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Ernst Kani

E-mail: Kani at mast.queensu.ca
Web: http://www.mast.queensu.ca/~kani/

Mailing Address:
Department of Mathematics, Jeffery Hall,
99 University Avenue, Queen's University,
Kingston, Ontario, K7L 3N6, Canada

Office: Jeffery Hall, Room 211
Office Phone: (613) 533-2435
Office Fax: (613) 533-2964

Courses (Fall/Winter 2023/24)

Course Notes

Research Interests

Key words: Arithmetic (Diophantine) Geometry; elliptic curves, abelian varieties, Galois representations, modular forms, moduli problems, Hurwitz spaces, curves of genus 2.

Summary: 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


Publications (after 1995)

Publications (before 1995)

Books and Lecture Notes

Selected Presentations   -- see also the complete list

Graduate Students

Undergraduate Student Research Projects

Selected Graduate Courses

A Personal Note

My non-mathematical interests include white-water canoeing, wilderness camping/canoeing and skiing.