(Richard) Patrick Morton

Professor of Mathematical Sciences, IUPUI

Ph.D., University of Michigan (1979); B.A., University of Arizona (1975)

e-mail:pmorton@math.iupui.edu

 

Main Interests - Number theory, curve theory, Galois theory, geometry, mathematical logic

 

Recent Publications and Preprints

8. Corrigendum to "On certain algebraic curves related to polynomial maps," (Math. Comp. 103), to appear in Compositio Mathematica.

7. Affine maps in triangle geometry and Feuerbach's theorem, preprint, 2009.

6. (with Igor Minevich) Synthetic cevian geometry, preprint, 2009.

5. The cubic Fermat equation and complex multiplication on the Deuring normal form, preprint, 2007.

4. Legendre polynomials and complex multiplication, I, preprint, 2007, submitted.

3. Ogg's theorem via explicit congruences for class equations, preprint, 2006, submitted.

2. Explicit identities for invariants of elliptic curves, Journal of Number Theory 120 (2006), 234-271.

1. (with John Brillhart) Class numbers of quadratic fields, Hasse invariants of elliptic curves, and the supersingular polynomial, Journal of Number Theory 106 (2004), 79-111.

 

Ongoing Projects

1. Solutions of the cubic Fermat equation in Hilbert class fields of imaginary quadratic fields.

2. On Hasse's version of the Grunwald-Hasse-Wang theorem.

3. Galois groups of iterated Euclidean constructions.

4. Legendre polynomials and complex multiplication.

 

Papers on the algebra and number theory of dynamical systems

1. Arithmetic properties of periodic points of quadratic maps, Acta Arithmetica 62 (1992), 343-372.

2. (with Pratiksha Patel) The Galois theory of periodic points of polynomial maps, Proc. London Math. Soc. 68 (1994), 225-263.

3. Characterizing cyclic cubic extensions by automorphism polynomials, J. Number Theory 49 (1994), 183-208.

4. (with Anjula Batra) Algebraic dynamics of polynomial maps on the algebraic closure of a finite field I, Rocky Mountain J. of Math. 24 (1994), 453-481.

5.(with Anjula Batra) Algebraic dynamics of polynomial maps on the algebraic closure of a finite field II, Rocky Mountain J. of Math. 24 (1994), 905-932.

6. (with Joseph Silverman) Rational periodic points of rational functions, International Mathematical Research Notices 2 (1994), 97-110.

7. (with Franco Vivaldi) Bifurcations and discriminants for polynomial maps, Nonlinearity 8 (1995), 571-584.

8. (with Joseph Silverman) Periodic points, multiplicities, and dynamical units, J. fr reine und angewandte Mathematik 461 (1995), 81-122.

9. On certain algebraic curves related to polynomial maps, Compositio Mathematica 103 (1996), 319-350.

10. Periods of maps on irreducible polynomials over finite fields, Journal of Finite Fields and their Applications 3 (1997), 11-24.

11. Arithmetic properties of periodic points of quadratic maps, II, Acta Arithmetica 87 (1998), 89-102.

12. Galois groups of periodic points, Journal of Algebra 201 (1998), 401-428.

 

Papers on digit patterns, automatic sequences, and Rudin-Shapiro coefficients

1. (with John Brillhart and John S. Lomont) Cyclotomic properties of the Rudin-Shapiro polynomials, J. fr reine und angewandte Mathematik 288 (1976), 37-65.

2. (with John Brillhart) ber Summen von Rudin-Shapiroschen Koeffizienten, Illinois J. of Math. 22 (1978), 126-148.

3. A note on recurrences satisfied by special classes of polynomials, Houston J. of Math. 7 (1981), 97-101.

4. (with John Brillhart and Paul Erds) On sums of Rudin-Shapiro coefficients, II, Pacific J. of Math. 107 (1983), 39-69.

5. (with David W. Boyd and Janice Cook) On sequences of 1's defined by binary patterns, Dissertationes Math. 283, Polish Scientific Publishers, 1989.

6. (with Wanda Mourant) Paper folding, digit patterns, and groups of arithmetic fractals, Proc. London Math. Soc. 59 (1989), 253-293.

7. Connections between binary patterns and paperfolding, Seminaire de Theorie des Nombres, Bordeaux 2 (1990), 1-12.

8.(with Wanda Mourant) Digit patterns and transcendental numbers, J. Australian Math. Soc. 51 (Series A) (1991), 216-236.

9. (with Jean-Paul Allouche and Jeffrey Shallit) Pattern spectra, substring enumeration, and automatic sequences, J. Theoretical Computer Science 94 (1992), 161-174.

10. (with John Brillhart) A Case Study in Mathematical Research: the Golay-Rudin-Shapiro sequence, American Mathematical Monthly 103 (1996), 854-869.  This article was awarded the Lester Ford Award for best expository paper of 1996 by the Mathematical Association of America in 1997.

 

Papers on number theory and diophantine problems (other than thesis papers)

1. (with Andrew Bremner) Polynomial relations in characteristic p, Quarterly J. Oxford (2) 29 (1978), 335-347.

2. (with Donald J. Lewis) On normal closures related to elliptic curves, J. Australian Math. Soc. 26 (Series A) (1978), 385-409.

3. A generalization of Zolotarev's theorem, American Math. Monthly 86 (1979), 374-375.

4. On the eigenvectors of Schur's matrix, J. Number Theory 12 (1980), 122-127.

5. (with Andrew Bremner) The integer points on three related elliptic curves, Mathematics of Computation 39 (1982), 235-238.

6. (with Donald J. Lewis) Quotients of polynomials and a theorem of Pisot and Cantor, J. of Faculty of Science, University of Tokyo (Sec. IA) 28 (1982), 813-822.

7. (with A. Bremner, R. Calderbank, P. Hanlon, and J. Wolfskill) Two-weight ternary codes and the equation y^2=4.3^k + 13, J. Number Theory 16 (1983), 212-234.

8. (with Andrew Bremner) A new characterization of the integer 5906, Manuscripta Math. 44 (1983), 187-229.

9. (with Andrew Bremner and Donald J. Lewis) Some varieties with points only in a field extension, Archiv fr Math. 43 (1984), 344-350.

10. Musings on the prime divisors of arithmetic sequences, American Math. Monthly 97 (1990), 323-328.

11. (with A.R. Calderbank) Quasi-symmetric 3-designs and elliptic curves, SIAM J. of Discrete Math. 3 (1990), 178-196.

12. (with John Brillhart) Table Corrigedum: H. Weber, Lehrbuch der Algebra, vol. 3, Chelsea, N.Y., Mathematics of Computation 65 (1996), 1379.

 

Papers related to my Ph. D. thesis (University of Michigan, Ann Arbor, 1979)

1. On Redei's theory of the Pell equation, J. fr reine und angewandte Mathematik 307/308 (1979), 373-398.

2. Density results for the 2-classgroups of imaginary quadratic fields, J. fr reine und angewandte Mathematik 332 (1982), 156-187.

3. Density results for the 2-classgroups and fundamental units of real quadratic fields, Studia Scientiarum Mathematicarum Hungarica 17 (1982), 21-43.

4. The quadratic number fields with cyclic 2-classgroups, Pacific Journal of Mathematics 108 (1983), 165-175.

5. On the non-existence of abelian conditions governing the solvability of the -1 Pell equation, J. fr reine und angewandte Mathematik 405 (1990), 147-155.

6. Governing fields for the 2-classgroup of Q((-q_1 q_2 p)) and a related reciprocity law, Acta Arithmetica 55 (1990), 267-290.