On monodromy in families of elliptic curves over \mathbb{C}

On monodromy in families of elliptic curves over $\mathbb C$

Abstract.

We show that if we are given a smooth non-isotrivial family of elliptic curves over  with a smooth base  for which the general fiber of the mapping (assigning -invariant of the fiber to a point) is connected, then the monodromy group of the family (acting on of the fibers) coincides with ; if the general fiber has connected components, then the monodromy group has index at most  in . By contrast, in any family of hyperelliptic curves of genus , the monodromy group is strictly less than .

Some applications are given, including that to monodromy of hyperplane sections of Del Pezzo surfaces.

Key words and phrases:
Monodromy, elliptic curve, hyperelliptic curve, -invariant, braid monodromy, Del Pezzo surface
1991 Mathematics Subject Classification:
14D05, 14H52, 14J26
The study has been funded by the Russian Academic Excellence Project ’5-100’.

Introduction

It is believed that if fibers in a family of algebraic varieties “vary enough” then the monodromy group acting on the cohomology of the fiber should be in some sense big. Quite a few results have been obtained in this direction. See for example [4] for families of elliptic curves, [8] for families of hyperelliptic curves, [2] for families of abelian varieties (see also [9] for abelian varieties in the arithmetic situation). In the cited papers cohomology means “étale cohomology with finite coefficients”. In this paper we address the question of “big monodromy” for families of elliptic curves over  and singular cohomology.

The main result of the paper (Proposition 4.2) asserts that if is a smooth non-isotrivial family of elliptic curves over  and if the general fiber of its “-map” (assigning to each point of the base the -invariant of the fiber) is connected, then the monodromy group of the family is the entire group , and if the general fiber has connected components, then the monodromy group of the family is a subgroup of index at most in . Here, by monodromy group we mean that acting on of the fiber.

An immediate consequence of this proposition is that, in any non-isotrivial family of elliptic curves, the monodromy group has finite index in (Corollary 4.5). This requires some comments.

The above assertion is similar to a well-known result about elliptic curves over number fields, viz. to Serre’s Theorem 3.2 from Chapter IV of [14]. It is possible that one can prove our Corollary 4.5 by imitating, mutatis mutandis, Serre’s proof of this theorem or even derive it from Serre’s theorem or similar arithmetical results. One merit of the approach presented in this paper is that the proofs are very simple and elementary. One should add that the similarity between arithmetic and geometric situations is not absolute. For example, Theorem 5.1 from [8] could suggest that, over , the monodromy group for some families of hyperelliptic curves of genus should be the entire . However, as we show in Proposition 5.2, for any family of hyperelliptic curves of genus over  the monodromy group acting on of the fiber is a proper subgroup of .

Our main result has three simple consequences, which are presented in Section 4. First, any smooth (i.e., without degenerate fibers) family of elliptic curves over a smooth base with commutative fundamental group, must be isotrivial (Proposition 4.6). Second, for non-isotrivial families we obtain an upper bound on the index of the monodromy group in (Proposition 4.7). Third, in the case of smooth elliptic surfaces we use Miranda’s results from [12] to obtain an upper bound on the index of monodromy group in terms of singular fibers (it turns out that only fibers of the types  and  count); see Proposition 4.11.

In Section 5 we prove the above mentioned result about families of hyperelliptic curves of genus or higher.

In Section 6, we derive from our main result that the hyperplane monodromy group of a smooth Del Pezzo surface (or, for Del Pezzos of degree , the monodromy group acting on of smooth elements of the anticanonical linear system) is the entire (Proposition 6.1). I realize that it is not the only way to obtain this result. Observe that, in view of Proposition 5.2, Proposition 6.1 cannot be extended to surfaces with hyperelliptic hyperplane sections.

Sections 1 through 3 are devoted to auxiliary material (Proposition 3.3 may be of some independent interest).

Acknowledgements

I am grateful to Yu. Burman, Andrey Levin, Sergey Rybakov, Ossip Schwarzman, and Yuri Zarhin for useful discussions.

Notation and conventions

All our algebraic varieties are defined over  and reduced, so they are essentially identified with their sets of closed points; the only exception is the discussion of the notion of quadratic twist in Section 3. If is an algebraic variety, then is its smooth locus and is its singular locus.

When we say “a general has property ”, this always means “property holds for a Zariski open and dense set of ’s”. The word “generic” is used in the scheme-theoretic sense.

If is an algebraic variety and is a proper and flat morphism such that a general fiber of is, say, a smooth curve of genus , we will say that is a family of curves of genus . If, in addition, the morphism is smooth, we will say that (or just if there is no danger of confusion) is a smooth family, or a family of smooth varieties. If is a family over and is a morphism, then by we mean the pullback of  along .

By of an algebraic variety over  we always mean fundamental group in the classical (complex) topology.

As usual, we put , where is the identity matrix.

Finally, we fix some terminology and notation concerning elliptic curves.

Following Miranda [12], we distinguish between curves of genus  and elliptic curves: by elliptic curve over a field we mean a smooth projective curve over  of genus  with a distinguished -rational point.

Similarly, by a smooth family of curves of genus we will mean a smooth family such that its fibers are curves of genus , and by a smooth family of elliptic curves we mean a pair , where is a smooth family of curves of genus  and is a section.

To each curve  of genus  over a filed  one can assign its -invariant ; recall that if is (the smooth projective model of) the curve defined by the Weierstrass equation , then

(1)

Two curves of genus  over  are isomorphic if and only if their -invariants are equal.

We say that a family over is isotrivial if it becomes trivial after a pullback along a generically finite morphism . For families of curves of genus  this is equivalent to the condition that -invariants of all fibers are the same.

1. Generalities on monodromy groups

Suppose that is an irreducible variety and is a family of smooth varieties.

If , , and is an abelian group, then the fundamental group acts on .

Definition 1.1.

The image (corresponding to this action) of in will be called monodromy group of the family  at  and denoted (we suppress the mention of and ; there will be no danger of confusion).

Since is irreducible, is path connected. Hence, if we fix once and for all the group for some , then all the groups define the same conjugacy class of subgroups of ; this class (or, abusing the language, any subgroup belonging to this class) will be denoted by .

In the sequel we will be working with families of smooth curves of genus (in most cases will be equal to ) as fibers and monodromy action on of the fiber. Since monodromy preserves the intersection form, the subgroups , where is such a family, will be defined up to an inner automorphism of the group  ( if ).

Convention 1.2.

If is a non-smooth family, then by we mean , where is the Zariski open subset over which is smooth.

Below we list some simple properties of monodromy groups.

Proposition 1.3.

Suppose that is an irreducible variety, is a non-empty Zariski open subset, and is a smooth family over . Then .

Proof.

The result follows from the fact that, for any , the natural homomorphism is epimorphic (see for example [7, 0.7(B) ff.]). ∎

Proposition 1.4.

Suppose that and are smooth irreducible varieties and is a smooth family over . If is a dominant morphism such that a general fiber of has connected components, then is conjugate to a subgroup of , of index at most .

Corollary 1.5.

Suppose that and are smooth irreducible varieties and is a smooth family over . If is a dominant morphism such that a general fiber of is connected, then is conjugate to .

Proof of Proposition 1.4.

It follows from [16, Corollary 5.1] and the algebraic version of Sard’s theorem that there exists a Zariski open non-empty such that all the fibers of over points of are smooth and the induced mapping is a locally trivial bundle in the complex topology. Proposition 1.3 implies that . Since is (path) connected, each fiber of this bundle has connected components, and the base is locally path connected, is a subgroup of index at most  in for any . This implies the proposition. ∎

2. Some remarks on 3-braids

In this section, all topological terms will refer to the classical (complex) topology.

We begin with some remarks on 3-braids (not claiming to novelty).

Let stand for the configuration space of unordered triples of distinct points in the complex plane. It is well known that , where is braid group with strands. If is an unordered triple, we will write instead of .

For any triple , we denote by the elliptic curve which is the smooth projective model of the curve with equation . We are going to define a homomorphism

To wit, it is well known that any braid can be represented by a homeomorphism such that and is identity outside a bounded set. Putting , we extend to a homeomorphism from to itself by putting . If is the morphism induced by the projection , then there exists a unique homeomorphism such that and on , where is the compact set outside of which . The automorphism

does not depend on the choice of the representing , and we put .

Proposition 2.1.

If is the braid represented by the loop in defined by the formula , , then .

Proof.

To prove the proposition, we choose generators of and a basis in .

To fix generators of the braid group, we choose the points so that they are collinear and lies between and . Now let and  be the braids corresponding to the following closed paths in : in the path defining , the point stays where it is while and are swapped, and moving along small arcs close to the segment so that the composition of paths traveled by and defines a positively oriented simple closed curve. The braid is defined similarly, with the point staying put and the points and being exchanged; see Figure 1.

Figure 1. Two generators of . One may assume that homeomorphisms of representing these braids are identity outside the dashed ovals.

The group is generated by and , and these braids satisfy the relation .

For a basis in we choose the -cycles and that are obtained by lifting the closed paths and on Fig. 2 from to .

Figure 2. Projections to  of the cycles and . Point corresponds to their intersection point on , point  is just an apparent intersection point.

Abusing the language, we will denote the action on of a homeomorphism representing the braid  by the same letter , and similarly for . Since the homeomorphisms representing and can be chosen to be identity outside the corresponding dashed ovals on Fig. 1, it is clear that and . Taking into account that and preserve the intersection pairing on , one concludes that, in the basis , the action of and on is given by matrices of the form

The relation implies that

(2)

Equations (2) imply that either or . The first case is impossible: if both and act as identity, then the entire braid group acts identically, which is absurd since its action is non-trivial (see for example [1]). So, one of the integers and is equal to  and the other is equal to . Dualizing, we see that either acts on as (in the basis dual to ) and acts as , or vice versa.

Now it is well known that . Plugging the possible values of and , one obtains the result. ∎

Remark 2.2.

One can show that, with the choice of signs as on figures 1 and 2, one has

We do not need to be that precise.

3. Quadratic twists and monodromy

In this and the following section we will be studying monodromy groups acting on of fibers in families of smooth curves of genus . In such families, the monodromy group acting on of the fiber is contained in .

If is a smooth family of elliptic curves, then the morphism assigning the -invariant to a point , will be denoted by . Following Miranda [12, Lecture V], we will say that is the -map of the family  (in Kodaira’s paper [11], the morphism is called analytic invariant of the family ).

Notation 3.1.

If is a smooth family of elliptic curves over and if , then the monodromy representation will be denoted by .

Suppose now that is a family of elliptic curves over a smooth and connected base. Since its fiber over the generic point of is an elliptic curve over the field of rational functions , and since this elliptic curve can be reduced to the Weierstrass normal from, there exists a Zariski open subset such that the restriction is isomorphic to the family

(3)

where and are regular functions on , the fiber over being the smooth projective model of the curve defined by the equation , and discriminant of the right-hand side of (3) does not vanish on . Proposition 1.3 shows that , so, as far as monodromy groups are concerned, we may and will assume that and that the family is defined by (3) with non-vanishing discriminant.

Any such family of the form (3) defines a morphism assigning to each point the collection of roots of . If and if is the set of roots of the polynomial , then the morphism induces a homomorphism . If is the fiber of over , and if

is the homomorphism defined in Section 2, then the diagram

is commutative.

Suppose that and are two families of elliptic curves over a base . One says that and differ by a quadratic twist if their scheme-theoretic generic fibers (which are elliptic curves over the field of rational functions ) are isomorphic over a quadratic extension of . It it clear that this is the case if and only if there exists a morphism of degree  (not necessarily finite or étale) such that and are isomorphic smooth families. If the families and differ by a quadratic twist, then they can be represented by Weierstrass equations

(4)

where is a rational function on (see [15, Chapter X, Proposition 5.4]).

Being interested only in the monodromy groups and , we can, replacing by a Zariski open subset if necessary, assume that the families and are smooth; in particular, this implies that is a regular function on without zeroes.

Suppose that is a smooth algebraic variety, is a regular function on without zeroes, and is a point. In the definition that follows we regard as a complex manifold and as a holomorphic function on .

Definition 3.2.

In the above setting, by we denote the homomorphism defined as follows. If , , we put if the function changes after the analytic continuation along a loop representing , and we put otherwise. In other words, if a loop representing is of the form , , then , where is the number of times the loop winds around the origin.

We will say that is the quadratic character associated to .

Proposition 3.3.

In the above setting, suppose that and are smooth families of elliptic curves that differ by a quadratic twist as in (4). Then the monodromy homomorphism differs from by an inner automorphism of .

Proof.

Suppose that and are defined by the equations (4), where has no zeroes or poles on and discriminants of the left-hand sides of never vanish. If , , and are the roots of the polynomial , where , then roots of the polynomial are , , and .

In the argument that follows we will not distinguish between path and loops in and their homotopy classes; this will not lead to a confusion. That said, choose a base point and fix a path in joining the points (unordered triples) and . If , then

where is the loop defined by the formula

in which we use the following notation: if and , then is the unordered triple .

If the loop winds times around the origin, then Proposition 2.1 implies that , where is the identity matrix, whence the result. ∎

Lemma 3.4.

Suppose that is a group and that and are homomorphisms. Put , . Then one of the following cases holds:

(i) ;

(ii) there exists a subgroup , , such that ;

(iii) is the subgroup of generated by and .

Proof.

If the quadratic character is trivial, then and case (i) holds. Suppose now that is non-trivial; then is a subgroup of index in .

If , then the character factors through the group :

now if , then .

Suppose finally that . Then and

so is the subgroup generated by and and case (iii) holds. ∎

Suppose now that and are smooth families of elliptic curves over the same base . If we fix a base point , we can identify (not canonically) first integer cohomology groups of the fibers and and identify them both with .

Proposition 3.5.

Suppose that and are smooth families of elliptic curves over the same base and that and differ by a quadratic twist with a non-vanishing regular function  as in (4). Put (these subgroups are only defined up to a conjugation). Then:

(i) either and are conjugate, or one of these groups contains a subgroup of index that is conjugate to the other subgroup, or each contains a subgroup of index  and the subgroups and are conjugate.

(ii) if , then .

Proof.

Put . Proposition 3.3 implies that, conjugating the subgroups if necessary, one may assume that there exist homomorphisms and such that , .

We prove part (i) first; to that end, we invoke Lemma 3.4. If case (i) or case (iii) of this lemma holds, we are done. In case (ii) there exists a subgroup , , such that .

If , this implies that ; if , this implies that , so is a subgroup of index in ; in the remaining case , one has , and . Thus, part (i) is proved.

To prove part (ii), suppose that . If case (i) or case (iii) of Lemma 3.4 holds, it is clear that . If case (ii), observe that contains a unique subgroup of index : this follows from the fact that the abelianization of is . Expressing the corresponding epimorphism in terms of its action on the generators and , one sees that , whence , whence . ∎

Corollary 3.6 (from the proof).

Suppose that and are families of elliptic curves over the same smooth base that differ by a quadratic twist. Then

(i) if the monodromy group has finite index in , then either the indices and are equal or one of them is twice greater than the other;

(ii) images of and in are conjugate.∎

Proposition 3.7.

Suppose that and are smooth families of elliptic curves over the same base and that their -maps are equal and non-constant. Then

(i) either or one of these indices is twice greater than the other;

(ii) images of and in are conjugate;

(iii) if then .

Notation 3.8.

In the next section we will see that if the -map is not constant then the monodromy group has finite index in . In the statement of this proposition we allow indices of subgroups to be infinite and assume that .

Proof.

In view of Propositions 3.6 and 3.5 it suffices to show that and differ by a quadratic twist. To that end put (the field of rational functions). The (scheme-theoretic) generic fibers of the families and over are elliptic curves over . They have the same -invariant , and this -invariant is not equal to or  since is not constant. Hence, these elliptic curves differ by a quadratic twist by virtue of Proposition 5.4 from [15, Chapter X], and so are the corresponding families. ∎

4. Main result and applications

We begin with a folklore result for which I do not know an adequate reference.

Proposition 4.1.

Suppose that is a smooth family of curves of genus , where is a variety (i.e., a reduced scheme of finite type over ). Then the mapping from to that assigns -invariant to a point , is induced by a morphism from to .

Proof.

If (relative Picard variety, see [10, Section 5]), then the family has a section (to wit, the zero section) and induces the same mapping from to since if is a smooth curve of genus  over . Thus, without loss of generality one may assume that the family in question has a section; in this case see [6, § 5]. ∎

Proposition 4.2.

Suppose that is a smooth family of curves of genus  over a smooth and connected base  (the ground field is ); let be the -map, attaching to any point the -invariant of the fiber of over .

(i) If the morphism  is not constant and its general fiber is connected, then .

(ii) If the morphism  is not constant and its general fiber has connected components, then is a subgroup of index at most  in and the image of in is a subgroup of index at most in .

We begin with a lemma.

Lemma 4.3.

Suppose that is a smooth family of curves of genus . Then there exists a smooth family of elliptic curves such that the -maps are the same and is conjugate to , where is the automorphism defined by the formula .

Proof.

Put . As we have seen in the proof of Proposition 4.1, and the family has a section. Finally, , where by we mean the constant sheaf with the stalk  (see for example [13, § 9]), and this implies the assertion about monodromy. ∎

Proof of Proposition 4.2.

Lemma 4.3 implies that we may assume that the family in question has a section. Assuming that, put

and consider the smooth family of elliptic curves in which the fiber over is the smooth projective model  of the curve with equation and the section assigns to the “point at infinity” of this model. It is well known (see for example [1, Corollary to Theorem 1]) that .

Now put , and

Let be the restriction of the family  to ; put , and let be the restriction of to . Proposition 1.3 implies that and .

Observe that there exists an isomorphism such that the diagram

is commutative. Indeed, one can define by the formula , and the inverse morphism will be

Hence, in the fibered product

(we mean fibered product in the category of reduced algebraic variates, so is the scheme theoretic fibered product modulo nilpotents) the variety is isomorphic to (in particular, is smooth and irreducible) and fibers of are isomorphic to fibers of . Thus, the hypothesis implies that a general fiber of the morphism has connected components. On the other hand, any fiber of the morphism is irreducible since it is isomorphic to . Now Proposition 1.4 and Corollary 1.5 imply that for the pullback families and on , the group is a subgroup of index at most  in and (as usual, this equation holds up to a conjugation).

Since , Proposition 3.7 implies the result. ∎

Remark 4.4.

I do not know whether the bound in this proposition can be improved to for .

Proposition 4.2 implies the following fact.

Corollary 4.5.

If is a non-isotrivial smooth family of curves of genus , then its monodromy group is a subgroup of finite index in .

Here is the first application of what we proved.

Proposition 4.6.

If is a smooth algebraic variety with abelian fundamental group, then any smooth family of curves of genus  over must be isotrivial.

Proof.

Suppose that is a smooth family of curves of genus , where is smooth and irreducible and is abelian.

We are to show that the -map is constant. If this is not the case, then Corollary 4.5 asserts that the monodromy group of the family  has finite index in . Since has finite index in , one has . If is the image of in , then is an abelian subgroup of finite index in . The latter group is isomorphic to the free group  with two generators, and Schreier’s theorem on subgroups of free groups implies that contains no abelian subgroup of finite index. We arrived at the desired contradiction. ∎

For the case of non-commutative of the base, one can obtain an upper bound on the index of monodromy groups in non-isotrivial families.

Proposition 4.7.

Suppose that is a smooth non-isotrivial family of curves of genus  over a smooth base and that can be generated by elements. Then .

Corollary 4.8.

Suppose that is a non-isotrivial family of elliptic curves over a smooth curve of genus , with degenerate fibers. Then .

Proposition 4.7 is a consequence of the following elementary lemma.

Lemma 4.9.

Suppose that is a subgroup of finite index and that can be generated by elements. Then .

Proof of the lemma.

Throughout the proof, free group with generators will be denoted by .

Since can be generated by elements, there exists an epimorphism . Putting , one obtains the following commutative diagram of embeddings and surjections:

(5)

If is the order of , then , so by Schreier’s theorem . Since the morphism  is surjective, the group can be generated by elements. Put , and let be the natural projection. The subgroup can be also generated by elements; since , Schreier’s theorem implies that , where (indeed, the free group cannot be generated by elements). Applying Schreier’s for the third time, we obtain that , whence . It follows from the right-hand square of the diagram (5) that

whence the result. ∎

Proof of Proposition 4.7.

Put . Since can be generated by elements, the same is true for ; now Corollary 4.5 implies that , and Lemma 4.9 applies. ∎

Using Proposition 4.2 one can obtain other lower bounds for monodromy groups. Observe first that the named proposition immediately implies the following corollary, in the statement of which we use Convention 1.2.

Corollary 4.10.

If is a family of elliptic curves over a smooth projective curve  and if  is its -map, and if is not constant, then .

If is smooth and has a section, one can be more specific.

Proposition 4.11.

Suppose that is a minimal smooth elliptic surface with section (it means that is a smooth projective surface, is a smooth projective curve, the general fiber of is a smooth curve of genus , no fiber of contains a rational -curve, and has a section) and that is not constant.

Then

(6)

where if the fiber over is a cycle of smooth rational curves or the nodal rational curve if (type  in Kodaira’s classification [11, 12]), if the fiber over consists of smooth rational curves with intersection graph isomorphic to the extended Dynkin graph , (type in Kodaira’s classification), and otherwise.

Proof.

In view of Corollary 4.10 the index in the left-hand side of (6) is less or equal to , and equals by virtue of Corollary IV.4.2 from [12]. ∎

Similarly, one can express (and obtain a lower bound for ) using the information about the points where -invariant of the fiber (smooth or not) equals  or , see for example [12, Lemma IV.4.5, Table IV.3.1] and Table. In the notation if [12], -invariant is 1728 times less than that defined by (1); of course, this does not affect multiplicities of poles.

5. A remark on families of hyperelliptic curves

Proposition 5.1.

If is a smooth family of hyperelliptic curves of genus , then

Corollary 5.2.

If is a smooth family of hyperelliptic curves of genus , then is a proper subgroup of .

Proof of Proposition 5.1.

In this proof, will denote the monodromy group acting on the integer of a fiber of , and will stand for the monodromy group acting on cohomology with coefficients in .

Since the reduction modulo  mapping is surjective, one has

so it suffices to show that

(7)

To that end, let be a hyperelliptic curve of genus that is a fiber of ; denote its Weierstrass points by . It is well known (see for example [5, Lemma 2.1]) that the -torsion subgroup is generated by classes of divisors