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. .
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: .
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: .
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:
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 of pairs -orbits with discriminant form correspond to of the maximal abelian covering of -forms 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:
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 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 . of this ramified covering are the -forms of -coverings 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 of the Jacobian is sufficiently generic. Theorem -torsion4 is case (1).
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 of its Jacobian. If -covering is the discriminant form of the pair then the quadric intersection , in is a lift of to a of the Jacobian. This covering has trivial period-index obstruction in the sense described in -coveringReference CFO08 and, conversely, any lift to a with trivial period-index obstruction may be given by an intersection of quadrics which generate a pencil with discriminant form -covering The analogous statement holds for any . (see Reference Cre16). Using this and Theorem 6 we obtain the following:
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:
2. Proof of Theorem 6
For a -module let
the product running over all completions of For a finite group . and -module define
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).
2.2. A lemma
Identifying with gives an exact sequence,
where the integer is .