Colloquium: Tuesday, April 24, 2 pm. Speaker: Ishai Dan-Cohen, (Ben-Gurion). Title: “Rational motivic path spaces”.
Let X be a smooth compact algebraic curve defined by equations with rational coefficients. If the genus of X is not less than 2, then by Mordel’s conjecture = Faltings’ theorem, the set X(Q) of rational points of X is finite. This gives rise to the problem of computing such sets algorithmically. This quest for an “effective Mordel’s conjecture” is regarded as a central goal of arithmetic geometry. An approach pioneered by Minhyong Kim revolves around a certain conjecture; in Kim’s approach, rational points face an obstruction coming from the prounipotent completion of the fundamental group, and the conjecture asserts that this obstruction completely determines the set of integral points inside the set of p-adic points for an auxiliary prime p of good reduction. In joint work with Tomer Schlank, we divide Kim’s conjecture into a series of smaller conjectures with a homotopical flavor.