Ernst Kani's homepage

Queen's Math. & Stats - Ernst Kani's Home Page

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

Degrees

B.Sc. (University of Toronto)
M.Sc. (University of Toronto)
Dr. rer. nat. (University Heidelberg, Germany)

Office Hours (Fall 2010)

Mon 13:30-14:30, Wed 12:30-13:30, Thurs 12:30-13:30 (in Room 211, Jeffrey Hall)

Courses (Fall/Winter 2008/09)

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

Recent Preprints

Binary theta series and modular forms with complex multiplication. Revised Sep 2011, 18pp. (To appear in Int. J. Number Theory)
Abstract: (not available) Preprint: dvi file or pdf file.
The space of binary theta series. Revised March 2011, 33pp.
Abstract: (not available) Preprint: dvi file or pdf file.
Correspondneces on hyperelliptic curves and applications to the discrete logarithm (with G. Frey) June 2011, 19pp. (To appear in Springer LN Comp.Sci. 7053).
Abstract: (not available) Preprint: dvi file or pdf file.
Simple geometricially split abelian surfaces over a finite field (with K.J. Chou). Sept 2009, revised June 2010, 28pp.
Abstract: (not available) Preprint: dvi file or pdf file.
Jacobians isomorphic to a product of two elliptic curves and ternary quadratic forms. July, 2009, revised July 2011, 36pp.
Revision of: The existence of Jacobians isomorphic to a product of two elliptic curves.
Abstract: (not available) Preprint: dvi file or pdf file.
See also: IEM Preprint No. 3 (2009), IEM (Essen), Aug 2009, 36pp.
Idoneal Numbers and some generalizations. May 2009, revised June 2010, 34pp. (To appear in Ann. Sci. Math. Quebec)
Abstract: (not available) Preprint: dvi file or pdf file.
The moduli spaces of Jacobians isomorphic to a product of two elliptic curves. Jan 2008, 39pp.
Abstract: (not available) Preprint: dvi file or pdf file.
Abelian subvarieties and the Shimura construction. (Revised) Aug. 2005, 25pp.
Abstract: pdf file Preprint: dvi file or pdf file.
Mazur's question on mod 11 representations of elliptic curves (with O. Rizzo). Dec. 2000, 22pp.
Abstract: pdf file Preprint: dvi file or pdf file.

Course Notes

Math 812: Quadratic Forms and Elliptic Curves (Fall 08), Course notes, part I

Math 817: Modular Forms (Winter 05), Course notes

Selected Publications

Products of CM elliptic curves. Collectanea Math. 62 (2011), 297-339.
Abstract: (not available) Preprint: dvi file or pdf file.
See also: IEM Preprint No. 2 (2009), IEM (Essen), July 2009, 54pp.
Curves of genus 2 with elliptic differentials and associated Hurwitz spaces (with G.Frey).
In: Arithmetic, Geometry, Cryptography and Coding Theory (Lachaud et al, eds.) Contemp. Math. 487 (2009), 33-81.
Abstract: (not available) Preprint: dvi file or pdf file.
See also: IEM Preprint No. 6 (2008), IEM (Essen), Oct 2008, 50pp.
Endomorphisms of Jacobians of modular curves. Archiv Math. 91 (2008), 226 - 237.
Abstract: pdf file Preprint: dvi file or pdf file.
The number of genus 2 covers of an elliptic curve. Manuscripta math. 121 (2006), 51-80.
Abstract: pdf-file Preprint: dvi file or pdf file.
The surjectivity of mod m Galois representations. Appendix to: The surjectivity of the Galois representations associated to non-CM elliptic curves by A.C. Cojocaru.
Bull. Can. Math. Soc. 48 (2005), 16-31.
Abstract: (not available) Preprint: dvi file or pdf file.
The modular degree and the congruence number of a weight 2 cusp form (with A.C. Cojocaru). Acta Arith. 114 (2004), 159-167.
Preprint: dvi file or pdf file.

Hurwitz spaces of genus 2 covers of elliptic curves. Collectanea Math. 54 (2003), 1-51.
See also: IEM Preprint No. 9 (2001), IEM (Essen), June 2001, 55pp.
Abstract: pdf file Preprint: dvi file or pdf file.

Projective p-adic representations of the K-rational geometric fundamental group (with G. Frey). Archiv der Mathematik 77 (2001), 32-46.
Preprint: dvi-file or pdf-file

Curves with infinite K-rational geometric fundamental group (with G. Frey and H. Volklein).
In: Aspects of Galois Theory - Proceedings of the Conference on Galois Groups 1997, University of Florida, Gainesville, Florida. (H. Volklein , D. Harbater, P. Muller, J. Thompson, eds.) London Math. Soc. Lecture Notes 256 (1999), 85-118.
Preprint: dvi-file or pdf-file

Modular diagonal quotient surfaces (with W. Schanz). Mathematische Zeitschrift 227 (1998), 337--366.
Preprint: dvi-file or pdf-file

Diagonal quotient surfaces (with W. Schanz). Manuscripta Mathematica 93 (1997), 67-108.
Preprint: dvi-file or pdf-file

The existence of curves of genus 2 with elliptic differentials. J. Number Theory 64 (1997), 130-161.
Preprint: dvi-file or pdf-file

The number of curves of genus 2 with elliptic differentials. J. reine angew. Math. 485 (1997), 93-121.
Preprint: dvi-file or pdf-file

Fermat's Last Theorem (Coleman-Ellis Lecture, November 1996). Queen's Mathematical Communicator, Summer 1997, 1-8.
Preprint: dvi-file or pdf-file

Elliptic curves on abelian surfaces. Manuscripta Mathematica 84 (1994), 199-223.
Preprint: dvi-file or pdf-file

Idempotent relations among arithmetic invariants attached to number fields and algebraic varieties (with M. Rosen). J. Number Theory 46 (1994), 230-254.

Discriminants of Hermitian R[G]-modules and Brauer's class number relation.
In: Algebra and Number Theory - Proc. Conf. at IEM Essen, 1992 (G. Frey, J. Ritter, eds.) W. de Gruyter, Berlin, 1994, pp. 43-135.
Preprint: dvi-file or pdf-file

Curves of genus 2 covering elliptic curves and an arithmetical application (with G. Frey).
In: Arithmetic Algebraic Geometry (G. van der Geer, F. Oort, J. Steenbrink, eds.), Progress in Math. 89, Birkhauser Boston, 1990, 153-176.

Idempotent relations and factors of Jacobians (with M. Rosen). Math. Ann. 284 (1989), 307-327.

Bounds on the number of non-rational subfields of a function field. Inventiones mathematicae 85 (1986), 185-198.

Relations between the genera and between the Hasse-Witt invariants of Galois coverings of curves. Canad. Math. Bull. . 28 (1985), 321-327.

On Castelnuovo's equivalence defect. J. reine angew. Math.. 352 (1984), 24-70.

Selected Presentations -- see also the complete list

Binary Theta Series and Modular Forms with Complex Multiplication (50 minute talk)
AMS Conference, Special Session in Number Theory , University Of Nebraska, Lincoln, 14-16 October, 2011.
Binary Theta Series and CM Modular Forms (40 minute talk)
Quebec-Maine Number Theory Conference , Laval University, Quebec, 2-3 October, 2010.
Curves of genus 2 and a Conjecture of Gauss (25 minute talk)
CMS Conference, Ottawa, Ontario , 6-8 December, 2008.
Curves of genus 2 and a Conjecture of Gauss (40 minute talk)
Quebec-Maine Number Theory Conference , Laval University, Quebec, 4-5 October, 2008.
A Galois theory for elliptic subfields of a genus 2 function field.
AMS Conference Durham, New Hampshire , 22-23 April, 2006.
The Hasse-Weil Zeta Function of a Quotient Variety
CMS Conference Waterloo, Ontario , 4-6 June, 2005.
Endomorphisms of Jacobians of Modular Curves and an Application
CMS Conference, Montreal, Quebec, 11-13 December, 2004.
Mazur's Question and Modular Diagonal Quotient Surfaces.
Quebec-Maine Conference U Laval, Quebec , 2-3 October, 2004.
Endomorphisms of Jacobians of Modular Curves
60th Birthday Conference of G. Frey, Essen, Germany , 7-10 July, 2004.
The number of covers of an elliptic curve.
AMS Conference Chapel Hill, North Carolina , 24-25 October, 2003.
Hurwitz spaces of covers of elliptic curves.
Quebec-Maine Conference U Laval, Quebec , 12-13 October, 2002.
The class number relations of Kronecker, Gierster and Hurwitz.
CMS Conference U Laval, Quebec , 15-17 June, 2002.
Hurwitz spaces of genus 2 covers of elliptic curves.
CMS Conference U Laval, Quebec , 15-17 June, 2002.
Modular Diagonal Quotient Surfaces.
Far Hills Workshop on Hilbert Modular Varieties. Far Hills Hotel (near Montreal), 3-6 January, 2002.
Hurwitz spaces of genus 2 covers of elliptic curves.
AMS Conference CA Irivine, California , 10-11 November, 2001.
Equivariant Atkin-Lehner Theory.
Mini Workshop on Arithmetic Geometry (G. Frey), Inst. Exp. Math., Univ. Essen, 4--6 June, 2000.
The state of the art of elliptic curve cryptography.
CITO Conference: Layers of Security CITO, Ottawa, 3 February, 2000.
p-adic representations of the K-rational geometric fundamental group.
Galois Actions and Geometry (P. Debes, H. Nakamura, A. Tamagawa), Math. Sci. Research Institute (MSRI), Berkeley, 11--15 October, 1999.
Mazur's question for mod 11 Galois representations.
Workshop on Algebraic Modular Forms and Modular Forms mod p (H. Darmon, M. Savin), CRM, University of Montreal, Montreal, 1--7 October, 1998.
Mazur's question for mod 11 Galois representations.
ICM-1998 Satellite Conference Algebraic and Arithmetic Geometry (H. Esnault, G. Frey, E. Viehweg), Universitaet Essen, 10--14 August, 1998.
Diagonal quotient surfaces and a question of Mazur.
Number Theory and Arithmetical Geometry: Arithmetical Applications of Modular Forms (G. Frey, J. Bost), San Feliu de Guixols, Spain, 24--29 October, 1997.
Diagonal quotient surfaces and a question of Mazur.
Algebraic Number Theory (C. Deninger, G. Frey, P. Schneider, A. Scholl), Oberwolfach, Germany, 20--26 July, 1997. Tagungsbericht Oberwolfach 28/1997, p.\ 7-8.
Fermat's Last Theorem.
Coleman-Ellis lecture, Queen's University, Kingston, 6 November 1996.
Applications of diagonal quotient surfaces to a problem of Mazur.
Conference on Arithmetic Algebraic Geometry (H. Koch, J. Kramer, E.-W. Zink), Berlin, Germany, 21--26 March, 1996. Max-Planck Institute f\"ur Mathematik Bonn Preprint No.\ 1996-51, 5pp.
Diagonal quotient surfaces.
Conference on Arithmetic Algebraic Geometry (H. Esnault, E. Viehweg, G. Frey), Institute for Experimental Mathematics, Essen, Germany, 21--24 November 1995.
The Noether formula.
Advanced Workshop on Arithmetic Algebraic Geometry II, Ain Shams University, Cairo, Egypt, 4--15 September 1993.
Curves of genus 2 with elliptic differentials and the Height Conjecture for elliptic curves.
Conference on Finiteness Theorems in Number Theory (G. Frey), Essen, Germany, 12-15 March 1991.
Inst. Exp. Math. Essen, Preprint No. 18-1991, pp. 30-39. See also Errata et Addenda.

Recent Graduate Students

C. Wan, Isomorphisms of Galois representation of elliptic curves over F_p, M.Sc.II (2004 Fall).
S. Mohit, The zeta-function of modular diagonal quotient surfaces, Ph.D. (2002 Spring).
A. VanTuyl, The field of N-torsion points of an elliptic curve over a finite field, M.Sc.II (1997 Fall).
G. Wiesend, G-equivariant Riemann-Roch Theorems, M.Sc. (1995 Spring).
A. Such, Endomorphisms of elliptic curves via p-divisible and formal groups. (1994 Spring).

Undergraduate Student Research Projects

R. Dewji (NSERC Summer Project, 2011(): The representatikon theory of SL_2(Z/NZ) and applications to modular forms
M. Ingle (NSERC Summer Project, 2008): The Galois groups of certain curve covers.
- A new formulation of Netto's argument ... by M. Ingle
P. Heakes, M. El-Jichi (4th year Math & Engineering Project, 2005/2006, co-supervised with F. Alajaji):
PK Cryptography: A runtime performance analysis of RSA, DL and ECC pertaining to smart cards.
C. Franc (NSERC Summer Project, 2005): Determining bases of spaces of modular forms
S. Tang, M. Whittaker (4th year Math & Engineering Project, 2004/2005, co-supervised with F. Alajaji):
An analysis of RSA, DL and ECC methods: Implementation and runtime performance.
T. Herland (4th year Math & Engineering Project, 2003/2004): Digital Signatures.
N. Alexander, J. Tytaneck (NSERC Summer Project, 2003): Cryptography: The Schoof-Elkies algorithm.
- Errata for Schoof's 1985 paper by N. Alexander and J. Tytaneck
M. Carmosini (NSERC Summer Project, 2000): Cryptography: The Schoof algorithm.
A. Durward (4th year Math & Engineering Project, 1999/2000): Techniques in Cryptography.
D. Tie Ten Quee & J. Wan (4th year Math & Engineering Project, 1999/2000): Public Key Encryption Protocols.
D. Stroszka (NSERC Summer Project, 1999): Finite field algorithms.
P Williams (Summer Project, 1990): Algebraic structures.
R. McCann (NSERC Summer Project, 1987): Plotting algebraic curves.

Recent Graduate Courses

Math 812: Quadratic Forms and Elliptic Curves (Fall 2008) Course notes, part I

Minicourse: Curves with elliptic differentials (Summer 2008)

Math 918: Galois representations and Algebraic Number Theory (Winter 2007)

Math 918: The arithmetic theory of modular forms II (Fall 2005)

Math 817: The arithmetic theory of modular forms I (Winter 2005) Course notes `

Math 893: Core Course in Algebra I (Fall 2004)

Math 812: Elliptic curves and modular forms (Fall 2003)

Math 918: Galois representations and modular diagonal quotient surfaces (Winter 2000)

Math 918: Atkin-Lehner Theory and the Shimura construction (Fall 1999)

Math 815: Number Theory and Cryptography (Fall 1999)

Seminar on Modular Forms (Winter 99)

Seminar on Hurwitz spaces (with S. Wevers) (Fall 98)

Math 912: Rational points on elliptic curves (Fall 98)

Math 811: Seminar on Galois Theory (with R. Murty) (Winter 98)

Math 812: Modular Forms and L-functions (with R. Murty) (Winter 97)

Math 917: Rational points on elliptic curves (Fall 94)

Math 918: The Lefschetz-Riemann-Roch Theorem (Winter 94)

Math 918: Curves of genus 2 with elliptic differentials (Winter 93)

Seminar on modular elliptic curves (Fall 92)

Relative Kurven [Relative curves] (University of Erlangen (Germany), Winter Semester 91/92)

Seminar on codes, lattices and elliptic surfaces (Winter 91)

A Personal Note

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