The difficulties in proving modularity lifting theorems over non-totally real fields - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T16:12:16Z http://mathoverflow.net/feeds/question/12416 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/12416/the-difficulties-in-proving-modularity-lifting-theorems-over-non-totally-real-fie The difficulties in proving modularity lifting theorems over non-totally real fields David Hansen 2010-01-20T15:17:55Z 2010-01-21T03:09:44Z <p>First of all, let me apologize in advance for the terseness of this question.</p> <p>It seems that by now there are well-developed techniques (the "Taylor-Wiles-Kisin" method) for proving modularity lifting theorems over totally real fields. For example, it is now known that many elliptic curves over totally real fields are modular. I am curious: what exactly is the stumbling block to proving such results over non-totally real fields? It seems to be common knowledge among experts that the usual techniques for comparing a universal deformation ring and a Hecke algebra in this situation break down quite badly, but I cannot find a reference for this in print. It would be great if someone could illustrate the problem here.</p> <p>I understand that there are some obvious difficulties. For example, for $GL_2$ over an imaginary quadratic field, the locally symmetric spaces on which the relevant modular forms live is $SL_2(\mathbb{C})/SU(2)$, and arithmetic quotients of this are certainly not Shimura varieties. I am asking about more fundamental obstructions to applying the Taylor-Wiles-Kisin method, i.e strange behavior in the Hecke algebra and the universal deformation ring... </p> http://mathoverflow.net/questions/12416/the-difficulties-in-proving-modularity-lifting-theorems-over-non-totally-real-fie/12429#12429 Answer by Emerton for The difficulties in proving modularity lifting theorems over non-totally real fields Emerton 2010-01-20T17:35:35Z 2010-01-21T03:09:44Z <p>Note: This is a fairly precise and detailed question about an important but technical aspect of algebraic number theory. My answer is written at a level that I think is appropriate for the question; it assumes some familiarity with the topic at hand.</p> <p><hr /></p> <p>The most basic difficulty is that there is non a map $R \rightarrow {\mathbb T}$ in general (i.e. one typically doesn't know how to create Galois representations attached to automorphic forms).</p> <p>The second difficulty is that in the TWK method, one must argue with auxiliary primes (the primes typically labelled $Q$), and show that as you add these primes, ${\mathbb T}$ grows in a reasonable way (basically, is free over $\mathcal O[\Delta_Q],$ where $\Delta_Q$ is something like the $p$-Sylow subgroup of $({\mathbb Z}/Q{\mathbb Z})^{\times}.)$</p> <p>One shows this (or some variant of it) by considering the analogous queston about cohomology of the arithmetic quotients. Suppose for a moment we are in the Shimura variety context, or perhaps the compact at infinity context. Then it will be the middle dimensional cohomology that is of interest, and if we localize at a non-Eisenstein maximal ideal we might hope to kill all other cohomology. Then we can replace a comuptation of middle dimensional cohomology by an Euler characteristic computation, and its easy to see that the Euler char. will multiply by $|\Delta_Q|$ when we add the auxiliary primes $Q$.</p> <p>But in more general contexts, there won't be a single middle dimension in which the maximal ideal of interest is supported (even if it is non-Eisenstein), and computing Euler characteristics will just give $0$, which is not much use. It's not clear that it's even true that adding the auxiliary primes forces the approriate growth of cohomology, and possible torsion in the cohomology just adds to the complication.</p> <p>There is much current work, by various groups of researchers, with various different approaches, aimed at breaking this barrier.</p> <p><hr /></p> <p>I should add that one can now handle certain questions about non-totally real field, say question related to conjugate self-dual Galois reps. over CM fields, because these are still related to a Shimura variety context. This plays a role in the recent progress on Sato--Tate for higher weight forms by Barnet-Lamb--Geraghty--Harris--Taylor and Barnet-Lamb--Gee--Geraghty, and is also the basis for a recent striking theorem of Calegari showing that if $\rho:G_{\mathbb Q} \to GL_2({\mathbb Q}_p)$ is ordinary at $p$ and de Rham with distinct Hodge--Tate weights (and probably $\overline{\rho}$ should satisfy some technical conditions), then $\rho$ is necessarily odd! </p> <p><hr /></p>