I am reading the fascinating paper of Deligne on "le groupe fondamental de la droite projective moins trois points", and other stuffs related to anabelian geometry. This suggested the following question, which seems natural to me and that I haven't seen answered in the literature (but this is perhaps a consequence of my still very low understanding of those matters).

Let $X$ be a proper and smooth, geometrically connected, variety over ${\mathbb Q}$. Let us assume that $X$ has a rational point $x$ over ${\mathbb Q}$. Let $\bar X = X \times_{\mathbb Q} {\bar {\mathbb Q}}$. Then there is a natural action of $G_{\mathbb Q}=Gal(\bar {\mathbb Q}/ {\mathbb Q})$ on the profinite fundamental group $\pi_1(\bar X,x)$. As in Deligne, let us make $\pi_1(\bar X,x)$ nilpotent of order $N$ (by quotienting it by $[\dots[[\pi_1,\pi_1],\pi_1],\dots]$ where the number of $[$ is $N$), then take its pro-$l$-Sylow (for a given prime $l$), and quotient it by its torsion (which is a normal subgroup): we obtain a nilpotent torsion free pro-$l$ group $P_{l,N}$ which inherits a continuous action of $G_{\mathbb Q}$. Such a group has a finite dimensional Lie algebra over $Z_l$, and let us call $V_{l,N}$ this Lie algebra tensorized by $Q_l$ (just for convenience, to have something over a field). Hence we have a finite dimensional Lie algebra $V_{l,N}$ on which $G_{{\mathbb Q}}$ acts continuously by Lie algebra automorphisms. In particular, forgetting the Lie algebra structure, $V_{l,N}$ is a continuous Galois representation of $G_{\mathbb Q}$.

(From here, I am really on uncharted territory for me, and I coulsd possibly have everything wrong). One of the first aim of Deligne's paper is to interpret $V_{l,N}$ as the $l$-adic realization of a mixed motive. In Deligne's original paper, a motive is just defined as a system of realizations, which means that what Deligne has to do, and does, is just to construct $l'$-adic analogs of $V_{l,N}$ (for $l'$ an other prime - the construction is just by replacing $l$ by $l'$ from the beginning) and a Betty analog, and a De Rham analog, and compatibilities between them. If I am not mistaken, the underlying motive should be mixed, not pure, even if we have assumed $X$ proper and smooth (excepted if $N=1$ where $V_{l,N}$ is just the $l$-adic homology of $X$ and hence shoould be be pure). In particular, we do not expect $V_{l,N}$ to be semi-simle as a Galois representation in general. First question: is the above correct? Second question: if it is,

Do we know any case of $X$, $x$ as above where $V_{l,N}$ is non semi-simple?

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    $\begingroup$ The object $V_{l,N}$ you consider seems very close to the polylogarithm on $X$ (which one can define under additional assumptions on $X$). It is my understanding that the polylogarithm provides explicit extensions which aren't split (in a suitable category, like variations of mixed Hodge structures). However I'm not sure of the link with semi-simplicity as a Galois representation. $\endgroup$ – François Brunault Jul 4 '11 at 13:03

As you say, in general the representation should not be semi-simple even if $X$ is smooth projective. One can construct explicit examples as follows:

Let $X$ be any smooth projective curve of genus $g=2$ over $\mathbb{Q}$ with a rational point such that for some prime $p$, $X$ has a regular model over $\mathbb{Z}_p$ with special fibre consisting of a union of two elliptic curves (one can do analogous things for larger $g$). Since the Jacobian of $X$ has good reduction at $p$ it follows that $H^1(X, \mathbb{Q}_l)$ is unramified at $p$.

By a theorem of Takayuki Oda (Galois action on the nilpotent completion of the fundamental group of an algebraic curve. Advances in number theory (Kingston, ON, 1991), 213–232) it follows that since $X$ does not have good reduction at $p$, the Galois representation on $V_{l,N}$ is ramified at $p$ for some $N$. Now the pronilponent completion has a filtration whose associated quotients are quotients of $H_1(X,\mathbb{Q}_l)^{\otimes m}$ for various $m$ and therefore unramified. It follows that $V_{l,N}$ cannot be semisimple.

  • $\begingroup$ Very nice, thank you. I have accepted this answer. I observe that the non-trivial extensions between the associated quotients of the filtration on $V_{l,N}$ that your example construct are not $f$ at $p$ in the sense of Bloch-Kato: that is, this is an extension of two unramified representations at $p$, that is itself ramified at $p$. Do you know if one can produce some non-trivial extensions that are $f$ in the sense of Bloch-Kato? And in general, do you know other type of examples of non semi-simple $V_{l,N}$? $\endgroup$ – Joël Jul 5 '11 at 6:20
  • $\begingroup$ Sorry, I don't know much about the condition $f$ of Bloch-Kato. The non-triviality of extensions is probably true much more generally but I don't know a precise statement. Also, there is a theorem of Hain and Pulte (independently) relating the non-triviality of a certain subextension (in the setting of mixed Hodge structures) for curves $X$ over $\mathbb{C}$ to the Abel-Jacobi image of the cycle $[X] - [X^-]$ in $J(X)$ which you might find interesting. See, for example, the article "The geometry of the mixed Hodge structure on the fundamental group" by Hain. $\endgroup$ – ulrich Jul 5 '11 at 8:32
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    $\begingroup$ There are indeed many fascinating questions in this direction, which I'll perhaps write about some other time. But here are a few comments: If you focus just on the local representation at $p\neq l$, then that will always be split in the unramified case. This is because the Frobenius weights of the graded pieces are strictly descreasing. On the other hand, if the curve has good reduction at $l$ and the reduction of the type discussed at $p$, then the global representation will be non-split because of the consideration at $p$, but the local representation at $l$ will be crystalline. $\endgroup$ – Minhyong Kim Jul 5 '11 at 12:14
  • $\begingroup$ Finally, when you consider the simplest case of the projective line minus three points with the standard tangential base-point considered by Deligne, then the $V_{l,N}$ will definitely not be semi-simple at $l$ for $N$ sufficiently large because the $l$-adic $L$-function (sorry, your choice of $l$ rather than $p$) has only finitely many zeros. This last point is, I believe, explained in Ihara's Annals paper on the subject from the 80's. $\endgroup$ – Minhyong Kim Jul 5 '11 at 12:19
  • $\begingroup$ Thanks Minhyong, that's helpful too. There are surely many fascinating questions in this field. In the projective line minus three points case, am I right if I say that the fact that some V_{l,N}'s are non semi-simple is not proven in Deligne's original paper, or am I missing something? It seems to me that Deligne explicitely construct some extension of \Q_l by \Q_l(3) (or 5,7,etc.) in some V_{l,N}, but that the fact that those extensions are non trivial is just conjectured (second part of conjecture 8.1). Is that correct? Anyway, I run to read Ihara's paper. $\endgroup$ – Joël Jul 5 '11 at 13:51

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