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

Courses (Fall/Winter 2023/24)

• Math 211: Algebraic Methods (Fall/Winter)
• Math 112: Introduction to Linear Algebra (Winter)

Course Notes

• Math 812: Quadratic Forms and Elliptic Curves (Fall 08), Course notes, part I
• Math 817: Modular Forms (Winter 05), Course notes

• Subcovers of curves and moduli spaces.   In: Geometry at the Frontier Comtemporary Math. 766 (2021), 229-250.
  Abstract:   n/a     Preprint: pdf file.
  
• Elliptic subcovers of curves of genus 2. I. The isogeny defect   Ann. math. Quebec. 43 (2019), 281-303.
  Abstract:   pdf file     Preprint:   dvi file   or   pdf file.
  
• Elliptic subcovers of curves of genus 2. II. The refined Humbert invariant   J. Number Theory 193 (2018), 302-335
  Abstract:   pdf file     Preprint:   dvi file   or   pdf file.
  
• Elliptic subcovers of hyperelliptic curves.   Math. Nachr. 290 (2017), 2890-2900.
  Abstract:   pdf file     Preprint:   dvi file   or   pdf file.
  
• The moduli spaces of Jacobians isomorphic to a product of two elliptic curves. Collectanea Math. 67 (2016), 21-54.
  Abstract:   pdf file     Preprint:   dvi file   or   pdf file.
• Normal forms of hyperelliptic curves of genus 3 (with G. Frey). Des. Codes Cryptogr. 77 (2015), 677-712.
  Abstract:   (not available)     Preprint:   dvi file   or   pdf file.
• Jacobians isomorphic to a product of two elliptic curves and ternary quadratic forms. J. Number Theory 139 (2014), 138--174.
  Abstract:   pdf file     Preprint:   dvi file   or   pdf file.
  
• Simple geometricially split abelian surfaces over a finite field (with K.J. Chou).   J. Ramanujan Math. Soc. 29 (2014), 31-62.
  Abstract:   pdf file     Preprint:   dvi file   or   pdf file.
  
• Binary theta series and modular forms with complex multiplication.   Int. J. Number Theory 10 (2014), 1025-1042.
  Abstract:   pdf file     Preprint:   dvi file   or   pdf file.
  
• The space of binary theta series.   Ann. Sci. Math. Quebec 36 (2012), 501-534 (published 2013).
  Abstract:   pdf file     Preprint:   dvi file   or   pdf file.
  
• Correspondences on hyperelliptic curves and applications to the discrete logarithm (with G. Frey).   Springer LN Comp.Sci. 7053 (2012), 1-19.
  Abstract:   (not available)     Preprint:   dvi file   or   pdf file.
  
• Idoneal Numbers and some generalizations.   Ann. Sci. Math. Quebec 35 (2011), 197-227.
  Abstract:   (not available)     Preprint:   dvi file   or   pdf file.
• 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.
• The Hilbert-Burch Theorem (with A.V. Geramita). In: The Curves Seminar at Queen's VI. Queen's Papers in Pure Appl. Math. No. 89 (1989), pp. A92-A96.
• Potential theory on curves. In: Theorie des Nombres/Number Theory. W. de Gruyter, Berlin, 1989, pp. 475-543.
• Idempotent relations and factors of Jacobians (with M. Rosen). Math. Ann. 284 (1989), 307-327.
• Unramified double covers of hyperelliptic Klein surfaces. C.R. Math. Rep. Acad. Sci. Canada 9 (1987), 133-138.
• The Galois-module structure of the space of holomorphis differentials. J. reine angew. Math. 368 (1986), 187-206.
• 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.
• Weil heights, Neron pairings and v-metrics on curves. Rocky Mountain J. 15 (1985), 417-449.
• Robert Arnold Smith 23.1937 - 30.3.1983 (Obituary). Rocky Mountain J. 15 (1985), 413-415.
• On Castelnuovo's equivalence defect. J. reine angew. Math. 352 (1984), 24-70.
• Nonstandard Methods in Diophantine Geometry. In: Journees Arithmetiques, 1980. LMS Lecture Notes 56 (1982), 322-342.
• Eine Verallgemeinerung des Satzes von Castelnuovo-Severi. J. reine angew. Math. 318 (1980), 178-220.
• Nonstandard Diophantine Geometry. In: Proc. Queen's Number Theory Conf., 1979. Queen's Papers in Pure and Appl. Math. 54 (1980), 129-172.

• The Collected Papers of Hans Arnold Heilbronn (with R.A. Smith). J. Wiley & Sons, New York, 1988. XVI + 608 pp.
• Math 211: Algebraic Methods 14th ed. (with N. Pullman, N. Rice). Campus Bookstore, Queen's University at Kingston, 2015). iii + 421 pp.
• Lectures on Applications of Modular Forms to Number Theory. Course Notes for Math 817: Modular Forms (Winter 2005), 94pp.
• Lectures on Quadratic Forms and Elliptic Curves I. Course notes {part I) for Math 812 (Fall 2008), 102pp.

Selected Presentations   -- see also the complete list

• Die Singularitaeten der Humbertflaechen. (60 minute talk)
  Tuebinger Colloquium Conference, Universitaet Erlangen, Erlangen, Germany, 10 June 2023.
• Kurven vom Geschlecht 2 auf abelschen Produktflaechen. (60 minute talk)
  Tuebinger Colloquium Conference, Universitaet Duisburg-Essen, Essen, Germany, 27 August 2022.
• Modulkurven auf Humbertflaechen. (60 minute talk)
  Tuebinger Colloquium Conference, Universitaet Heidelberg, Heidelberg, Germany, 4 May 2019.
• Subcovers of curves and moduli spaces. (45 minute talk)
  Geometry at the Frontier III Conference (Lange Fest)
  Universidad de la Frontera, Temuco/Pucon, Chile. 13 November 2018.
• Moduli problems attached to isomorphisms of elliptic Galois representations. (Keynote address, 90 minutes)
  Workshop Alpbach 2018: Galois representations and heights. Alpbach, Austria, 1 July 2018.
• Galoisdarstellungen von elliptischen Kurven mit komplexer Multiplikation. (60 minute talk)
  Tuebinger Colloquium Conference, Universitaet Erlangen, Erlangen, Germany, 8 May 2018.
• Intersections of Humbert surfaces and binary quadratic forms. (50 minute talk)
  Conference on Arithmetic Aspects of Explicit Moduli Problems.
  Banff International Research Station (BIRS), Banff, Canada. 30 May 2017.
• Durchschnitte von Humbertflaechen und binaere quadratische Formen. (60 minute talk)
  Tuebinger Colloquium Conference, Universitaet Augsburg, Augsburg, Germany, 6 May 2017.
• Intersections of Humbert surfaces and binary quadratic forms. (45 minute talk)
  Quebec-Maine Number Theory Conference, Laval University, Quebec, 9 October 2016.
• Perfekte Kuboide und die Kastenvarietaet. (60 minute talk)
  Tuebinger Colloquium Conference, Universitaet Duisburg-Essen, Essen, Germany, 21 May 2016.
• Subcovers of curves and moduli spaces. (60 minute talk)
  R.D. Accola Memorial Conference, Boston University, Boston, MA, USA, 19 March 2016.
• Perfect cuboids and the box variety. (55 minute talk)
  CMS Conference, Section in Arithmetic Geometry, McMaster University, Hamilton, Ontario, 7-8 December 2014.
• Perfect cuboids and the box variety. (45 minute talk)
  Quebec-Maine Number Theory Conference, Laval University, Quebec, 27 September 2014.
• Hurwitz spaces for hyperelliptic curve covers. (60 minute talk)
  Conference in Algebra and Number Theory, Tel Aviv University, Tel Aviv, Israel, 2 May 2014.
• 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.
• 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, 14-16 October 2011.
• Der Ring der Modularkorrespondenzen (60 minute talk)
  Tuebinger Colloquium Conference, IEM (=Institute of Experim. Math.), U Duisburg-Essen, 7 May 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.
• Conjectural applications of arithmetic surfaces to number theory.
  Advanced Workshop on Arithmetic Algebraic Geometry II, Ain Shams University, Cairo, Egypt, 4--15 September 1993.
• Heights on abelian varieties and the theorem of Mordell-Weil.
  Advanced Workshop on Arithmetic Algebraic Geometry II, Ain Shams University, Cairo, Egypt, 4--15 September 1993.
• 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.

Graduate Students

• R. Leyland, Isomorphisms of Galois represesentations of CM elliptic curves. Ph.D. (2020 Fall)
• N. Garcia-Fritz, Curves of low genus on surfaces and applications to Diophantine problems. Ph.D. (2015 Fall)
• R. Ejouamai, The Tate Conjectures for product and quotient surfaces. Ph.D. (2013 Fall)
• A. Muresan, Computing explicit isogenies of elliptic curves over finite fields. MSc.II (2011 Fall).
• K.J. Chou, Constructing pairing-friendly algebraic curves of genus 2 with small rho-values. Ph.D. (2011 Summer).
• 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. M. Sc. (1994 Spring).

Undergraduate Student Research Projects

• R. Dewji (NSERC Summer Project, 2011): The representation 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.

Selected Graduate Courses

• Math 913: The Arithmetic of Modular Forms II (Reading Course) (Winter 2013)
• Math 913: The Arithmetic of Modular Forms (Winter 2012)
• Minicourse: Modular forms and the action of SL_2(Z) (June/July 2011)
• Minicourse: The representation theory of finite groups (May 2011)
• Math 812: Quadratic Forms and Elliptic Curves (Fall 2008)   Course notes, part I
• Minicourse: Curves with elliptic differentials (May-July 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)
• Math 919: Algebraic Geometry I, II (Fall 88, Winter 89)
• Math 821: Riemann Surfaces (Winter 88)
• Introduction to Diophantine Geometry (Winter 87)
• Rigide analytische Geometrie [Rigid analytic geometry] I, II (U Heidelberg, (Oct 85 - July 86)
• Diophantische Geometrie [Diophantine Geometry] I, II (U Heidelberg, Summer, Fall 84)
• Arithmetik und Geometrie der Modulkurven [Arithmetic and geometry of modular curves] I, II (U Heidelberg, Fall 81, Winter 82)
• Algebraische Geometrie [Algebraic Geometry] I-IV (U Heidelberg, Fall 79, Winter 80, Fall 80, Winter 81)
• Diophantische Gleichungen [Diophantine Equations] (U Heidelberg, Winter 79)