Diophantine problems Michael Bennett Introduction What we know Effective methods for Diophantine problems Case studies Michael Bennett University of British Columbia BIRS Summer School : June 2012 Diophantine problems Michael What are Diophantine equations? Bennett Introduction According to Hilbert, a Diophantine equation is an equation of What we know the form Case studies D(x1; : : : ; xm) = 0; where D is a polynomial with integer coefficients. Hilbert's 10th problem : Determination of the solvability of a Diophantine equation. Given a diophantine equation with any number of unknown quantities and with rational integral numerical coefficients: To devise a process according to which it can be determined by a finite number of operations whether the equation is solvable in rational integers. Diophantine problems Michael Bennett What are Diophantine equations? (take 2) Introduction What we know A few more opinions : Case studies Wikipedia pretty much agrees with Hilbert Mordell, in his book \Diophantine Equations", never really defines the term (!) Wolfram states \A Diophantine equation is an equation in which only integer solutions are allowed". Diophantine problems Michael Bennett Introduction Back to Hilbert's 10th problem What we know Case studies Determination of the solvability of a Diophantine equation. Given a diophantine equation with any number of unknown quantities and with rational integral numerical coefficients: To devise a process according to which it can be determined by a finite number of operations whether the equation is solvable in rational integers. But . Hilbert's problem is still open over Q (and over OK for most number fields) It is not yet understood what happens if the number of variables is \small" Diophantine problems Michael Bennett Matiyasevich's Theorem Introduction (building on work of Davis, Putnam and Robinson) is that, in What we know Case studies general, no such process exists. Hilbert's problem is still open over Q (and over OK for most number fields) It is not yet understood what happens if the number of variables is \small" Diophantine problems Michael Bennett Matiyasevich's Theorem Introduction (building on work of Davis, Putnam and Robinson) is that, in What we know Case studies general, no such process exists. But . It is not yet understood what happens if the number of variables is \small" Diophantine problems Michael Bennett Matiyasevich's Theorem Introduction (building on work of Davis, Putnam and Robinson) is that, in What we know Case studies general, no such process exists. But . Hilbert's problem is still open over Q (and over OK for most number fields) Diophantine problems Michael Bennett Matiyasevich's Theorem Introduction (building on work of Davis, Putnam and Robinson) is that, in What we know Case studies general, no such process exists. But . Hilbert's problem is still open over Q (and over OK for most number fields) It is not yet understood what happens if the number of variables is \small" Diophantine problems Michael Bennett Perhaps we should simplify (?) things... Introduction What we know Case studies 1 We could try to answer Hilbert's question for plane curves; i.e. try to decide whether an equation of the shape f(x; y) = 0 has integral or rational solutions. 2 We could try to bound the number of such solutions. 3 We could try to find an algorithm for explicitly solving such equations. Diophantine problems Michael Bennett However.... Introduction What we know Case studies A complete answer to Problem 1 is unavailable { no algorithm is known to determine whether a curve has a rational point! Problem 2 has some partial answers (R´emond). Problem 3 is open, even in the case of genus 2... Sticking to curves { let C be a nonsingular algebraic curve of genus g over, say, Q. Then the set of rational points on C is 1 infinite or empty if g = 0 (conic section) 2 empty or C is an elliptic curve, if g = 1 (so that the rational points form a finitely generated abelian group, via Mordell), or 3 at most finite if g > 1 (Faltings' theorem n´eeMordell's conjecture). Diophantine problems Michael Bennett So, what can we prove? Introduction What we know Case studies 1 infinite or empty if g = 0 (conic section) 2 empty or C is an elliptic curve, if g = 1 (so that the rational points form a finitely generated abelian group, via Mordell), or 3 at most finite if g > 1 (Faltings' theorem n´eeMordell's conjecture). Diophantine problems Michael Bennett So, what can we prove? Introduction What we know Sticking to curves { let C be a nonsingular algebraic curve of Case studies genus g over, say, Q. Then the set of rational points on C is 2 empty or C is an elliptic curve, if g = 1 (so that the rational points form a finitely generated abelian group, via Mordell), or 3 at most finite if g > 1 (Faltings' theorem n´eeMordell's conjecture). Diophantine problems Michael Bennett So, what can we prove? Introduction What we know Sticking to curves { let C be a nonsingular algebraic curve of Case studies genus g over, say, Q. Then the set of rational points on C is 1 infinite or empty if g = 0 (conic section) 3 at most finite if g > 1 (Faltings' theorem n´eeMordell's conjecture). Diophantine problems Michael Bennett So, what can we prove? Introduction What we know Sticking to curves { let C be a nonsingular algebraic curve of Case studies genus g over, say, Q. Then the set of rational points on C is 1 infinite or empty if g = 0 (conic section) 2 empty or C is an elliptic curve, if g = 1 (so that the rational points form a finitely generated abelian group, via Mordell), or Diophantine problems Michael Bennett So, what can we prove? Introduction What we know Sticking to curves { let C be a nonsingular algebraic curve of Case studies genus g over, say, Q. Then the set of rational points on C is 1 infinite or empty if g = 0 (conic section) 2 empty or C is an elliptic curve, if g = 1 (so that the rational points form a finitely generated abelian group, via Mordell), or 3 at most finite if g > 1 (Faltings' theorem n´eeMordell's conjecture). Diophantine problems Michael Bennett Introduction So, what can we prove? What we know Case studies Sticking to curves { let C be a nonsingular algebraic curve of genus g over, say, Q. Then the set of integral points on C is 1 infinite or empty, or somewhere in between, if g = 0 2 at most finite if g > 0 (Siegel's theorem). Solutions over Q =) solutions over R and Qp for all p. Diophantine problems Michael Bennett A Diversion : Local-Global Principles Introduction What we know Case studies Consider x2 + y2 = −1; x2 + y2 = 3; and x2 + y2 = 5: Diophantine problems Michael Bennett A Diversion : Local-Global Principles Introduction What we know Case studies Consider x2 + y2 = −1; x2 + y2 = 3; and x2 + y2 = 5: Solutions over Q =) solutions over R and Qp for all p. We are interested in effective methods (and not just for curves!). Diophantine problems Michael Bennett So we understand curves, right? Introduction What we know Case studies The problem is that the theorems of Faltings and Siegel and ineffective, in that their proofs do not provide a way to determine the implicit finite set (in case the genus of the curve satisfies g > 1, or g > 0, respectively). Diophantine problems Michael Bennett So we understand curves, right? Introduction What we know Case studies The problem is that the theorems of Faltings and Siegel and ineffective, in that their proofs do not provide a way to determine the implicit finite set (in case the genus of the curve satisfies g > 1, or g > 0, respectively). We are interested in effective methods (and not just for curves!). 1 They arise \naturally" in other areas of math. 2 They provide valuable test-cases for theorems and conjectures coming from, say, algebraic geometry. 3 It beats working. Diophantine problems Michael Bennett A Diversion : Motivation Introduction What we know Case studies Why do we study Diophantine equations? 2 They provide valuable test-cases for theorems and conjectures coming from, say, algebraic geometry. 3 It beats working. Diophantine problems Michael Bennett A Diversion : Motivation Introduction What we know Case studies Why do we study Diophantine equations? 1 They arise \naturally" in other areas of math. 3 It beats working. Diophantine problems Michael Bennett A Diversion : Motivation Introduction What we know Case studies Why do we study Diophantine equations? 1 They arise \naturally" in other areas of math. 2 They provide valuable test-cases for theorems and conjectures coming from, say, algebraic geometry. Diophantine problems Michael Bennett A Diversion : Motivation Introduction What we know Case studies Why do we study Diophantine equations? 1 They arise \naturally" in other areas of math. 2 They provide valuable test-cases for theorems and conjectures coming from, say, algebraic geometry. 3 It beats working. Ramanujan's Question of 1913 (Journal of the Indian Mathematical Society) : Are the numbers in bold the only squares in the sequence? Diophantine problems Case study : The Ramanujan-Nagell equation Michael Bennett n Introduction Consider the sequence of integers 2 − 7, n ≥ 3 : What we know Case studies 1; 9; 25; 57; 121; 249; 505; 1017; 2041; 4089; 8185; 16377; 32761; 65529; 131065; 262137; 524281; 1048569; 2097145; 4194297;::: Diophantine problems Case study : The Ramanujan-Nagell equation Michael Bennett n Introduction Consider the sequence of integers 2 − 7, n ≥ 3 : What we know Case studies 1; 9; 25; 57; 121; 249; 505; 1017; 2041; 4089; 8185; 16377; 32761; 65529; 131065; 262137; 524281; 1048569; 2097145; 4194297;::: Ramanujan's Question of 1913 (Journal of the Indian Mathematical Society) : Are the numbers in bold the only squares in the sequence? Diophantine problems Michael A Class of Diophantine Equations Bennett Introduction This is an example of a What we know Case studies Polynomial-Exponential Equation.