surfaces, are also closely related to following topic(s):

5) Modular forms and Galois representations