Hilbert 10: Winter 2025
Together with Francesco A. Zuccon, we are organizing a reading seminar on Hilbert's 10th problem. The seminar will cover recent work (along with its prerequisites) settling Hilbert 10 in the negative for any infinite commutative ring R that is finitely generated over Z.
We meet on Wednesdays from 4pm to 6pm in the CICMA room on the 9th floor of Concordia's library building.
Schedule:
Lecture 0: Introduction. (Carlo Pagano, Jan 15).
Lecture 1: Diophantine sets, part 1. (Francesco Zuccon, Jan 22).
Lecture 2: Diophantine sets, part 2. (Carlo Pagano, Jan 29).
Lecture 3: Diophantine sets, part 3. (Francesco Zuccon, February 5).
Lecture 4: Turing machines. (Carlo Pagano, February 12).
Lecture 5: H10/Z is undecidable, part 1. (Francesco Zuccon, February 19).
Lecture 6: H10/Z is undecidable, part 2. (Carlo Pagano, March 5).
Lecture 7: Plan of proof for H10/R and intro to (2-)Selmer groups. (Carlo Pagano, March 12).
Lecture 8: Shlapentokh's theorem: defining number rings in quadratic extensions via elliptic curves. (Carlo Pagano, March 19).
Lecture 9: Description of 2-Selmer in quadratic twist families, part 1. (Francesco Zuccon, March 26).
Lecture 10: Description of 2-Selmer in quadratic twist families, part 2. (Carlo Pagano, April 2).
Lecture 11: Description of 2-Selmer in quadratic twist families, part 3. (Francesco Zuccon, April 9).
Lecture 12: The auxiliary twist and Mazur-Rubin's theorem. (Francesco Zuccon, April 16).
Lecture 13: The last 4 primes and end of proof. (Carlo Pagano, April 23).
Lecture 14: H10/R is undecidable. (Francesco Zuccon, April 30).
References:
[1] Hilbert's tenth problem is unsolvable (M. Davis).
[2] Hilbert's tenth problem: what can we do with Diophantine equations? (Y. Matiyasevich).
[3] Ranks of twists of elliptic curves and Hilbert's tenth problem (B. Mazur and K. Rubin).
[4] Hilbert tenth's problem via additive combinatorics (P. Koymans and C. Pagano).
[5] Hilbert tenth's problem over rings of number theoretic interest (B. Poonen).
[6] Existential definability and diophantine stability (B. Mazur, K. Rubin and A. Shlapentokh).