Most binary forms come from a pencil of quadrics

By Brendan Creutz

Abstract

A pair of symmetric bilinear forms and determine a binary form . We prove that the question of whether a given binary form can be written in this way as a discriminant form generically satisfies a local-global principle and deduce from this that most binary forms over are discriminant forms. This is related to the arithmetic of the hyperelliptic curve . Analogous results for nonhyperelliptic curves are also given.

1. Introduction

A pair of symmetric bilinear forms and in variables over a field of characteristic not equal to determine a binary form

of degree . The question of whether a given binary form can be written as a discriminant form in this way is studied in Reference BG13Reference Wan13Reference BGW15Reference BGW16. We prove that the property of being a discriminant form generically satisfies a local-global principle and deduce from this that most binary forms over are discriminant forms.

It is easy to see that a binary form is a discriminant form if and only if the forms are too, for every . When , it thus suffices to consider integral binary forms, in which case we define the height of to be and consider the finite sets of integral binary forms of degree with . We prove:

Theorem 1.

For any ,

Moreover, these values tend to as .

When is even the corresponding is strictly less than , as can been seen by local considerations. For example, a square free binary form over is a discriminant form if and only if it is not negative definite (see Reference BGW16, Section 7.2), a property which holds for a positive proportion of binary forms over . One is thus led to ask whether local obstructions are the only ones. This question was posed in Reference BS09, Question 7.2, albeit using somewhat different language. When , the answer is yes and turns out to be equivalent to the Hasse principle for conics, and in this case the limit appearing in Theorem 1 is the probability that a random conic has a rational point, which is (see Reference BCF15, Theorem 1.4).

When is a number field, define the height of to be the height of the point in weighted projective space , and set to be the finite set of degree binary forms over of height at most . We prove the following:

Theorem 2.

Let be a number field. For any ,

It is known that a square free binary form is a discriminant form over if the smooth projective hyperelliptic curve with affine model given by has a rational point Reference BGW16, Theorem 28. In particular binary forms of odd degree are discriminant forms. Results of Poonen and Stoll allow one to compute the proportion of hyperelliptic curves over of fixed genus that have points everywhere locally Reference PS99aReference PS99b. This gives lower bounds on the proportion of binary forms of fixed even degree that are locally discriminant forms. Computing these bounds and applying Theorem 2, one obtains Theorem 1.

Theorem 2 states that the property of being a discriminant form satisfies a local-global principle generically. This is rather surprising given that such a local-global principle does not hold in general. For example, there is a positive density set of positive square free integers such that the binary form

is a discriminant form locally, but not over (see Reference Cre13, Theorem 11). Of course, the forms appearing in Equation 1.1 are not generic. It is well known (as was first proved over by van der Waerden Reference vdW36) that 100% of degree univariate polynomials over a number field have Galois group . Therefore Theorem 2 is a consequence of the following:

Theorem 3.

Suppose is a binary form of degree over a global field of characteristic not equal to and such that has Galois group . If is a discriminant form everywhere locally, then is a discriminant form over .

A square free binary form of even degree gives an affine model of a smooth hyperelliptic curve with two points at infinity. As shown in Reference BGW16 the -orbits of pairs with discriminant form correspond to -forms of the maximal abelian covering of of exponent unramified outside the pair of points at infinity. Geometrically these coverings arise as pullbacks of multiplication by on the generalized Jacobian where is the modulus comprising the points at infinity. The Galois-descent obstruction to the existence of such coverings over (and hence to the existence of a pencil of quadrics over with discriminant form ) is an element of (Reference BGW16, Theorems 13 and 24). The Galois action on factors faithfully through the Galois group of , so Theorem 3 follows from:

Theorem 4.

Suppose is a hyperelliptic curve of genus over a global field of characteristic not equal to and that . Then ; i.e., an element of is trivial if it is everywhere locally trivial.

This result is all the more surprising given that the analogous statement for the usual Jacobian is not true! There exist hyperelliptic curves of genus with Jacobian , generic Galois action on and such that . A concrete example is given in Reference PR11, Example 3.20; see also Example 16 below. This leads one to suspect that there may exist locally solvable hyperelliptic curves whose maximal abelian unramified covering of exponent does not descend to (or, equivalently, that the torsor parameterizing divisor classes of degree is not divisible by in the group ). This can happen when the action of Galois on is not generic; an example is given in Reference CV15, Theorem 6.7. But Theorem 4 implies that it cannot happen when the Galois action is generic:

Theorem 5.

Suppose is an everywhere locally soluble hyperelliptic curve satisfying the hypothesis of Theorem 4 and let denote the base change to a separable closure of . Then

(a)

the maximal unramified abelian covering of of exponent descends to , and

(b)

the maximal abelian covering of of exponent unramified outside descends to .

Proof.

The covering in (a) is the maximal unramified subcovering of that in (b), while (b) follows from Theorem 4 and the discussion preceding it.

Theorem 4 generalizes to the context considered in Reference Cre16, which we now briefly summarize. Given a curve , an integer and a reduced base point free effective divisor on , multiplication by on the generalized Jacobian factors through an isogeny whose kernel is dual to the Galois module . In the situation considered above and is multiplication by on (in this case the duality is proved in Reference PS97, Section 6). Via geometric class field theory the isogeny corresponds to an abelian covering of of exponent unramified outside . The maximal unramified subcoverings of the -forms of this ramified covering are the -coverings of parameterized by the explicit descents in Reference BS09Reference Cre14Reference BPS16. The Galois-descent obstruction to the existence of such a covering over is the class in of the coboundary of from the exact sequence .

The following theorem says that the group is trivial provided the action of Galois on the -torsion of the Jacobian is sufficiently generic. Theorem 4 is case (1).

Theorem 6.

Suppose is a global field of characteristic not dividing , is a smooth projective and geometrically integral curve of genus over , and and are as above. In all of the cases listed below, .

(1)

and .

(2)

, is a canonical divisor and .

(3)

and .

(4)

, for some prime and integer and neither of the following holds:

(a)

The action of the absolute Galois group, , on is reducible.

(b)

The action of on factors through the symmetric group .

Remark 7.

The statement and proof of the theorem depend only on the cohomology of the -module and its dual . If one likes, this can be taken as the definition of , and the isogeny can be ignored.

In the case of genus one curves, the corresponding coverings can be described using the period-index obstruction map in Reference CFO08. For example, a genus hyperelliptic curve can be made into a -covering of its Jacobian. If is the discriminant form of the pair , then the quadric intersection in is a lift of to a -covering of the Jacobian. This covering has trivial period-index obstruction in the sense described in Reference CFO08 and, conversely, any lift to a -covering with trivial period-index obstruction may be given by an intersection of quadrics which generate a pencil with discriminant form . The analogous statement holds for any (see Reference Cre16). Using this and Theorem 6 we obtain the following:

Theorem 8.

Fix . For 100% of locally solvable genus one curves of degree there exists a genus one curve of period and index dividing such that in the group parameterizing isomorphism classes of torsors under the Jacobian of .

When case (3) of Theorem 6 shows that we may replace with “all”. This was first proved in the author’s PhD thesis Reference Cre10, Theorem 2.5. It would be interesting to determine if this is always true when is prime. In this case it is known that there always exists such that Reference Cas62, Section 5 (but not for composite Reference Cre13). However, it is unknown whether may be chosen to have index dividing .

The proportion of locally solvable genus one curves of degree has been computed by Bhargava-Cremona-Fisher Reference BCF16. As of cubic curves satisfy the hypothesis in case (4) of Theorem 6, this yields the following:

Theorem 9.

At least 97% of cubic curves admit a lift to a genus one curve of period and index such that in the Weil-Châtelet group of the Jacobian.

2. Proof of Theorem 6

For a -module let

the product running over all completions of . For a finite group and -module define

Lemma 10.

Suppose is a finite -module and let be the Galois group of its splitting field over . Then

(1)

is contained in the image of under the inflation map,

(2)

if , then , and

(3)

if , then .

Proof.

(1) (2) because the inflation map is injective, and (2) (3) by Tate’s global duality theorem. We prove (1) using Chebotarev’s density theorem as follows.

Let and for each place of , choose a place of above and let be the decomposition group. The inflation-restriction sequence gives the following commutative and exact diagram:

Since splits over , we have . The map is therefore injective by Chebotarev’s density theorem. Hence . By a second application of Chebotarev’s density theorem, the groups range (up to conjugacy) over all cyclic subgroups of . From this it follows that .

Recall that . Since (see Reference Cre16) it suffices to prove, under the hypothesis of Theorem 6, that

2.1. Proof of Theorem 6 case (1)

By assumption, the complete linear system associated to gives a double cover of which is not ramified at . Changing coordinates if necessary, we may arrange that is the divisor above . Then is the hyperelliptic curve given by , where is a binary form of degree with nonzero discriminant. The ramification points of form a finite étale subscheme of size which may be identified with the set of roots of .

As described in Reference PS97, Section 5 (see also Reference BGW16, Proposition 22), we may identify with the subsets of of even parity, while corresponds to subsets modulo complements and corresponds to even subsets modulo complements. Parity of intersection defines a Galois equivariant and nondegenerate pairing,

(See Reference PS97, Section 6 or Reference Cre16.) The induced pairing on is the Weil pairing (written additively). Fixing an identification of the roots of with the set , the action of on factors through the symmetric group . The following lemma proves Theorem 6(1).

Lemma 11.

.

Proof.

For , let denote the transposition and let (recall is identified with the even subsets of ). We use to denote the image of in . We note that for any ,

This is because is a transposition, addition is given by the symmetric difference, and the pairing is given by parity of intersection.

Now suppose is a -cocycle in which represents a class in . By our assumption, the restriction of to the subgroup is a coboundary. Hence there is some such that Since form a basis for and is nondegenerate, we can find such that for all . From this it follows that , for all . In other words, simultaneously plays witness to the fact that is a coboundary on each of the subgroups . But then must be a coboundary, since the generate .

2.2. A lemma

Identifying with gives an exact sequence,

where the integer is .

Lemma 12.

Let and . The map induces a surjection

Proof.

Cohomology of -modules gives an exact sequence,

Since , we see that is contained in the image of . Hence, we may lift any to some . Now sits in an exact sequence . We must show it is possible to choose such that , where is as in the inflation-restriction sequence