Title: Congruences between Bernoulli-Hurwitz numbers.
Abstract: If we average 1/z^k over the nonzero points of the lattice ZZ[i] we get some complex numbers. Turns out they satisfy lots of congruences! We explain how to see this using the Weierstrass p function and some Fourier (no-)theory.
Title: "The Art of Artin L-functions"
Abstract: I will describe the context and motivation which led Artin to define his eponymous L-functions, as well as predict and prove their "functorial" properties. The Flying Spaghetti Monster willing, we will examine specific examples of the factorization of (higher-dimensional) Artin L-functions into (1-dimensional) Hecke L-functions.Notes.
No No Theory Seminar this week.
Title: On BSD for Elliptic Curves with CM
Abstract: In 1987, Karl Rubin showed for the first time that the Tate-Shafarevich group of certain elliptic curves with CM were finite. This gave the first examples of elliptic curves for which BSD could be confirmed. I aim to use his result to prove (or at least give numeric evidence for) BSD for specific curves. There will be 3 parts to the talk: first, the basics of elliptic curves with CM following Silverman, the second on the BSD conjecture, and third applying Rubin's result to specific curves.
Title: The Taylor-Wiles-Kisin patching method (or "How much theory can Jeff get away with in his 'No' Theory seminar talk?")
Abstract: I will prove Fermat's Last Theorem. This may involve a bit of theory, but I will try to keep things in the spirit of No Theory seminar as much as possible. My basic goal here is to sketch the main ideas of the proof, while not going into too much detail in the more technical parts of the argument.
This will obviously need quite a bit of theory, but the "main" part of the argument, namely the Taylor-Wiles(-Kisin) patching method, doesn't really use much beyond some commutative algebra.
Some of the more technical parts of the talk will need things like properties of modular curves and Hecke algebras, Galois cohomology, Chebotarev density, Galois deformations, and algebraic topology. I will try to keep these as separate from the rest of the talk as possible, and not go into too much detail on them, so you are welcome to ignore those parts of the talk and just take the results as black boxes.
Title: Some examples about lifting elliptic curves.
Abstract: I will explain Serre-Tate theory about infinitesimal deformation of abelian varieties. We can apply this theory to infinitesimal deformation of ordinary and supersingular elliptic curves. In the ordinary case, there is a natural group structure on the 1-parameter deformation space, so that a canonical lifting can be singled out. We will see many examples of canonical liftings of ordinary elliptic curves (thanks to many known examples of elliptic curves with CM). In the supersingular case, the deformation space is still 1 dimensional, but we can't single out a canonical lifting.
There are other parameters that can be given to the 1-parameter deformation space of ordinary or supersingular elliptic curves. For example, we can consider the j-invariant, and the parameter coming from some period map. I was trying to compute some examples of the supersingular case, but my computation wasn't very successful. I will try to explain the difficulties that I met, and explain about what kind of results I was expecting to get.
Title: The No Theory of Quaternion Algebras.
Abstract: Quaternion algebras, a family of non commutative 4-dimensional algebras over a field, are like the rebellious teenage siblings of quadratic extensions. In this introduction, I'll explain the classification of quaternion algebras over Q, tell the story of how I got de-confused about a Galois cohomology computation, talk about some sweet local-to-global properties, and wrap it up with some pictures and wild speculations (not as wild nor ambitious as Prof. Venkatesh's.)
No No Theory Seminar this week.
Title: Ideal class groups and Sha
Abstract: We will take a cohomological perspective that will allow us to prove that the ideal class group of a number field is analogous to the Tate-Shafarevich group of an elliptic curve. We will begin with a brief overview of homogeneous spaces, Galois cohomology, and the failure of the Hasse principle that will lead us to the definition of Sha. Then we will repeat the construction of Sha replacing the elliptic curve with the group of units in the algebraic integers over a number field K and prove that we get the ideal class group of K, and then discuss several questions about importing the interpretation of Sha to the world of number fields.
Title: A tour of the Modular curve Y_1(13).
Abstract: Spend an hour with me on the modular curve Y_1(13). Let's look at its cusps. Its differentials. Feel its handles. Were you looking for rational points? Get out of here.