This is the first meeting of Oregon Number Theory Days, a triannual number theory seminar rotating between the University of Oregon, Oregon State University, and Portland State University. The main speaker will be Kirsten Eisentrager of Penn State University, who will give two talks. Lunch will be provided and there will also be a talk by graduate student Travis Scholl of the University of Washington. If you are interested in attending this meeting, please register by following the instructions at http://people.oregonstate.edu/~petschec/ONTD/
Kirsten Eisenträger Lecture I, Undecidability in number theory: In 1900 Hilbert presented his now famous list of 23 open problems. The tenth problem in its original form was to find an algorithm to decide, given a multivariate polynomial equation with integer coefficients, whether it has a solution over the integers. Hilbert's Tenth Problem remained open until 1970 when Matiyasevich, building on work by Davis, Putnam and Robinson, proved that no such algorithm exists, i.e. Hilbert's Tenth Problem is undecidable. In this talk we will give an overview of Hilbert's Tenth Problem and related problems in undecidability. We will also discuss how elliptic curves can be used to prove the undecidability of Hilbert's Tenth Problem for various rings and fields.
Kirsten Eisenträger Lecture II, Existentially and universally definable subsets of global fields: In 1970, Matiyasevich, building on work by Davis, Putnam and Robinson, proved that Hilbert's Tenth Problem over the integers is undecidable. The analogue of Hilbert's Tenth Problem over the rationals remains open. A diophantine definition of the integers over the rationals, together with a standard reduction argument, would show that Hilbert's Tenth Problem over the rationals is undecidable. Such a diophantine definition of the integers over the rationals seems out of reach right now, but recently, Koenigsmann proved that it is possible to define the integers inside the rationals using only universal quantifiers. If Mazur's conjecture holds then this may be the best possible result. In this talk we use class field theory to extend Koenigsmann's result to global function fields of odd characteristic and show that rings of S-integers can be defined universally. We also prove that the set of non-norms and the set of non-squares in global fields are diophantine. The result for non-squares gives a new proof of a theorem by Poonen who used results on the Brauer-Manin obstruction to prove that the set of non-squares in global fields is diophantine.
Travis Scholl, Isolated elliptic curves in cryptography: We call an elliptic curve "isolated" if it does not admit many efficiently computable isogenies. The motivation for isolated curves comes from elliptic curve cryptography. We present some examples, and results that suggest that random isolated curves are rarely susceptible to known attacks on elliptic curve cryptosystems.