Bjorn Poonen

Bjorn Poonen
Distinguished Professor in Science
Office: Room 2-243 in the Simons Building at 182 Memorial Dr. (Building 2)
Phone: (617) 258-8164
Fax: (617) 253-4358
email: (comments on sending me email)
Office hours through May 13, 2025: See this document.

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Introduction to my research
Curriculum vitae and publication list
MathSciNet search for Poonen (this link works only if your institution subscribes)
My research is supported by the National Science Foundation and the Simons Foundation. I am part of the Simons Collaboration on Arithmetic Geometry, Number Theory, and Computation.
Students, postdocs, and other researchers supervised or sponsored
There is currently one Ph.D. student working with me.

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Courses and seminars

• 18.725, Algebraic geometry I, Fall 2024.
• 18.A34, Mathematical problem solving, Fall 2024.
• 18.03, Differential equations, Spring 2025.
• Past courses

• MIT number theory seminar (co-organized with Andrew Sutherland and Wei Zhang)
• BC-MIT number theory seminar (co-organized with Sol Friedberg, Ben Howard, Dubi Kelmer, Spencer Leslie, Keerthi Madapusi Pera, Andrew Sutherland, and Wei Zhang)
• STAGE (co-organized with Kenta Suzuki and Xinyu Zhou)

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Websites I am helping to develop

• researchseminars.org (with Edgar Costa, David Roe, and Andrew Sutherland) (featured in Nature)
• MIT number theory

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Conferences I am helping to organize

• Summer Research Institute in Algebraic Geometry, Colorado State University, July 14-August 1, 2025 (co-organized with Renzo Cavalieri, Gavril Farkas, Angela Gibney, Christopher Hacon, Andrei Okounkov, Karen Smith, and Chenyang Xu).
• Past conferences

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Editorial positions

• , founding managing editor, 2007-2021; editor 2021-
• Involve (journal for high-quality research involving students), editorial board member, 2007-
• Past editorial positions

Before sending me any papers about the Goldbach conjecture, the Riemann hypothesis, the infinitude of twin primes, the nonexistence of odd perfect numbers, Fermat's last theorem, or generalizations of any of these, please follow the advice by Henry Cohn at this website.

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Other professional activities

• AMS Leroy P. Steele Prize Selection Committee, 2024-2027.
• AMS Frank Nelson Cole Prize Selection Committee, 2025-2026.
• American Academy of Arts and Sciences Membership Panel for Mathematics, Applied Mathematics, and Statistics, 2023-.
• SASTRA Ramanujan Prize Committee, 2023-2025.
• AMS Colloquium Lecture Committee, 2022-2025. Chair during 2022-23.
• ICM Algebraic and Complex Geometry panel, 2022.
• AMS Current Events Bulletin Committee, 2019-2021.
• AMS Graduate Studies in Mathematics Editorial Committee, 2018-2022.
• ICCM best paper award, global committee member, 2017-.
• MIT Presidential Committee for Distinguished Fellowships, 2023-2027.
• MIT Office of Minority Education Faculty Advisory Committee, 2021-.
• MIT First-Year Advisor, 2013-2015, 2016-2017, 2019-.
• MIT Pure Mathematics Committee, 2008-2015 and 2016-.
• MIT Mathematics Department Nominations Committee, 2008-2015 and 2016- (chair 2017-).
• MIT Mathematics Department Education Committee, 2008-2009 and 2016-.
• MIT Mathematics Major Advisor, 2009-.
• MIT Mathematics Department Teaching Assignment Area Captain for Logic, 2008-.
• Girls' Angle Advisory Board, 2009-.
• Museum of Mathematics Advisory Council, 2008-.
• Arizona Winter School, co-PI 2002-2006, Advisory Board 2006-.
• Past professional activities

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Some of my papers (disclaimer)

• Research articles
  • Brauer-Manin obstructions requiring arbitrarily many Brauer classes (.pdf) (with Jennifer Berg, Carlo Pagano, Michael Stoll, Nicholas Triantafillou, Bianca Viray, and Isabel Vogt)
  • Obstructions to applying the Baker-Bilu method for determining integral points on curves (.pdf) (with Aaron Landesman)
  • Lattices in Tate modules (.pdf) (with Sergey Rybakov)
  • Abelian varieties of prescribed order over finite fields (.pdf) (with Raymond van Bommel, Edgar Costa, Wanlin Li, and Alexander Smith)
  • Space vectors forming rational angles (.pdf) (with Kiran Kedlaya, Alexander Kolpakov, and Michael Rubinstein) [press release and video for a general scientific audience] (featured in Quanta Magazine) (follow-up on further work by MIT undergraduates)
  • The valuation of the discriminant of a hypersurface (.pdf) (with Michael Stoll)
  • The local-global principle for integral points on stacky curves (.pdf) (with Manjul Bhargava)
  • Linear independence in linear systems on elliptic curves (.pdf) (with Bradley W. Brock, Bruce W. Jordan, Anthony J. Scholl, and Joseph L. Wetherell)
  • The exceptional locus in the Bertini irreducibility theorem for a morphism (.pdf) (with Kaloyan Slavov)
  • The S-integral points on the projective line minus three points via étale covers and Skolem's method (.pdf)
  • Heuristics for the arithmetic of elliptic curves (.pdf)
  • Gonality of dynatomic curves and strong uniform boundedness of preperiodic points (.pdf) (with John Doyle)
  • Statistics of K-groups modulo p for the ring of integers of a varying quadratic number field (.pdf) (with Bruce W. Jordan, Zev Klagsbrun, Christopher Skinner, and Yevgeny Zaytman)
  • The analytic class number formula for 1-dimensional affine schemes (.pdf) (with Bruce W. Jordan)
  • Using zeta functions to factor polynomials over finite fields (.pdf)
  • Abelian varieties isogenous to a power of an elliptic curve (.pdf) (with Bruce W. Jordan, Allan G. Keeton, Eric Rains, Nicholas Shepherd-Barron, and John T. Tate)
  • Local arboreal representations (.pdf) (with Jacqueline Anderson, Spencer Hamblen, Laura Walton)
  • A heuristic for boundedness of ranks of elliptic curves (.pdf) (with Jennifer Park, John Voight, Melanie Matchett Wood) (featured in Quanta Magazine)
  • A computable functor from graphs to fields (.pdf) (with Russell Miller, Hans Schoutens, and Alexandra Shlapentokh)
  • Automorphisms of Harbater-Katz-Gabber curves (.pdf) (with Frauke Bleher, Ted Chinburg, and Peter Symonds)
  • Galois points on varieties (.pdf) (with Moshe Jarden)
  • Bertini irreducibility theorems over finite fields (.pdf) (with François Charles)
  • p-adic interpolation of iterates (.pdf)
  • Most odd degree hyperelliptic curves have only one rational point (.pdf) (with Michael Stoll)
  • Modeling the distribution of ranks, Selmer groups, and Shafarevich-Tate groups of elliptic curves (.pdf) (with Manjul Bhargava, Daniel M. Kane, Hendrik W. Lenstra jr., and Eric Rains)
  • Berkovich spaces embed in Euclidean spaces (.pdf) (with Ehud Hrushovski and François Loeser)
  • Computing Néron-Severi groups and cycle class groups (.pdf) (with Damiano Testa and Ronald van Luijk)
  • Uniform boundedness of rational points and preperiodic points (.pdf)
  • Generalized explicit descent and its application to curves of genus 3 (.pdf) (with Nils Bruin and Michael Stoll)
  • Extending self-maps to projective space over finite fields (.pdf)
  • Convergence of the restricted Nelder-Mead algorithm in two dimensions (.pdf) (with Jeffrey C. Lagarias and Margaret H. Wright)
  • Self cup products and the theta characteristic torsor (.pdf) (.dvi.gz) (with Eric Rains)
  • Random maximal isotropic subspaces and Selmer groups (.pdf) (.dvi.gz) (with Eric Rains)
  • Néron-Severi groups under specialization (.pdf) (.dvi.gz) (with Davesh Maulik)
  • Automorphisms mapping a point into a subvariety (.pdf) (.dvi.gz) (with an appendix by Matthias Aschenbrenner)
  • Multivariable polynomial injections on rational numbers (.pdf) (.dvi.gz)
  • Curves over every global field violating the local-global principle (.pdf) (.dvi.gz)
  • Insufficiency of the Brauer-Manin obstruction applied to etale covers (.pdf) (.dvi.gz)
  • Existence of rational points on smooth projective varieties (.pdf) (.dvi.gz)
  • The set of nonsquares in a number field is diophantine (.pdf) (.dvi.gz)
  • The p-adic closure of a subgroup of rational points on a commutative algebraic group (.pdf) (.dvi.gz)
  • • Independence of points on elliptic curves arising from special points on modular and Shimura curves, I: global results (.pdf) (.dvi.gz) (with Alexandru Buium)
    • Independence of points on elliptic curves arising from special points on modular and Shimura curves, II: local results (.pdf) (.dvi.gz) (with Alexandru Buium)
  • Characterizing integers among rational numbers with a universal-existential formula (.pdf) (.dvi.gz)
  • The Brauer-Manin obstruction for subvarieties of abelian varieties over function fields (.pdf) (.dvi.gz) (with José Felipe Voloch)
  • Smooth hypersurface sections containing a given subscheme over a finite field (.pdf) (.dvi.gz)
  • The moduli space of commutative algebras of finite rank (.pdf) (.dvi.gz)
  • Isomorphism types of commutative algebras of finite rank over an algebraically closed field (.pdf) (.dvi.gz)
  • First-order characterization of function field invariants over large fields (.pdf) (.dvi.gz) (with Florian Pop)
  • Gonality of modular curves in characteristic p (.pdf) (.dvi.gz)
  • Uniform first-order definitions in finitely generated fields (.pdf) (.dvi.gz)
  • Twists of X(7) and primitive solutions to x^2+y^3=z^7 (.pdf) (with Edward F. Schaefer and Michael Stoll)
  • Heuristics for the Brauer-Manin obstruction for curves (.pdf) (.dvi.gz)
  • Diophantine definability of infinite discrete nonarchimedean sets and Diophantine models over large subrings of number fields (.pdf) (.dvi.gz) (with Alexandra Shlapentokh)
  • Multiples of subvarieties in algebraic groups over finite fields (.pdf) (.dvi.gz)
  • Unramified covers of Galois covers of low genus curves (.pdf) (.dvi.gz)
  • Orbits of automorphism groups of fields (.pdf) (.dvi.gz) (with Kiran S. Kedlaya)
  • Random diophantine equations (.pdf) (.dvi.gz) (with José Felipe Voloch, and containing appendices by Jean-Louis Colliot-Thélène and Nicholas M. Katz)
  • Sums of values of a rational function (.pdf) (.dvi.gz)
  • Hilbert's Tenth Problem and Mazur's Conjecture for large subrings of Q (.pdf) (.dvi.gz)
  • Finiteness results for modular curves of genus at least 2 (.pdf) (.dvi.gz) (with Matt Baker, Enrique Gonzalez-Jimenez, and Josep Gonzalez)
  • Curves of every genus with many points, II: asymptotically good families (.pdf) (.dvi.gz) (with Noam Elkies, Everett Howe, Andrew Kresch, Joseph Wetherell, and Michael Zieve)
  • The Grothendieck ring of varieties is not a domain (.pdf) (.dvi.gz)
  • Using elliptic curves of rank one towards the undecidability of Hilbert's Tenth Problem over rings of algebraic integers (.pdf) (.dvi.gz) (Springer LNCS link)
  • Bertini theorems over finite fields (.pdf) (.dvi.gz)
  • Squarefree values of multivariable polynomials (.pdf) (.dvi.gz)
  • Everywhere ramified towers of global function fields (.pdf) (with Ivan Duursma and Michael Zieve)
  • The conjugate dimension of algebraic numbers (.pdf) (.dvi.gz) (with Neil Berry, Arturas Dubickas, Noam D. Elkies, and Chris Smyth)
  • Zeros of sparse polynomials over local fields of characteristic p (.pdf) (.dvi.gz)
  • Points having the same residue field as their image under a morphism (.pdf) (.dvi.gz)
  • Néron-Tate projection of algebraic points (.pdf) (.dvi.gz)
  • Computing torsion points on curves (.pdf) (.dvi.gz) (PARI/GP code, some examples)
  • Random polynomials having few or no real zeros (.pdf) (.dvi.gz) (with Amir Dembo, Qi-Man Shao, and Ofer Zeitouni)
  • The following three papers are the renamed and updated versions of the preprints "Curves over finite fields without extra automorphisms" and "Hypersurfaces over finite fields without extra automorphisms" referred to in Katz and Sarnak, "Random matrices, Frobenius eigenvalues, and monodromy":
  • Varieties without extra automorphisms I: curves (.pdf) (.dvi.gz)
  • Varieties without extra automorphisms II: hyperelliptic curves (.pdf) (.dvi.gz)
  • Varieties without extra automorphisms III: hypersurfaces (.pdf) (.dvi.gz)
  • An explicit algebraic family of genus-one curves violating the Hasse principle (.pdf)
  • Algebraic families of nonzero elements of Shafarevich-Tate groups (with Jean-Louis Colliot-Thélène)
  • Mordell-Lang plus Bogomolov
  • A local-global principle for densities (with Michael Stoll)
  • The Cassels-Tate pairing on polarized abelian varieties (with Michael Stoll)
  • Large torsion subgroups of split Jacobians of curves of genus two or three (.pdf) (.dvi.gz) (with Everett Howe and Franck Leprévost)
  • The number of intersection points made by the diagonals of a regular polygon (.pdf) (with Michael Rubinstein. Auxiliary programs: ngon.c, ngon.m)
  • Drinfeld modules with no supersingular primes (.pdf) (.dvi.gz)
  • Explicit descent for Jacobians of cyclic covers of the projective line (.pdf) (.dvi.gz) (with Edward F. Schaefer)
  • The worst case in Shellsort and related algorithms (.pdf) (.dvi.gz)
  • Maximally complete fields (.pdf) (this is based on my undergraduate thesis)
• Expository articles
  • Press release: Tetrahedra with rational dihedral angles (.pdf) (with K. Kedlaya, A. Kolpakov, and M. Rubinstein)
  • Why all rings should have a 1
  • Lattice polygons and the number 12 (with Fernando Rodriguez-Villegas)
  • A brief summary of the main statements of class field theory (.pdf) (.dvi.gz)
  • Elliptic curves (.pdf) (.dvi.gz)
  • Average rank of elliptic curves (.pdf)
  • Sieve methods for varieties over finite fields and arithmetic schemes (.pdf) (.dvi.gz)
  • Selmer group heuristics and sieves (.pdf)
  • Computing rational points on curves (.pdf) (.dvi.gz)
  • Computational aspects of curves of genus at least 2 (.pdf) (.dvi.gz)
  • Hilbert's Tenth Problem over rings of number-theoretic interest (.pdf) (.dvi.gz)
  • Undecidability in number theory (.pdf) (.dvi.gz) (awarded the 2011 Chauvenet Prize)
  • Undecidable problems: a sampler (.pdf)
  • The method of Chabauty and Coleman (.pdf) (.dvi.gz) (with William McCallum)
  • A p-adic approach to rational points on curves (.pdf) (intended for a general mathematical audience)
  • p-adic approaches to rational and integral points on curves (.pdf) (intended for arithmetic geometers)
  • Introduction to Drinfeld modules
• Books
  • Rational Points on Varieties (.pdf). If your use of this book is more than just incidental, please purchase the official print or electronic version. (errata) (awarded the 2023 AMS Doob Prize)
  • Arithmetic of Higher-Dimensional Algebraic Varieties, edited with Yuri Tschinkel.
  • The William Lowell Putnam Mathematical Competition 1985-2000: Problems, Solutions, and Commentary, with Kiran Kedlaya and Ravi Vakil.
• Course notes
  • 18.02 Multivariable calculus
  • 18.03 Differential equations (received the MIT School of Science Prize in Undergraduate Teaching for this class)
  • 18.786 Number theory II (only the month on Tate's thesis)
  • Real representations
  • Jordan-Hölder theorem (one-page proof)
  • Comments on Serre, Finite groups: an introduction (textbook used in 18.704, Fall 2018)
• Slides
  • Elliptic curves (aimed at high school students)
  • Introduction to rational points
  • Hilbert's tenth problem
  • The projective line minus three fractional points
  • Undecidability everywhere (the third of the 2008 Cantrell Lectures given at the University of Georgia)
  • The first-order theory of finitely generated fields
  • Uniform boundedness of rational points
  • Bertini irreducibility theorems via statistics
  • The origins of the Tate conjecture
• Miscellaneous
  • Practical suggestions for mathematical writing
  • Practical suggestions for mathematical speaking
  • Remarks and errata

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Math pronunciation

• Arithmetic is a noun. Arithmetic is an adjective.
• It's class field theory, not class field theory, since it's the theory of class fields, not a special kind of field theory.
• Homogeneous, not homogenous, is the word used in mathematics, and both of the letters e are pronounced "ee"!
• Robin Hartshorne's last name is pronounced Harts-horne, with no "sh"; he told me himself. (A hart is a male deer.)

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