# An open image theorem for a general class of abelian varieties

###### Abstract.

Let be a number field and be a polarized abelian variety with absolutely trivial endomorphism ring. We show that if the Néron model of has at least one fiber with potential toric dimension one, then for almost all rational primes , the Galois group of the splitting field of the -torsion of is .

###### 2000 Mathematics Subject Classification:

11G10,14K15## 1. Introduction

Let be a number field and a polarized abelian -fold with trivial -endomorphism ring. For each rational prime , let denote the -torsion of and the Galois group of the splitting-field extension . If , then is an elliptic curve and a well-known theorem of Serre asserts that is isomorphic to for all sufficiently large [S2, Theorem 2]. In a series of lectures and letters Serre (cf. [S4, Corollaire au Théorème 3]) later showed how to extend the result to the case when is odd, 2, or 6: if is sufficiently large, then . However, for general it is an open problem to show that is as big as possible for almost all . In this paper we show that this is true when we assume an additional hypothesis on the reduction of , and our main theorem is the following.

###### Theorem 1.

Suppose satisfies the following property:

(T) : there is a finite extension so that the Néron model of over the ring of integers has a semistable fiber with toric-dimension one.

If is sufficiently large with respect to and , then .

The rest of this paper is devoted to a proof of the theorem. In section 2 we prove the theorem modulo a result of Serre on the rigidity of inertial tori, and in section 3 we prove the necessary rigidity result.

An example due to Mumford shows that one cannot remove hypothesis (T) from the statement of our theorem [M, Section 4]. More precisely, Mumford constructed an abelian 4-fold with absolutely trivial endomorphism ring such that does not contain for infinitely many . His 4-fold does not satisfy hypothesis (T) because it has potentially good reduction everywhere; for infinitely many there are no non-trivial unipotent elements satisfying , while potential positive-dimension toric reduction would give rise to such elements. More importantly, Mumford’s example fails to satisfy the conclusion of the theorem because its so-called Mumford-Tate group is strictly smaller than for those 4-folds addressed in the theorem (cf. [S3, annotation 4]). However, the groups are as big as possible for almost all once one takes into consideration the upper bound imposed by the Mumford-Tate group, and for a general polarized abelian -fold it is conjectured that is almost always as big as possible given the constraints imposed by the endomorphism ring and Mumford-Tate group.

While Mumford’s example shows that an arbitrary abelian -fold will not satisfy the hypothesis (nor the conclusion) of the theorem, one can ask for the likelihood that a “random” will have absolutely trivial endomorphism ring and satisfy hypothesis (T). For , a necessary and sufficient condition is that -invariant does not lie in the ring of integers of . For , if or and is a degree- polynomial in whose splitting field has Galois group , then Zarhin showed that the endomorphism ring of the Jacobian of the hyperelliptic curve is absolutely trivial [Z, Theorem 2.1]. If moreover there is a prime in such that the reduction of modulo (is defined and) has distinct zeros (over an algebraic closure), one of which is a double zero, then the Jacobian satisfies (T).

In an appendix to this paper E. Kowalski shows that most monic polynomials in with integral coefficients satisfy both these properties, thus for most hyperelliptic curves over the Jacobian satisfies the hypotheses of the theorem. Of course, for most polarized abelian -folds do not arise as the Jacobians of hyperelliptic curves, so it is an open problem to determine how often the hypotheses of the theorem are satisfied in general.

### 1.1. Notation

We use the notation to mean that there is a constant which depends on the object and satisfies .

## 2. Proof of Main Theorem

Up to replacing by a finite extension we may assume satisfies (T) for . We fix an odd prime which is relatively prime to the polarization degree of . We regard as a vector space over and write for the Weil pairing; the pairing exists because is polarized and it is non-degenerate because is prime to the polarization degree. If is a subspace, then we write for the complement of with respect to . We identify with the similitude subgroup of and with the isometry group. There is short exact sequence

such that, for every and , we have . The action of is compatible with , so there is an embedding and the theorem asserts it is an isomorphism if . To prove the theorem we will construct a subgroup which we can show satisfies for , from which it will follow easily that for .

###### Lemma 2.

If , then is an irreducible -module.

###### Proof.

If is a -submodule, then the isogeny is defined over . If were isogenies for distinct , then would be an endomorphism outside of because , so distinct give rise to distinct elements of the -isogeny class of . However, Faltings’ theorem implies there can only be finitely many abelian varieties in the -isogeny class of (cf. [D, Corollaire 2.8]), so there are only finitely many such that is reducible. ∎

We say is a transvection if it is unipotent and has codimension one. The -conjugate of a transvection is a transvection, and we write for the normal subgroup generated by the subset of transvections in . The proof of the following lemma shows that condition (T) is sufficient, but not necessary, to show that is non-trivial for almost all .

###### Lemma 3.

If , then is non-trivial.

###### Proof.

Suppose is a prime in over which has toric-dimension one; exists because satisfies (T) for . Then the monodromy about is a transvection provided does not divide the order of the component group of the Néron model of over (cf. 2.1, 2.5 and 3.5 of [G]). ∎

Suppose is non-trivial. Let be a non-trivial irreducible -submodule and be the stabilizer of .

###### Lemma 4.

If is an irreducible -module, then and .

###### Proof.

This follows from (the proof of) lemma 3.2 of [H] because for all . ∎

Suppose is an irreducible -module, and let be the subgroup generated by the transvections which act non-trivially on . The image of in is non-trivial, irreducible, and generated by transvections, so is all of by [ZS, Main Theorem].

###### Lemma 5.

If as cosets, then the commutator is trivial.

###### Proof.

Note, if and only if . If is a transvection which acts non-trivially on , then . Thus if , then lies in , so acts trivially on . In particular, if , are transvections and , then for each , at least one of acts trivially on , so the restrictions of to commute (for every coset ), hence they commute on all of . ∎

The lemma implies for all and is the central product . Therefore, if we write , then is isomorphic to the wreath product and . The next step is to show that .

Let be the kernel of .

###### Lemma 6.

Let be a prime in and let be the ramification index of over . If and is the inertia subgroup of a prime in over , then the image of in is trivial.

###### Proof.

If , then has no elements of order , so the image in of the -Sylow subgroup of must be trivial. In particular, if does not lie over , then is an -group because it is trivial or generated by a unipotent element, so the image of in is trivial. Therefore we may suppose lies over . Let be a complement of the -Sylow subgroup; it is a cyclic subgroup of order prime to . By section 1.13 of [S1] (following [R]), there is a finite extension and a surjective homomorphism so that the representation has amplitude at most ; see section 3 for the definition of amplitude. The kernel of has index at most in and it commutes with (because it lies in ), so lemma 9 of section 3 implies all of commutes with because . In particular, the centralizer of in is , so , the image of , and lie in . ∎

The lemma implies that the fixed field of the kernel of is unramified and has uniformly bounded degree over . By a theorem of Hermite, there are only finitely many such extensions, so up to replacing by a finite extension we may assume that the image of in is trivial (for all ). Therefore and hence because acts irreducibly, so and . Once we know that is big, the following lemma completes the proof of the theorem.

###### Lemma 7.

If and , then .

###### Proof.

If , then and are disjoint extensions of . On the other hand, if , then is the Galois group of , so must be all of for . ∎

Remark: Most of the above carries through if we replace by a global field of characteristic . One key difference is that is no longer equal to for all , but it does contain for all . Another difference is that the argument in lemma 6 is made simpler by the fact that there are no inertial tori to contend with for .

## 3. Rigidity of Tori

Let be the multiplicative group of a finite extension . We regard as the set of -points of the algebraic torus given by the Weil restriction of scalars of the split one-dimensional torus . The fundamental characters corresponding to the embeddings form a basis for the character group . We define the amplitude of a character as , and we say is -restricted if for all and for some (cf. [S1, section 1.7] and [S4, annotation 5]).

Let be a finite extension and be a finite-dimensional vector space. We say a representation of algebraic groups over is -restricted if all its characters are -restricted. Every representation extends uniquely to an -restricted representation , and we define the amplitude of as the maximum of the amplitudes of its characters.

###### Lemma 8.

Let be a semisimple element and be a representation of amplitude . If commutes with , then commutes with in .

###### Proof.

Up to a base change by a finite extension of , we may assume that is diagonalizable in . Thus there is a decomposition so that acts on via an element and preserves the decomposition because it commutes with . The amplitude of the restriction is at most . If we write for the -restricted representation corresponding to , then commutes with because is a scalar. In particular, the composition of the product representation with the obvious embedding is an -restricted representation extending , so it must be and hence commutes with . ∎

The power of the previous lemma is that it allows us to show that representations are ‘rigid’ if they have sufficiently small amplitude (cf. [S4]).

###### Lemma 9.

Let be a semisimple element and be a representation of amplitude . If is a subgroup such that commutes with in and , then commutes with in .

###### Proof.

Let and denote the composition of with the th-power-map . By assumption has amplitude , hence the corresponding -restricted representation is the composition of the -restricted representation with the th-power-map . Moreover, commutes with in , so by the previous lemma commutes with in . In particular, is surjective because , hence and a foriori commute with . ∎

## 4. Acknowledgements

We gratefully acknowledge helpful conversations with Serre, and in particular, for clarifications with regard to initial tori. We also acknowledge helpful conversations with N.M. Katz and E. Kowalski.

## References

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## Appendix: Most hyperelliptic curves have big monodromy

Emmanuel Kowalski^{1}^{1}1 ETH Zürich - DMATH,

Let be a number field and its ring of integers. Let be a monic squarefree polynomial of degree or for some integer , and let be the (smooth, projective) hyperelliptic curve of genus with affine equation

and its jacobian.

In the previous text, C. Hall has shown that the image of the Galois representation

on the -torsion points of is as big as possible for almost all primes , if the following two (sufficient) conditions hold:

(1) the endomorphism ring of is ;

(2) for some prime ideal , the fiber over of the Néron model of is a smooth curve except for a single ordinary double point.

These conditions can be translated concretely in terms of the polynomial , and are implied by:

(1’) the Galois group of the splitting field of is the full symmetric group (this is due to a result of Zarhin [Z], which shows that this condition implies (1));

(2’) for some prime ideal , factors in as where are relatively prime polynomials such that for some and is squarefree of degree ; indeed, this implies (2).

In this note, we show that, in some sense, “most” polynomials satisfy these two conditions, hence “most” jacobians of hyperelliptic curves have maximal monodromy modulo all but finitely many primes (which may, a priori, depend on the polynomial, of course!).

More precisely, for and as above, let us denote

and let the height be defined on by

where is the norm from to and is any reasonable height function on , e.g., choose a -basis of , where , and let

for all .

Let denote the finite set

(4.1) |

We have , where

Say that has big monodromy if the Galois group of its splitting field is . We will show:

###### Proposition 10.

Let and be as above. Then

for all , where the implied constant depends on and .

Say that has *ordinary ramification* if it
satisfies condition (2’) above.

###### Proposition 11.

Let and be as above, and assume . There exists a constant , depending on and , such that we have

for , where the implied constant depends on and .

Finally, say that has big monodromy if the image of is as big as possible for almost all primes .

###### Corollary 12.

Assume that . Then we have

###### Remark 13.

Quantitatively, we have proved that the rate of decay of this probability is at least a small power of power of logarithm, because of Proposition 11. With more work, one should be able to get equal or very close to , but it seems hard to do better with the current ideas (the problem being in part that we must avoid for which the discriminant is a unit in , which may well exist, and sieve can not detect them better than it does discriminants which generate prime ideals, the density of which could be expected to be about ).

For both propositions, in the language of [K1], we consider a sieve with data

and we claim that the “large sieve constant” for the sifting range

satisfies

where the implied constant depends only on . Indeed, this follows from the work of Huxley [Hu], by combining in an obvious manner his Theorem 2 (which is the case , arbitrary) with his Theorem 1 (which is the case , arbitrary).

Concretely, this implies that for arbitrary subsets in the image of under reduction modulo — the latter is simply the set of monic polynomials of degree in , and has cardinality — we have

(4.2) |

where the sum is over squarefree ideals in with norm at most , and therefore also

(4.3) |

Proposition 10 is a result of S.D. Cohen [C]; it is also a simple application of the methods of Gallagher [G] (one only needs (4.3) here), the basic idea being that elements of the Galois group of the splitting field of a polynomial are detected using the factorization of modulo prime ideals. We recall that the first quantitative result of this type (for ) is due to van der Waerden [vdW], whose weaker result would be sufficient here (though the proof is not simpler than Gallagher’s).

###### Proof of Proposition 11.

Let be a prime ideal, and let be the set of polynomials which are monic of degree and factor as described in Condition (2’). We claim that, for some constant , (depending on and ), we have

(4.4) |

for all prime ideals with norm , for some depending on and .

Indeed, for , we have clearly

for , this holds with the convention that is irreducible of degree , and for , we must subtract 1 from the second term on the right. If , we are done, otherwise it is well-known that

as , hence the lower bound (4.4) follows by combining these two facts (showing we can take for any constant if is chosen large enough; using more complicated factorizations of the squarefree factor of degree , one could get arbitrarily close to ).

Now we apply (4.3) with this choice of subsets for with norm , and with for other . We take , assuming that , i.e., that is large enough. Since, if does not have ordinary ramification, we have by definition for any , it follows by simple computations that

where the implied constant depends on and

where now restricts the sum to squarefree ideals not divisible by a prime ideal of norm , and where is the number of prime ideals dividing .

It is then a standard fact about sums of multiplicative functions that

for large enough (depending on ; recall that ), and this leads to the proposition, since and are comparable in logarithmic scale. ∎

## References

- [C] S.D. Cohen: The distribution of the Galois groups of integral polynomials, Illinois J. Math. 23 (1979), 135–152.
- [G] P.X. Gallagher: The large sieve and probabilistic Galois theory, in Proc. Sympos. Pure Math., Vol. XXIV, Amer. Math. Soc. (1973), 91–101.
- [Hu] M.N. Huxley: The large sieve inequality for algebraic number fields, Mathematika 15 (1968) 178–187.
- [K1] E. Kowalski: The large sieve and its applications: arithmetic geometry, random walks, discrete groups, Cambridge Univ. Tracts (to appear).
- [vdW] B.L. van der Waerden: Die Seltenheit der reduziblen Gleichungen und der Gleichungen mit Affekt, Monath. Math. Phys. 43 (1936), 133–147.
- [Z] Y.G. Zarhin: Hyperelliptic Jacobians without complex multiplication, Math. Res. Lett. 7 (2000), no. 1, 123–132.