Mathematical Diary

William Stein

April 4, 1996

1 4/3/96

1.1 Lam’s Rep. Theory

We showed that S_{4} is an M-group. This was accomplished by inducing up 1 dimensional representations of certain subgroups H and using Frobenius reciprocity. Next we proved that G′ ∩ Z(G) ∩ H ⊂ H′ where G′ denotes the commutator subgroup of G. Lam said he couldn’t prove this in a purely group theoretic way but he proved it easily using induced characters.

1.2 Hour Break

I talked to Saul about Ribet’s class. We discussed the construction of the ρ_{λ} and the construction of the π via functions on GL_{2}(A).

1.3 Ribet

We learned more about the structure of the representations π_{p} corresponding (via Langlands) to the local representations ρ_{λ}|_{D_{p}}. π_{p} can be one of three things. 1) a principal series representation P S(α, β), 2) a special representation κ ⊗ st, or 3) cuspidal (some of these come from quadratic extensions K/Q_{p}, the reset are called extraordinary).

We also talked about twisting a modular form by a Dirichlet character κ:

f ⊗ κ = ∑_{1}^{∞} κ(n)a_{n}q^{n} ∈ S_{k}(Γ_{1}(NM^{2}), εκ^{2}).

The fact is: there is a newform g = ∑ b_{n}q^{n}, b_{n} = a_{p}κ(p) for all p |̸NM such that if ρ_{λ} corresponds to f then ρ_{λ} ⊗ κ corresponds to g.

Next we had to talk about CM modular forms in order to look at what can happen to ρ_{λ}|D_{p} when λ|p = ℓ. For f to be CM means that ρ_{λ} becomes abelian on some finite index subgroup.

Finally, we began asking questions about the reduction of the ρ_{λ} mod ℓ.

1.4 Wetherall

We went to lunch. He told me about a problem of Ed Schaeffer. “If C is a curve with points of degrees 2 and 3 does C necessarily have a rational points?” A possibly helpful fact is that if P has degree 2 and Q has degree 3 then D = galois orbit of Q − galois orbit of P is a divisor of degree 1. Thus R ↦ R − D gives a map C ↪ Jac(C).

1.5 Hartshorne

We studied curves of genus 3. They break into 2 disjoint classes. 1) those whose canonical sheaf is very ample (these are all plane curves of degree 4) and 2) hyperelliptic genus 3 curves (which Hartshorne claims all live on a quadric surface). We did the exercise to prove dim |D| ≤ deg D. The solution didn’t use Riemann-Roch like I thought it would.

1.6 Vojta

Let X be a pgsav (proper generically smooth arithmetic variety). Let D be a Cartier divisor on X. Then there is a way to use the arithmetic structure of X to define a function g_{D} on X − Supp D. g_{D} is in fact a Weil function.

1.7 Atiyah

I went to a talk by Sir Michael on topological quantum field theory. He gave an axiomatic formulation which resembled homology theory. A TQFT seems to be a map from manifolds of dimension d (with boundary) to Hilbert Spaces which turns disjoint union into tensor products instead of direct sums. The orientation reversed version of a manifold M must map to the dual hilbert space. This gives rise to lots of interesting bilinear pairings. Number theory even comes into it because there is some numerical invariant which often turns out to be the value of Riemann’s zeta function at some point.