Mark Kisin gave a talk at the number theory seminar last week where the following problem arose:
Let \(W\) be the Galois representation associated to the Tate module of an abelian variety \(A\) over a number field, and suppose that \(W = U \otimes V\). Now suppose that the Galois action on \(U\) is unramified at all primes above \(p\). Can you prove that the Galois action on \(U\) has finite image?
Of course this is a special case of the unramified Fontaine-Mazur conjecture. But here the representation \(U\) literally “comes from an abelian variety” although as a tensor factor rather than a direct factor. At first sight it seems like it should be much easier than the actual Fontaine-Mazur conjecture if you just find the right trick, but I don’t see how to do it! Here at least is a very special case.
Lemma: Suppose that \(A/K\) has ordinary reduction at a set of primes of density one, and
that \(U\) is a representation which is unramified at all primes dividing \(p\) of odd dimension which occurs as a tensor factor of \(W = H^1(A) = U \otimes V\). Then, after some finite extension of \(K\), \(U\) contains a copy of the trivial representation.
Proof: One may as well assume by induction that the action of the Galois group
on \(U\) is absolutely irreducible of odd dimension \(d\) and remains so for every finite extension (otherwise decompose it into such pieces and take one of odd dimension).
Now choose a prime \(v\) of \(K\). Let \(\alpha_i\) be the eigenvalues of Frobenius at \(v\) on \(U\),
and let \(\beta_j\) be the corresponding eigenvalues on \(V\). We know that \(\alpha_i \beta_j\) are algebraic numbers which are Weil numbers of norm \(N(v)\). The ratios of any two roots thus are also algebraic numbers with absolute value \(1\) at all real places, and so \(\alpha_i/\alpha_1\) has this property.
Let’s suppose that the ratios \(\alpha_i/\alpha_1\) are actually roots of unity for a set \(v\) of density one. Since \(W\) is be defined over a fixed finite extension \(E = \mathbf{Q}_p\), the degrees of these ratios has uniformly bounded order over \(E\), and the the orders of these roots of unity also have uniformly bounded order. But then (projectively) only finitely many characteristic polynomials will arise from Frobenii for a set of (edit: density one), which would imply that \(U\) has finite projective image, from which it easily follows that \(U\) becomes trivial over a finite extension (remember the determinant is unramified so of finite image). Hence it suffices to show that the \(\alpha_i/\alpha_1\) are all algebraic integers and then use Kronecker’s theorem.
For finite places not dividing \(N(v)\) this is clear because the valuations of the \(\alpha_i \beta_j\) are all trivial and so are their ratios. For finite places dividing \(N(v)\) now suppose in addition that \(A\) is ordinary. Fix a place above \(v\). If the \(\alpha_i/\alpha_1\) have valuation given by \(a_i\), and \(\beta_j/\beta_1\) have valuation \(b_i\), it follows that the quantities \(a_i + b_j\) take on precisely two values, zero and either \(1\) or \(-1\), and they take on each of these values exactly half the time. But then either \(a_i\) is constant and thus (considering \(i = 1\)) equal to \(0\), or the \(b_j\) are all zero, and then half the \(a_i\) are zero and half are \(1\) or \(-1\). But that’s clearly only possible if \(U\) has odd dimension. So done!
I suspect the case that \(\dim(U)=2\), even with an ordinary hypothesis, is probably quite hard. But I would be happy to be mistaken.
This step:
> But then (projectively) only finitely many characteristic polynomials will arise from Frobenii for a set of positive density, which would imply that U has finite projective image
seems dubious. Consider the group O(2), where a positive density set of elements have the same characteristic polynomial but which isn’t finite.
Of course one can fix this by strengthening the hypothesis to the usual strengthened conjecture that after passing to a finite field extension the set of ordinary primes has density 1. But maybe this isn’t necessary, as if I remember correctly this positive density condition implies the group is virtually upper-triangular and the unramified Fontaine-Mazur conjecture in the upper-triangular case is easy.
Actually I think the results of Richard Pink in his paper ell-adic algebraic monodromy groups, cocharacters, and the Mumford-Tate conjecture allow the ordinariness assumption to be removed entirely: A “weak Hodge cocharacter” is a cocharacter of the p-adic monodromy group of A whose weights are 0 with multiplicity g and 1 with multiplicity g. Such a cocharacter must clearly act by scalars on U if dim U is odd. But the result of the Pink is that these characters generate the identity component of the ell-adic monodromy group, so the identity component of the ell-adic monodromy group consists of scalars, and then finiteness of the determinant does the trick.
For the dim U=2 case, the first hard case may be the an abelian variety of genus 4 whose Galois representation has monodromy group isogenous to SL_2 x SL_2 x SL_2 acting by the tensor product of standard representations. The individual SL_2 factors will give two-dimensional representations, which of course in real life will be ramified at p, but it doesn’t seem possible to prove that by reasoning about the Frobenius at other places. If I understand right, the results of Pink only help if we already know the Frobenius torus of the abelian variety is maximal at p.
(Update Will’s objection was to a point (now fixed) where I said “positive density” but had certainly intended to say “density one” …)
The \(O(2)\) phenomonon is simply coming from the fact that the component group is not connected. That’s why there is the assumption that the density of ordinary primes is one! That condition is stable under finite extensions. (If the image of Galois on \(U\) was connected then ordinary for a positive density of primes would be sufficient, but that condition is not stable under finite extensions.)
I did avoid mentioning Pink’s paper in part because in Kisin’s talk he *used* the Mumford-Tate conjecture as an ingredient to avoid having to address this Fontaine-Mazur question. Pink also proves some very nice results “at almost all primes”.
The relevant paper from the talk:
https://people.math.harvard.edu/~kisin/dvifiles/northcott.pdf
Interesting question. In the tensor decomposition W = U \otimes V, are the representations U and V both self-dual (up to a twist)?
My conjectures with Wee Teck and Dipendra involve representations of the form
W = U \otimes V, with U symplectic and V orthogonal of even dimension, so U is a parameter for SO(2n+1) and V is a parameter for SO(2m). I was never able to characterize via invariant theory which symplectic representations W have that form. Will’s product W = 2 x 2 x 2 = 2 x 4 in Sp(8) is one of them.