My recent trip to Berkeley did not result in a chance to test whether the Cheeseboard pizza maintained its ranking, but did give me the opportunity to attend the latest Bay Area Number Theory and Algebraic Geometry day, on a (somewhat disappointingly) rainy Saturday in Evans Hall. The weather was somewhat better on Sunday, however, allowing myself to make the trip to Mint Plaza for the following cup, which should bear some resemblance to the banner picture on this site. (Unfortunately, they were no longer serving their mini-Brioche buns.)
But now on to the good stuff, a report on some of the talks:
Jaclyn Lang gave a talk on her work concerning the image of big Galois representions. The setup is roughly as follows. Let
\(\overline{\rho}: G_{\mathbf{Q}} \rightarrow \mathrm{GL}_2(\mathbf{F}_p)\)
be an absolutely irreducible odd Galois representation over a finite field (hence modular). Suppose this Galois representation was the residual representation associated to an ordinary modular form that lived inside a Hida family that was smooth over weight space. Then one might expect that the corresponding representation
\(\overline{\rho}: G_{\mathbf{Q}} \rightarrow \mathrm{GL}_2(\mathbf{Z}_p[[T]])\)
to have image which was as big as possible, namely, containing \(\mathrm{SL}_2(\mathbf{Z}_p[[T]]).\) This can’t always be the case, however; for example, the residual representation (or the entire family) could be dihedral. However, if the residual representation contains \(\mathrm{SL}_2(\mathbf{F}_p),\) and one additionally assumes that the image of inertia at p is sufficiently large, then this is indeed the case (probably this assumes that the residue characteristic is at least 5). There have been a number of generalizations of this result due to Hida and others which improves the result by weaken the various hypotheses; for example, allowing coefficients, allowing the residual representation to be dihedral, and weakening the ramification hypothesis at p. For these results, one can’t expect that the image is full, but rather that the image contains an appropriate congruence subgroup of \(\mathrm{SL}_2.\) I like to think of this as follows: at classical specializations, one knows that the image (if it is not CM or weight one) will have open image; the results of this talk and of previous work show that index can be controlled in families. Actually, this is not quite true, because another obstruction to having open image even classically is the existence of inner twists. The main result of the talk was to deal with this issue of inner twists, and hence also allow for a generalization of the results not only to smooth Hida families but to any irreducible component of any Hida family. (More details to be found here.)
A natural question: one output of Lang’s result is to give an ideal \(\mathfrak{b}\) of the Hida family for which the image of these Galois representations contains the \(\mathfrak{b}\)-congruence subgroup (after accounting for inner twists). In characteristic zero, my impression from the talk was that one can identify the support of this ideal as coming from CM points and classical weight one modular forms. On the other hand, apparently there is also a version of this result in the reducible case (due to Hida and with extra hypotheses); in that case the zeros should correspond to the reducible locus, or equivalently, the zeroes of the p-adic zeta function. However, a stronger result is true, namely, that \(\mathfrak{b}\) can essentially be identified with this p-adic zeta function. So, returning back to the residually irreducible case, the natural question is: can the support of \(\mathfrak{b}\) contain the prime p?
Kestutis Cesnavicius gave a talk on the Manin-Stevens and Manin constants for elliptic curves, with emphasis on the prime p=2. He raised the following question: Suppose that \(N\) is odd. Is there a surjection from the space of weight 2 classical modular cusp forms of level \(\Gamma_0(N)\) with coefficients in \(\mathbf{Z}_2\) to the space of weight 2 Katz cusp forms of the same level over \(\mathbf{F}_2?\) The issue here is that the latter space is really the cohomology of the associated stack, not the course moduli space. Unfortunately, this question distracted me a little as I tried to find a counter-example (I failed). A result of Serre and Carayol basically implies that the result can only fail after localizing at a non-Eisenstein maximal ideal \(\mathfrak{m}\) of the Hecke algebra \(\mathbf{T}\) if the corresponding representation \(\overline{\rho}_{\mathfrak{m}}\) is induced from \(\mathbf{Q}(\sqrt{-1}).\) (Analogously, for \(p =3,\) when the representation is induced from \(\mathbf{Q}(\sqrt{-3}).\)) This is related to the classic failure of the first version of Serre’s conjecture for \(p =3\) at level \(\Gamma_1(13).\) However, as Serre quickly realized, this failure ultimately comes from a failure to lift mod-p forms as above, except in this case from the intermediate curve \(X_H(13),\) not from \(X_0(13).\) I ultimately convinced myself that lifting was always possible unless \(\mathfrak{m}\) was not only Eisenstein but also the ideal containing \(T_{p}\) for all odd p not dividing N. I think this must be related to Ken’s result on component groups of Neron models, and how the non-Eisenstein parts arise for \(X_H(N)\) but not for \(X_0(N)\) or \(X_1(N).\) (More details here.)
The final speaker of the day was Daquin Wan. The key question that arose in his talk was the following. Suppose that \(D(k,T)\) is the characteristic power series of the \(U\) operator on the space of overconvergent p-adic modular forms in integral weight k. Can one show that
\(L(k,T) = \displaystyle{\frac{D(k+2,T)}{D(k,pT)}}\)
has infinitely many zeros and infinitely many poles? One actually has to assume that \(k \ne 0\) here, since otherwise the result is false, as this will be a polynomial of dimension the space of weight two forms. One feels that p-adic Langlands should be able to say enough about slopes in these weights to obtain a contradiction, but I don’t unfortunately see how to do it. The main point of the talk was two-fold. There is an argument of Coleman that shows that \(D(k,T)\) is not itself a polynomial. This argument can be generalized to prove that \(L(k,T)\) is not a rational function. Second, the product \(L(k,T) L(-k,p^k T)\) is actually a rational function because of the properties of the theta operator. So one deduces that at least one of these functions had infinitely many poles and the other had infinitely many zeroes. This also relies on a previous result of Wan that these functions are meromorphic. (Oh, I should mention that this was joint work with … and here I didn’t take notes for a talk two weeks ago … Liang Xiao? Please correct me if I’m wrong)
(Romyar Sharifi also talked, but since I am actively trying to understand something about that talk on a more technical level, so I will have to return to a discussion of it later.)