Topics in Arithmetic Dynamics

18.784: Seminar in Number Theory, Spring 2025

Welcome to 18.784: Seminar in Number Theory! This course is a Communication Intensive in Mathematics (CI-M) for undergraduates at MIT.

This semester, we will explore arithmetic dynamics — an area of mathematics that studies how numbers move. The subject lies at the intersection of dynamical systems and number theory, involving a breadth of mathematical disciplines including:

Algebraic geometry
Algebraic number theory
Algorithmic number theory
Analytic number theory
Combinatorics
Complex analysis
p-adic analysis
Representation theory

Students will prepare a short paper and present specific topics in arithmetic dynamics. Grading is primarily based on the paper, presentations, and active participation during the semester.

Prerequisites for the course are Algebra 1 (18.701) or (Modern Algebra (18.703) and Linear Algebra (18.06 or 18.700)).
Additional mathematical background, such as elementary number theory (18.781) and Galois theory (18.702), will be helpful but not required.

Course files:

Google forms:

Intro questionnaire
Presentation feedback forms and sign-up sheets are posted in the course Slack

The seminar meets in 2-151 from 1:00pm—2:30pm (ET) on Tuesdays and Thursdays.
My office hours are in 2-238 from 2:30pm—3:30pm (ET) on Thursdays until March 6 and Tuesdays starting March 11.

To contact me, please email me or message me in the course Slack.
To contact our communications instructor, Emily Robinson, you can email her at erobin73@mit.edu.

Course schedule

Date	Speaker	Title	References
February 4	Robin Zhang	Organizational meeting
(Intro questionnaire due 11:59pm)	Emily Robinson	Communications workshop: preparing for presentations	[Ruf]
February 6	All students	3-minute introductory talks (favorite definition / example / theorem)
	Robin Zhang	What is arithmetic dynamics	[Sil1]§Introduction [Sil4]§1 [BIJMST]§1-2
February 11	Youssef Marrakchi & Alejandro Reyes	What is modular arithmetic	[IR]§3-4, 7 [Ros]§4-5, 9 [Smi] Ch. II
	Matija Likar & Jake Ross	What is a Cauchy sequence	[Bri]§1-2, 5 [Hun]§2.6, 3.7, 13.1-13.4
February 13	Jeff Lin & Ray Wang	What is a complex function	[Dun]§2-4 [SS1] Ch. 1-3
	Maximus Lu & Angeline Peng	What is Galois theory	[DF]§13-14 [Goo] [Lei]
February 18	President's Day Tuesday (no meeting)	President's Day Tuesday (no meeting)
February 20	Clarise Han & Isaac Rajagopal	What is a p-adic number	[Con2] [Gou]§1-3
	Lucy Epstein & Andrew Tung	What is an algebraic curve	[Ful]§1.1-1.7, 2.1-2.4, 3.1 [Gat] Ch. 0-1
February 25	Angeline Peng	Elliptic curves and their torsion points	[SS2]§Ogg's conjecture [ST]§1.1-1.2, 2.1-2.2, 2.5
	Youssef Marrakchi & Jake Ross	Complex rational maps and the projective line	[Ful]§4.1 [GG]§1.1.1 [Sil1]§1.1 [Sil4]§2.1-2.2
February 27	Emily Robinson	Communications workshop: effective mathematics presentations
March 4	Lucy Epstein & Maximus Lu	Dynamics in the digits of the rationals and modular arithmetic	[Con2] [Mor1]§1-7 [Viv]§1-2
	Isaac Rajagopal & Andrew Tung	Complex critical points and the Riemann-Hurwitz formula	[Sil1]§1.2 [Sil4]§2.3
March 6	Clarise Han & Jeff Lin	Rational periodic points of quadratic polynomials of periods 1–3	[Ros]§13.1 [WR]§1-3
	Matija Likar	Complex periodic points and multipliers	[Sil1]§1.3 [Sil4]§3
March 11	Alejandro Reyes & Ray Wang	Cyclotomic and dynatomic polynomials	[Con1] [Mor2]§1 [Sil1]§4.1
	Robin Zhang	Introduce paper assignment
March 13	Emily Robinson	Communications workshop: reading research papers
March 18	Alejandro Reyes	Asymptotics in dynamics over finite fields	[AG] [BIJMST]§18 [BOSZ] [Gre] [Hea] [Juu]
	Youssef Marrakchi	Graphs and zeta functions	[Bri1]§1 [Bri2]§1 [Fri] [Rue1] [Rue2]§1-7 [Ter]§1-2, 4, 6
March 20 (Paper topic proposal due 11:59pm)	Jeff Lin	Dynatomic modular curves	[BL]§1 [Mor2]§1 [Sil1]§4.1-4.2
	Clarise Han	Dynamical pseudorandom number generators	[BOSZ] [GIGS] [OS]
March 25	Spring Break (no meeting)	Spring Break (no meeting)
March 27	Spring Break (no meeting)	Spring Break (no meeting)
April 1	Andrew Tung	Moduli spaces of rational maps and dynamical systems	[Man]§4-7 [Sil1]§4.3-4.4
	Matija Likar	Arboreal and dynatomic Galois representations	[BJ] [Jon] [BIJMST]§5-6
April 3	Angeline Peng	Height functions and Northcott's theorem	[BIJMST]§14-15 [Sil1]§3.1-3.2, 3.4 [Sil4]§4.1, 5.1-5.2
	Lucy Epstein	Rational periodic points of quadratic polynomials of large period	[FPS]§1 [Mor4]§1-2 [Pan1]§2.1-2.2 [Pan2] [Poo] [Sto]§1
April 8	Ray Wang	Julia sets and Mandelbrot sets	[Bel] [Dev] [PR]§Frontiers of Chaos, 4, B. B. Mandelbrot, A. Douady
	Isaac Rajagopal	p-adic periodic points of quadratic polynomials	[BIJMST]§22 [Kru]§1 [WR]§5-6
April 10	Maximus Lu	The uniform boundedness conjecture for abelian varieties	[Dem]§1 [KW]§1 [Sil2]§VIII.7 [SS]§Ogg's conjecture [ST]§2.5
	Jake Ross	Dynatomic Galois groups and dynatomic fields	[Kru2] [Sil1]§3.9 [Mor2] [MP] [VH]§1-4
April 15 (Paper first draft due 1pm)	Emily Robinson	Communications workshop: rhetorical strategies for writing about mathematics effectively
April 17	— (solo presentation)	—	—
	— (solo presentation)	—	—
April 22	— (solo presentation)	—	—
	— (solo presentation)	—	—
April 24	— (solo presentation)	—	—
	— (solo presentation)	—	—
April 29 (Paper second draft due 11:59pm)	— (solo presentation)	—	—
	— (solo presentation)	—	—
May 1	— (solo presentation)	—	—
	— (solo presentation)	—	—
May 6	All students	Peer review on papers
May 8	— (solo presentation)	—	—
	— (solo presentation)	—	—
May 13 (Paper final draft due 11:59pm)	All students	Course retrospective

Books and theses

[BL] Xavier Buff and Tan Lei. "The quadratic dynatomic curves are smooth and irreducible". In Araceli Bonifant, Misha Lyubich, and Scott Sutherland (eds.). Frontiers in complex dynamics, 49–72, Princeton Math. Ser., 51, Princeton Univ. Press, 2014.
[DF] David S. Dummit and Richard M. Foote. Abstract Algebra. 3rd Edition. John Wiley & Sons, 2004.
[Ful] William E. Fulton. Algebraic Curves. 3rd Edition, 2008.
[GG] Ernesto Girondo and Gabino González-Diez. Introduction to Compact Riemann Surfaces and Dessins d'Enfants. London Mathematical Society Student Texts, 79. Cambridge University Press, 2012.
[Gou] Fernando Gouvêa. p-adic numbers. Springer, 1997.
[Gre] George Grell. Iterated Galois Groups of Quadratic Rational Functions. PhD Thesis – University of Rochester, 2019.
[IR] Kenneth Ireland and Michael Rosen. A Classical Introduction to Modern Number Theory. Second Edition. Springer, 1990.
[Pan1] Chatchawan Panraksa. Arithmetic dynamics of quadratic polynomials and dynamical units. PhD Thesis – University of Maryland, 2011.
[PR] Heinz-Otto Peitgen and Peter H. Richter. The Beauty of Fractals. Springer, 1986.
[Ros] Kenneth H. Rosen. Elementary Number Theory and Its Applications. Sixth Edition. Addison–Wesley, 2011.
[Rue1] David Ruelle. "Dynamical Zeta Functions: Where Do They Come from and What Are They Good for?" In Konrad Schmüdgen (ed.). Mathematical Physics X, 43–51, Springer, 1992.
[Rue2] David Ruelle. Dynamical zeta functions for piecewise monotone maps of the interval. CRM Monograph Series, 4, American Mathematical Society, 1994.
[Sil1] Joseph H. Silverman. The Arithmetic of Dynamical Systems. Springer, 2007.
[Sil2] Joseph H. Silverman. The Arithmetic of Elliptic Curves. Second Edition. Springer, 2009.
[SS1] Elias M. Stein and Rami Shakarchi. Complex Analysis. Princeton University Press, 2003.
[ST] Joseph H. Silverman and John Tate. Rational Points on Elliptic Curves. Second Edition. Springer, 2015.

Articles

[AG] Amir Akbary and Dragos Ghioca. Periods of orbits modulo primes. J. Number Theory, 129(11):2831–2842, 2009.
[Bri1] Andrew Bridy. Transcendence of the Artin-Mazur zeta function for polynomial maps of A₁(F_p). Acta Arith., 156(3):293–300, 2012.
[Bri2] Andrew Bridy. The Artin-Mazur zeta function of a dynamically affine rational map in positive characteristic. J. Théor. Nombres Bordeaux 28(2):301–324, 2016.
[BIJMST] Robert L. Benedetto, Patrick Ingram, Rafe Jones, Michelle Manes, Joseph H. Silverman and Thomas J. Tucker. Current trends and open problems in arithmetic dynamics. Bull. Amer. Math. Soc., 56(4):611–685, 2019.
[BJ] Nigel Boston and Rafe Jones. Arboreal Galois representations. Geom. Dedicata, 124:27–35, 2007.
[Dev] Robert L. Devaney. The Mandelbrot Set, the Farey Tree, and the Fibonacci Sequence Amer. Math. Monthly, 106(4):289–302, 1999.
[Fri] David Fried. The zeta functions of Ruelle and Selberg. I. Ann. Sci. École Norm. Sup. (4), 19(4):491–517, 1986.
[FPS] E. V. Flynn, Bjorn Poonen, and Edward F. Schaefer. Cycles of quadratic polynomials and rational points on a genus 2 curve. Duke Math. J., 90(3): 435–463, 1997.
[GIGS] Jaime Gutierrez, Álvar Ibeas, Domingo Gómez-Pérez, and Igor E. Shparlinski. Predicting masked linear pseudorandom number generators over finite fields. Des. Codes Cryptogr., 67:395–402, 2013.
[Hea] D. R. Heath-Brown. Iteration of quadratic polynomials over finite fields. Mathematika, 63(3):1041–1059, 2017.
[Jon] Rafe Jones. Galois representations from pre-image trees: an arboreal survey. Publications mathématiques de Besançon: Algèbre et théorie des nombres, 107–136, 2013.
[Juu] Jamie Juul. The image size of iterated rational maps over finite fields. Int. Math. Res. Not., 2021(5):3362–3388, 2021.
[Kru1] David Krumm. A local-global principle in the dynamics of quadratic polynomials. Int. J. Number Theory, 12(8):2265–2297, 2016.
[Kru2] David Krumm. Galois groups in a family of dynatomic polynomials. J. Number Theory, 187:469–511, 2018.
[KW] Sheldon Kamienny and Joseph L. Wetherell. On torsion in abelian varieties. Comm. Algebra, 26(5):1675–1678, 1998.
[Mor1] Pieter Moree. Artin's Primitive Root Conjecture – A Survey. Integers, 12(6):1305–1416, 2012.
[Mor2] Patrick Morton. On certain algebraic curves related to polynomial maps. Compos. Math., 103(3):319–350, 1996.
[Mor3] Patrick Morton. Galois Groups of periodic points. J. Algebra, 201(2):401–428, 1998.
[Mor4] Patrick Morton. Arithmetic properties of periodic points of quadratic maps, II. Acta Arith., 87(2):89–102, 1998.
[MP] Patrick Morton and Pratishka Patel. The Galois theory of periodic points of polynomial maps. Proc. London Math. Soc., 68(2): 225–263, 1994.
[Pan2] Chatchawan Panraksa. Rational periodic points of x^d+c and Fermat-Catalan equations. Int. J. Number Theory, 18(5):1111–1129, 2022.
[Poo] Bjorn Poonen. The classification of rational preperiodic points of quadratic polynomials over ℚ: a refined conjecture. Math. Z., 228:11–29, 1998.
[OS] Alina Ostafe and Igor E. Shparlinski. On the degree growth in some polynomial dynamical systems and nonlinear pseudorandom number generators. Math. Comp., 79:501–511, 2010.
[Sil3] Joseph H. Silverman. Integer points, Diophantine approximation, and iteration of rational maps. Duke Math J., 71(3): 793–829, 1993.
[Sto] Michael Stoll. Rational 6-cycles under iteration of quadratic polynomials. LMS J. Comput. Math., 11:367–380, 2008
[SS2] Norbert Schappacher and René Schoof. Beppo Levi and the arithmetic of elliptic curves. Math. Intelligencer, 18(1):57–69, 1996
[VH] Frank Vivaldi and Spyridon J. Hatjispyros. Galois theory of periodic orbits of rational maps. Nonlinearity, 5(4):961–978, 1992.
[WR] Ralph Walde and Paulo Russo. Rational periodic points of the quadratic function Q_c(x) = x² + c. Amer. Math. Monthly, 101(4):318–331, 1994.

Notes

[BOSZ] Nigel Boston, Alina Ostafe, Igor Shparlinski, and Michael Zieve. The act of iterating rational functions over finite fields. Banff International Research Station, 2013.
[Con1] Keith Conrad. Cyclotomic extensions. University of Connecticut.
[Con2] Keith Conrad. The p-adic Expansion of Rational Numbers. University of Connecticut, 2006.
[Bel] Jim Belk. Julia Sets and the Mandelbrot Set. Bard College, 2015.
[Bri] Paige Bright. Introduction to Metric Spaces. MIT OpenCourseWare, 2023.
[Dem] Spencer Dembner. Torsion points on elliptic curves and Mazur's theorem. University of Chicago REU, 2019.
[Dun] Jörn Dunkel. 18.04: Complex analysis with applications. MIT, 2019.
[Gat] Andreas Gathmann. Plane Algebraic Curves. RPTU Kaiserslautern, 2023.
[Goo] Dan Goodman. An Introduction to Galois Theory. NRICH, 2011.
[Hun] John K. Hunter. An Introduction to Real Analysis. UC Davis, 2014.
[Lei] Tom Leinster. Galois theory. University of Edinburgh, 2024.
[Man] Michelle Manes. The moduli space of rational functions of degree d. 2006.
[Sil4] Joseph H. Silverman. Lecture Notes on Arithmetic Dynamics. Arizona Winter School, 2010.
[Smi] Tim Smits. Introduction to Number Theory. UCLA, 2023.
[Viv] Franco Vivaldi. An introduction to Arithmetic Dynamics. The University of London, 2011.

Communications resources

[Ser] Jean-Pierre Serre. How to write mathematics badly (56 minutes). Harvard University, 2003. Transcript.
[Ruf] Susan Ruff. Preparing a seminar chalk talk. Massachusetts Institute of Technology, 2019.

YouTube videos

Inside Dynamical Systems and the Mathematics of Change (2 minutes). Quanta Magazine, 2020.
63 and -7/4 are special (12 minutes). Numberphile, 2014.
What's so special about the Mandelbrot Set? (17 minutes). Numberphile, 2019.
This equation will change how you see the world (the logistic map) (19 minutes). Veritasium, 2020.
Fractals are typically not self-similar (20 minutes). 3Blue1Brown, 2017.
The Simplest Math Problem No One Can Solve - Collatz Conjecture (22 minutes). Veritasium, 2021.
Number Theory and Dynamics, by Joseph Silverman (53 minutes). UConn Mathematics, 2018.
Fractals: The Colours of Infinity (54 minutes). Arthur C. Clarke, 1995.