At the beginning of this month I defended my PhD thesis, and the amount of work (and pages!) that went into my thesis is the reason this blog hasn’t been updated since last May. For anyone interested, here is a link to it on Columbia’s academic commons, but let me give a brief synopsis of it here.

The main result of my thesis concerns the Bloch–Kato Selmer group of the symmetric cube of the Galois representation attached to a cuspidal eigenform of level ; under some standard conjectures I construct nontrivial elements in such Selmer groups.

In a little more detail, let be a cuspidal eigenform of level and weight , and its Galois representation as constructed by Deligne. We also have the unitary cuspidal automorphic representation attached to , and it turns out that since has level , the -function

always vanishes at the central point (in fact its functional equation has sign ). The Bloch–Kato conjecture then predicts (after unraveling normalizations involving Tate/determinant twists) that the Bloch–Kato Selmer group

is nontrivial, and I prove this assuming a version of the global Langlands correspondence for cohomological automorphic representations for the exceptional group , as well as specific cases of the Arthur conjectures. Let me describe the general method through which the argument goes now.

The Skinner–Urban method constructs nontrivial elements in the Bloch–Kato Selmer groups of certain Galois representations and was pioneered in this paper and this paper of Skinner and Urban. I usually like to summarize it using the following diagram, whose notation I will explain below.

Here we are starting with a nice automorphic representation of a reductive -group . When made precise, this adjective “nice” with which I qualified should imply that has attached to it a -adic Galois representation which itself is nice in an appropriate sense (i.e., it is geometric in the sense of Fontaine–Mazur); this is the content of the arrow labelled “Langlands” in the diagram. If we assume for simplicity that is split, then will be a continuous representation of the absolute Galois group of into the group of -points of the dual reductive group to .

Now let be a finite dimensional algebraic representation of . Then we can consider the composition , which is a nice representation . It has attached to it an -function which should satisfy a functional equation. Let us assume things are normalized so that the center of the functional equation for is located at . Then the purpose of the Skinner–Urban method is to prove implications of the form

which are predicted by the Bloch–Kato conjecture.

The way the method does this is by traversing the top arch of the diagram. The first thing to do is to embed inside a bigger reductive group as the Levi of some parabolic . (Realistically, one usually embeds times some factors in as a Levi, but these factors cause no trouble and I will ignore them.)

On the right side of the diagram, to the parabolic will correspond a parabolic in the dual group of . This parabolic is just the one whose Levi contains the coroots corresponding to the roots contained in . Its Levi will indeed be the dual group to .

Now let us traverse the arch. The first step is to ascend the arrow labelled “Eisenstein.” This means we must construct a functorial lift of to an automorphic representation of . This must be done by some process which resembles parabolic induction, usually with Eisenstein series (though it is sometimes better to use certain CAP representations).

Now the functorial lift must be constructed in anticipation of the next step, which is to -adically deform in a generically cuspidal family of automorphic representations . One can usually construct such a family if one knows that satisfies certain properties locally at the archimedean place. For example, often knowing that is a certain type of discrete series at is enough. This family will vary in weight and is usually constructed as a piece of an eigenvariety (whence the nomenclature). It should be parametrized by, say, an affinoid rigid analytic space over .

The next step is to move from the left, automorphic side to the right, Galois side along the top of the diagram. What this means is that, assuming that we have an appropriate Langlands correspondence for , we must construct a rigid analytic family of Galois representations which interpolates those attached to the classical, cuspidal members of . Ideally, having such a family of Galois representations would mean is that is a representation which is continuous in a certain sense, where is the affinoid ring of analytic functions on . In reality, we cannot expect the construction of to be so direct, but roughly whatever object we get in its place should have properties which resemble the ones just listed.

The final step in this process is to descend the arrow labelled “Ribet.” This involves constructing a particular lattice in (again in an appropriate sense of the word “lattice”) and specializing it at the point in corresponding to . If done correctly, one will not quite reobtain the representation , but instead one will get a representation which does not factor through . The failure to factor through the Levi should be measured by a nontrivial cocycle which gives a Galois cohomology class in . One must then show that this class satisfies the appropriate local conditions, putting it in the correct Bloch–Kato Selmer group.

The one thing left that I should explain is where the representation shows up in this process. In order to carry out the Skinner–Urban method, certain pieces of numerology must be satisfied by the objects at play, and among them is the following. The Levi of will act on the unipotent radical of by the adjoint action. The Jordan–HÃ¶lder constituents of will therefore be representations of , and we must require (among other things) that the last constituent be isomorphic to . (One then sees on the automorphic side, for example, in the constant terms of Eisenstein series constructed from by the Langlands–Shahidi method.)

In my next post (or maybe posts, depending on how long they get) I want to describe how the first step of this method (that is, the construction of the functorial lift ) goes in the particular case of my thesis, and at the same time, correct an error I made in my previous post. In short, the error is that there is actually no Eisenstein class in degree for as constructed in that post, contrary to what I thought (and also contrary to what my advisor says in an example in his eigenvarieties paper!). I’ll try to explain where this error comes from and how its resolution affects my work and the aforementioned example in my advisor’s paper.