Langlands in dimension 2: the Yoshida conjecture

Background: One prominent part of the Langlands program is the conjecture that all motives are automorphic. It is of interest to consider special cases that are more precise, if less sweeping. The idea is to formulate generalized modularity conjectures that are as concrete as the Shimura-Taniyama-Weil conjecture (now the elliptic modularity theorem). The latter gained much in precision, for example, by Weil's experimental observation of the link between the conductor of the elliptic curve and the level of the weight two modular form. As a next step up the dimension ladder it is natural to consider abelian surfaces over ${\mathbb Q}$, in which case one encounters

Yoshida's conjecture: Any irreducible abelian surface $A$ defined over ${\mathbb Q}$ and with End$(A)={\mathbb Z}$ is modular in the sense that associated to each is a holomorphic Siegel modular cusp eigenform $F$ of genus 2, weight 2, and some level $N$, such that its spinor L-function $L_{\rm spin}(F,s)$ agrees with that of the abelian surface $$L(H^1(A),s) ~=~ L_{\rm spin}(F,s).$$

Questions:

1. Has Yoshida's conjecture been proven for some classes of abelian surfaces over ${\mathbb Q}$?

2. Are there lists of abelian surfaces and associated Siegel modular forms that extend the very useful lists constructed by Cremona and Stein for elliptic curves and their associated modular forms?

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+1 for the background! –  Qiaochu Yuan Oct 1 '10 at 15:48
The answer to question 1 is yes: you want to look at the works of Tilouine, Genestier-Tilouine and most recently V.Pilloni. The article of J.Tilouine which appeared in Compositio Math 142 would seem to be a good starting point for you. Note that typically, these results show p-adic modularity, in the sense that they do not establish that the form is really classical, as you wish. The so-called classicity property is much harder, but might be within reach. –  Olivier Oct 1 '10 at 17:54
For the applications I have in mind p-adic modularity unfortunately doesn't suffice. –  Laie Oct 1 '10 at 19:31
One reason that proving modularity (in the classical, not $p$-adic sense) is harder for abelian surfaces than for elliptic curves is that the Hodge structure of an abelian surface is not regular. We say that a Hodge structure is regular if all $h^{p,q}$ are $\leq 1$. Most modularity theorems have as a hypothesis that the Hodge structure attached to the motive one is trying to show is modular is regular. These don't apply to an abelian surface, since for an abelian surface $h^{1,0} = h^{0,1} = 2$. So, from the point of view of current methods, this is an intrinsically harder question ... –  Emerton Oct 2 '10 at 0:06
@Francois: If one can find the candidate Siegel modular form (like Poor/Yuen can do sometimes) then perhaps one can use the strategy I outline in the comments to David Hansen's answer to answer Q1. –  Kevin Buzzard Oct 5 '10 at 20:07

There's no need to require irreducibility. If $A=E_1 \times E_2$ is a product of elliptic curves the conjecture is true, by modularity of elliptic curves combined with Yoshida's lifting from pairs of classical cusp forms to Siegel modular forms of genus two. If $K$ is a real quadratic field and $E/K$ is a modular elliptic curve, then Yoshida's conjecture is true for the surface $A=\mathrm{Res}_{K/\mathbb{Q}}(E)$. This follows from a theorem of Johnson-Leung and Roberts; see arxiv 1006.5105. Presumably any individual $A$ can be done "by hand" using Faltings-Serre plus some serious computational cleverness.

Sorry if you already know all these things. :)

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Do you have any examples which do not come for GL_2 in some way? I.e., where the image of the Galois representation is as large as possible given that it comes from an irreducible abelian surface? –  Marty Oct 1 '10 at 16:49
Recent work of Brumer/Kramer arxiv.org/abs/1004.4699 and Poor/Yuen (cited therein) give computations of ab surfaces and Siegel MFs which don't come from GL_2, I believe. –  Kevin Buzzard Oct 3 '10 at 10:11
I think that these are interesting papers, but my reading was that there is no proof of an example in these references. At this point they have some computations of local factors of Siegel forms and abelian surfaces that agree. –  Laie Oct 4 '10 at 15:11
@Laie: I wonder whether in fact one could come up with a proof. If Brumer/Kramer/Poor/Yuen give some candidate pairs (ab surface, Siegel modular form) then probably by Weissauer's construction of Galois reps attached to SMFs (which may only apply in the regular case but which you can probably extend to more general cases in the ordinary situation using Hida theory) you can attach a p-adic Galois representation to the SMF. Now the job is to check that this representation is the Tate module of the ab surface. But there's this Faltings-Serre method, which Livne used to prove that... –  Kevin Buzzard Oct 5 '10 at 20:03
...a certain Galois representation was modular using only a finite amount of computation and a trick involving a Frattini subgroup. One could try doing the same trick in this situation. I don't know whether at the end of the day one would have to check too many Frobenii, but it's perhaps worth a try. This might make a nice masters thesis or chapter of a PhD thesis (or even a PhD thesis if you pushed it far enough). –  Kevin Buzzard Oct 5 '10 at 20:05

And so it turns out that I was in the audience of a seminar talk just today on this very subject. The opinion I expressed in comments is apparently not too far from the truth: V.Pilloni and B.Stroh now have some variants of Coleman's criterion for classicality for $\textrm{GSp}_{4}$, and so full modularity for some abelian surfaces is now accessible. So the answer to your question 1 seems to be yes, but you need to rely on cutting edge results to really get these kind of results.

Check out the joint works of V.Pilloni and B.Stroh.

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What is Coleman's criterion for the simpler GL2 case? I ask as a novice in p-adic modular forms. :) –  David Hansen Oct 6 '10 at 0:27
The $p$-adic valuation of the $U_p$-eigenvalue of the overconvergent $p$-adic eigenform of weight $k$ should be $< k-1$. –  Emerton Oct 6 '10 at 3:35
This is great, but doesn't it beg the question of proving modularity of the residual representation? Are there any known instances of modularity for a continuous odd representation $\rho: G_{\mathbb{Q}} \to GSp_4(\mathbb{F}_p)$ with full image? –  David Hansen Oct 6 '10 at 4:27
I just realized the answer is likely yes - just take a Siegel modular form which isn't CAP or Yoshida, and look at the mod-$p$ reduction of the $p$-adic Galois representation attached to it by Laumon/Weissauer/Sorenson, for $p$ "generic". But does this obviously surject onto a $GSp_4(\mathbb{F}_{p^n})$? –  David Hansen Oct 6 '10 at 4:36
Why the insistence on full image? I doubt anything nearly as strong is needed to apply R=T theorems, for example. –  TSG Oct 6 '10 at 12:18