Bjorn Poonen Department of Mathematics   MIT, 2-244   77 Massachusetts Avenue   Cambridge, MA 02139-4307, USA

Office: Building 2, Room 244 Phone: (617)-258-8164  Fax: (617)-253-4358  email:

• 18.02, Multivariable calculus, Fall 2009.
• 18.782, Introduction to arithmetic geometry (new course), Fall 2009.
• Past courses

• MIT number theory seminar
• BC-MIT number theory seminar (organized by Ben Brubaker, Sol Friedberg, Ben Howard, and myself)
• STAGE

• Past conferences

• Algebra & Number Theory, managing editor, 2006-
• Involve (new journal for high-quality research involving students), editorial board member, 2007-
• Journal of the American Mathematical Society, associate editor, 2000-2003 and 2004-2010.
• A K Peters Research Notes in Mathematics book series, editorial board member, 2005-
• LMS Journal of Computation and Mathematics, editorial board member, 2007-
• Past editorial positions

• MIT Graduate Co-Chair (with Gigliola Staffilani), 2009-
• Putnam Competition Committee, 2007-2010.
• Past professional activities

• Research articles
  • Néron-Severi groups under specialization (.pdf) (.dvi.gz) (with Davesh Maulik and Claire Voisin)
  • Automorphisms mapping a point into a subvariety (.pdf) (.dvi.gz)
  • 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)
  • 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)
  • 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 (.ps) (.dvi.gz + planecubics.eps)
  • Algebraic families of nonzero elements of Shafarevich-Tate groups (with Jean-Louis Colliot-Thélène)
  • Mordell-Lang plus Bogomolov
  • The Cassels-Tate pairing on polarized abelian varieties (with M. 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)(.ps) (with M. 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 E. Schaefer)
  • The worst case in Shellsort and related algorithms (.pdf) (.dvi.gz)
• Expository articles
  • Lattice polygons and the number 12 (with F. Rodriguez-Villegas)
  • A brief summary of the main statements of class field theory (.pdf) (.dvi.gz)
  • Elliptic curves (.pdf) (.dvi.gz)
  • 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)
  • Sieve methods for varieties over finite fields and arithmetic schemes (.pdf) (.dvi.gz)
  • The method of Chabauty and Coleman (.pdf) (.dvi.gz) (with William McCallum)
  • Undecidability in number theory (.dvi.gz)
• Books
  • The William Lowell Putnam Mathematical Competition 1985-2000: Problems, Solutions, and Commentary, with Kiran Kedlaya and Ravi Vakil.
  • Arithmetic of Higher-Dimensional Algebraic Varieties, edited with Yuri Tschinkel.
• Slides
  • 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

Open Question: Let Z be the set of integers, and let N be the set of nonnegative integers.  Is there a polynomial f(x,y) such that f(Z x Z) = N?