Title: The algebra of Hecke operators mod 2.
Abstract: We introduce modular forms of level 1 mod 2 and their associated algebra of Hecke operators. This algebra is isomorphic to a power series ring in two variables. On our way to proving this, we'll come across a surprising fact- the action of Hecke operators on a modular form is essentially determined by the binary expansion of its weight. The talk shouldn't contain significantly more theory than the abstract.
Title: Zero-cycles on surfaces.
Abstract: We will review Mumford's proof that any surface with non-zero geometric genus has Chow group of zero cycles infinite dimensional in some sense. A conjecture of Spencer Bloch claims that if a surface has geometric genus zero, then its Chow-0 should be finite dimensional. This conjecture is still open for surfaces of general type. We will compute the Chow-0 for some of the surfaces with geometric genus 0, and then if time permits, I will sketch the proof of Bloch's conjecture for surfaces with kodaira dimension <2.
Title: The Gross-Hopkins period map.
Abstract: In this talk we will explain the construction of the Gross-Hopkins period map. First, an abstract approach using Dieudonné crystals will be sketched, in order to stress the analogy with the classical period mapping. Then we will explain some details of the explicit construction presented in the paper “Equivariant vector bundles on the Lubin-Tate moduli space”.
Title: The Hasse Principle for Finite Schemes over Q -- DOES IT HOLD?!
Abstract: ... which is just a fancy way to ask, does a polynomial with a root in every completion of Q have a root in Q? Find out next time, at No Theory Seminar!
Title: How Chabauty, Coleman and Kim solve equations
Abstract: We will solve some Diophantine equations using the classical methods of Mordell-Weil descent and Chabauty-Coleman, and try to get a flavour of the work of Minhyong Kim that provides an ambitious and far-reaching generalisation of these techniques. Time permitting, I will talk about the fascinating recent work of Jennifer Balakrishnan and Netan Dogra on explicit non-abelian Chabauty for higher genus curves.
No No Theory Seminar this week.
Title: Points of order 13 on elliptic curves.
Abstract: Mazur has the famous theorem which roughly says that elliptic curves defined over Q has no rational points of order N>=13, for any prime N. It's super hard and definitely full of theory... But we can study a special case of N=13 by looking at the rational points of J_1(N), which Ogg computed has torsion part = Z/19Z. We'll use descent method to show these are all of its rational points by looking at the action of a certain dihedral group. Then we are happy! The method can be used for certain larger values of N as well. The proof follows the paper of Mazur and Tate with the same title.
Title: Valence Formulas for Classical Modular Forms.
Abstract: It is well known that the ring of modular forms of SL2(Z) is M(SL2(Z)) = C(G2, G4). A standard argument uses path integration around a fundamental domain to produce a valence formula (restricting the zeros of a modular form). In low level, (when X(Gamma) has genus 0), this method can generalize to give a full description of the ring M(Gamma). We will do this for Gamma = Gamma_0(2), Gamma_0(3), Gamma_0(4), and possibly Gamma_1(4).