Queen's Math. & Stats  Ernst Kani's Home Page
Ernst Kani
Email:
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) 5332435
Office Fax: (613) 5332964

Degrees
 B.Sc. (University of Toronto)
 M.Sc. (University of Toronto)
 Dr. rer. nat. (University Heidelberg, Germany)
Office Hours (Winter 2013)
Mon 11:3012:30, Wed 10:3011:30, Fri 9:3010:30
(in Room 211, Jeffrey Hall)
Courses (Fall/Winter 2013/14)
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 interconnected 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:
pdf file
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.
(To appear in J. Number Theory .)
Revision of:
The existence of Jacobians isomorphic to a product of two elliptic curves.
Abstract:
pdf file
Preprint:
dvi file
or
pdf file.
See also: IEM Preprint No. 3 (2009),
IEM (Essen),
Aug 2009, 36pp.

The moduli spaces of Jacobians isomorphic to a product of two elliptic curves.
Revised August 2013, 40pp.
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
Selected Publications

Simple geometricially split abelian surfaces over a finite field
(with K.J. Chou).
J. Ramanujan Math. Soc. 29 (2014), 3162.
Abstract:
pdf file
Preprint:
dvi file
or
pdf file.

The space of binary theta series.
Ann. Sci. Math. Quebec 36 (2012), 501534.
Abstract:
pdf file
Preprint:
dvi file
or
pdf file.

Correspondneces on hyperelliptic curves and applications to the discrete logarithm
(with G. Frey)
Springer LN Comp.Sci. 7053 (2012), 119.
Abstract:
(not available)
Preprint:
dvi file
or
pdf file.

Idoneal Numbers and some generalizations.
Ann. Sci. Math. Quebec 35 (2011), 197227.
Abstract:
(not available)
Preprint:
dvi file
or
pdf file.

Products of CM elliptic curves.
Collectanea Math. 62 (2011), 297339.
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), 3381.
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), 5180.
Abstract:
pdffile
Preprint:
dvi file or
pdf file.

The surjectivity of mod m Galois representations.
Appendix to:
The surjectivity of the Galois representations
associated to nonCM elliptic curves by A.C. Cojocaru.
Bull. Can. Math. Soc. 48 (2005), 1631.
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), 159167.
Preprint: dvi file
or
pdf file.

Hurwitz spaces of genus 2 covers of
elliptic curves.
Collectanea Math. 54 (2003), 151.
See also: IEM Preprint No. 9 (2001),
IEM (Essen),
June 2001, 55pp.
Abstract: pdf file
Preprint: dvi file
or
pdf file.

Projective padic representations of the
Krational
geometric fundamental group (with G. Frey).
Archiv der Mathematik 77 (2001),
3246.
Preprint:
dvifile
or
pdffile

Curves with infinite Krational 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), 85118.
Preprint:
dvifile
or
pdffile

Modular diagonal quotient surfaces (with W. Schanz).
Mathematische Zeitschrift 227
(1998), 337366.
Preprint:
dvifile
or
pdffile

Diagonal quotient surfaces (with W. Schanz).
Manuscripta Mathematica
93 (1997), 67108.
Preprint:
dvifile
or
pdffile

The existence of curves of genus 2 with elliptic differentials.
J. Number Theory 64 (1997),
130161.
Preprint:
dvifile
or
pdffile

The number of curves of genus 2 with elliptic differentials.
J. reine angew. Math. 485 (1997),
93121.
Preprint:
dvifile
or
pdffile

Fermat's Last Theorem (ColemanEllis Lecture, November 1996).
Queen's Mathematical Communicator, Summer 1997, 18.
Preprint:
dvifile
or
pdffile

Elliptic curves on abelian surfaces.
Manuscripta Mathematica
84 (1994), 199223.
Preprint:
dvifile
or
pdffile

Idempotent relations among arithmetic invariants attached to
number fields and algebraic varieties (with M. Rosen).
J. Number Theory
46 (1994), 230254.

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. 43135.
Preprint: dvifile
or
pdffile

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, 153176.
 Idempotent relations and factors of Jacobians
(with M. Rosen).
Math. Ann. 284 (1989), 307327.

Bounds on the number of nonrational subfields of a function field.
Inventiones mathematicae 85 (1986), 185198.
 Relations between the genera and between the
HasseWitt
invariants of Galois coverings of curves.
Canad. Math. Bull. . 28 (1985), 321327.

On Castelnuovo's equivalence defect.
J. reine angew. Math.. 352 (1984), 2470.
Selected Presentations
 see also the complete list

Tensor products of Galois representations.
(50 minute talk)
Ram Murty's 60th Birthday Conference
CRM, Montreal, P.Q. 15 Oct 2013.

Humbertsche Schemata und Durchschnitte von Humbert Flaechen.
(60 minute talk)
Tuebinger Colloquium Conference,
U Erlangen, Erlangen, Germany. 8 June 2013.

Generalized Humbert Schemes and Intersections of Humbert Surfaces.
(45 minute talk)
AMS Conference, Special Session in Computational Advances in
Special Functions and Tropical Geometry ,
University of Iowa, Ames, Iowa, 27 April 2013.

Tensor products of Galois representations.
CMS Conference, Special Session in
Algebraic Number Theory ,
Montreal, P.Q., 9 December 2012.

Tensorprodukte von Galoisdarstellungen.
Tuebinger Colloquium Conference ,
U Augsburg, Augsburg, Germany. 2 June,
2012.

Korrespondenzen und hyperelliptische Kurven.
Tuebinger Colloquium Conference ,
U Augsburg, Augsburg, Germany. 2 June 2012.

The Ring of Modular Correspondences.
Hawaii Conference in Algebraic Number Theory, Arithmetic Geometry,
and Modular Forms (HCANTAGMF)
University of Hawaii, Honolulu, 6 March 2012.

Binary theta series and CM modular forms.
AMS Conference, Special Session in
Number Theory ,
University of Hawaii, Honolulu, 3 March 2012.

Der Ring der Modularkorrespondenzen. (60 minute talk)
Tuebinger Colloquium Conference, IEM (=Institute of Experim. Math.), U
DuisburgEssen, 7 May 2011

Binary Theta Series and Modular Forms with Complex Multiplication
(50 minute talk)
AMS Conference, Special Session in Recent Directions in Number Theory,
University Of Nebraska, Lincoln, 1416 October 2011.

Binary Theta Series and CM Modular Forms (40 minute talk)
QuebecMaine Number Theory Conference ,
Laval University, Quebec, 23 October 2010.

Curves of genus 2 and a Conjecture of Gauss (25 minute talk)
CMS Conference, Ottawa, Ontario ,
68 December 2008.

Curves of genus 2 and a Conjecture of Gauss (40 minute talk)
QuebecMaine Number Theory Conference ,
Laval University, Quebec, 45 October 2008.

A Galois theory for elliptic subfields of a genus 2 function field.
AMS Conference Durham, New Hampshire ,
2223 April 2006.

The HasseWeil Zeta Function of a Quotient Variety
CMS Conference Waterloo, Ontario ,
46 June 2005.

Endomorphisms of Jacobians of Modular Curves and an Application
CMS Conference, Montreal, Quebec,
1113 December 2004.

Mazur's Question and Modular Diagonal Quotient Surfaces.
QuebecMaine Conference U Laval, Quebec ,
23 October 2004.

Endomorphisms of Jacobians of Modular Curves
60th Birthday Conference of G. Frey, Essen, Germany ,
710 July 2004.

The number of covers of an elliptic curve.
AMS Conference Chapel Hill, North Carolina , 2425 October, 2003.

Hurwitz spaces of covers of elliptic curves.
QuebecMaine Conference U Laval, Quebec ,
1213 October, 2002.

The class number relations of Kronecker, Gierster and Hurwitz.
CMS Conference U Laval, Quebec , 1517 June, 2002.

Hurwitz spaces of genus 2 covers of elliptic curves.
CMS Conference U Laval, Quebec , 1517 June, 2002.

Modular Diagonal Quotient Surfaces.
Far Hills Workshop on Hilbert Modular Varieties.
Far Hills Hotel (near
Montreal), 36 January, 2002.

Hurwitz spaces of genus 2 covers of elliptic curves.
AMS Conference CA Irivine, California , 1011 November, 2001.

Equivariant AtkinLehner Theory.
Mini Workshop on Arithmetic Geometry
(G. Frey), Inst. Exp. Math., Univ. Essen, 46 June, 2000.

The state of the art of elliptic curve cryptography.
CITO Conference: Layers of Security CITO, Ottawa,
3 February, 2000.

padic representations of the Krational geometric
fundamental
group.
Galois Actions and Geometry
(P. Debes, H. Nakamura, A. Tamagawa), Math. Sci. Research Institute
(MSRI), Berkeley, 1115 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, 17 October, 1998.

Mazur's question for mod 11 Galois representations.
ICM1998 Satellite Conference Algebraic and Arithmetic Geometry
(H. Esnault, G. Frey, E. Viehweg), Universitaet Essen, 1014 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,
2429 October, 1997.

Diagonal quotient surfaces and a question of Mazur.
Algebraic Number Theory (C. Deninger, G. Frey, P. Schneider,
A. Scholl),
Oberwolfach, Germany, 2026 July, 1997.
Tagungsbericht Oberwolfach 28/1997, p.\ 78.

Fermat's Last Theorem.
ColemanEllis 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, 2126 March, 1996.
MaxPlanck Institute f\"ur Mathematik Bonn Preprint No.\ 199651, 5pp.
 Diagonal quotient surfaces.
Conference on Arithmetic Algebraic Geometry
(H. Esnault, E. Viehweg, G. Frey),
Institute for Experimental Mathematics, Essen, Germany, 2124 November 1995.

The Noether formula.
Advanced Workshop on Arithmetic Algebraic Geometry II,
Ain Shams University, Cairo, Egypt, 415 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, 1215 March 1991.
Inst. Exp. Math. Essen, Preprint No. 181991, pp. 3039.
See also
Errata et Addenda.
Recent Graduate Students

R. Ejouamai, The Tate Conjectures for product and quotient surfaces,
Ph.D. (2013 Summer)

A. Muresan, Computing explicit isogenies of elliptic curves over finite
fields. MSc.II (2011 Fall).

K.J. Chou, Constructing pairingfriendly algebraic curves of genus 2 with
small rhovalues, Ph.D. (2011 Summer).

C. Wan, Isomorphisms of Galois representation of elliptic curves over F_p,
M.Sc.II (2004 Fall).

S. Mohit, The zetafunction of modular diagonal quotient surfaces,
Ph.D. (2002 Spring).

A. VanTuyl, The field of Ntorsion points of an elliptic curve over a
finite field, M.Sc.II (1997 Fall).

G. Wiesend, Gequivariant RiemannRoch Theorems, M.Sc. (1995 Spring).

A. Such, Endomorphisms of elliptic curves via pdivisible 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. ElJichi (4th year Math & Engineering Project, 2005/2006,
cosupervised 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,
cosupervised 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 SchoofElkies 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: AtkinLehner 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 Lfunctions (with R. Murty)
(Winter 97)

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

Math 918: The LefschetzRiemannRoch 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 nonmathematical interests include whitewater canoeing,
wilderness camping/canoeing and skiing.