DESCENT ON PICARD GROUPS USING FUNCTIONS ON CURVES

SAMIR SIKSEK

Abstract. Let k be a perfect field, X a smooth curve over k, and denote by X^{c} the subset of closed points of X. We show that for any non-constant element f of the function field k(X) there exists a natural homomorphism Pic(X) → k^{∗}/G_{f} (k)

where

G_{f} (k) := ∏_{P∈X^{c}} Norm_{k(P)/k}(k(P)^{∗})^{ord_{P} (f)}.

We explain how this generalizes the usual results on descents on Jacobians and Picard groups of curves.

1. Introduction

Let k be a perfect field and X a smooth curve over k, by which we mean a complete, non-singular and absolutely irreducible curve over k. Denote the function field of X over k by k(X), the Jacobian by J_{X}, and the Picard group by Pic(X) (see Section 2 for the definition of the Picard group). The groups J_{X} and Pic(X) are often studied (particularly when k is a number field) by constructing homomorphisms from either of these groups to groups of finite exponent. For example:

(1) Descent algorithms on curves are usually concerned with the construction of homomorphisms from the Jacobian J_{X}(k) to groups of the form L^{∗}/L^{∗}^{q}, for suitably defined finite k-algebras L, and positive integers q. As observed by Schaefer ([6]), virtually all such descents utilize functions on the curve whose divisors are q-divisible.

(2) Let Y be a non-singular projective variety and f a non-constant element of the function field k(Y) whose divisor is a norm for some finite extension K/k. Then there is a homomorphism CH_{0}(Y) → k^{∗}/ Norm(K^{∗}) (see for example [1, pages 447-448]). If Y is our (smooth) curve X then we have a homomorphism Pic(X) → k^{∗}/ Norm(K^{∗}).

In view of this it is natural to ask the following question: given an arbitrary nonconstant function f ∈ k(X), is there an induced homomorphism of J_{X}(k) or Pic(X) into some group of finite exponent as above? In this paper we answer this question affirmatively. It is our hope that these homomorphisms will find interesting applications in the study of the arithmetic of curves.

Before stating the main theorem of this paper we set some notation. A closed point P of X corresponds to a discrete valuation ring O_{P} of k(X) containing k, with maximal ideal 𝔪_{P}. The residue field of P is by definition k(P) := O_{P}/𝔪_{P}, and is a finite extension of k. The degree of P is given by |P| := [k(P) : k]. If g ∈ k(X) is regular at point P, that is g ∈ O_{P}, then the value of g at P, denoted by g(P), is defined to be the image of g in k(P); it thus makes sense to speak of the Norm_{k(P)/k}(g(P)) ∈ k. If K is a finite extension of k, and n is an integer then, as usual, we let

Date: 7, January 2002.

2000 Mathematics Subject Classification. Primary 11G30, Secondary 11G25.

Key words and phrases. curves, descents, Jacobians, Picard groups.

Norm_{K/k}(K^{∗})^{n} := {α^{n} : α ∈ Norm_{K/k}(K^{∗})}.

Clearly Norm_{K/k}(K^{∗})^{0} = {1}. For a closed point P ∈ X let ord_{P} : k(X)^{∗} → ℤ be the corresponding valuation. We denote the set of closed points on X by X^{c}; this of course is all of X except for the generic point. Now suppose f ∈ k(X) is a non-constant function on the curve, and we let

(1)  G_{f}(k) := ∏_{P∈X^{c}} Norm_{k(P)/k}(k(P)^{∗})^{ord_{P} (f)}.

The product makes sense since all but finitely many of the terms are {1}, and the result is clearly a subgroup of k^{∗}. Our main theorem defines a homomorphism from the Picard group Pic(X) to the quotient group k^{∗}/G_{f}(k) (see Section 2 for the definition of the Picard group). In essence, this means that we are doing descent on the Picard group of the curve.

Theorem 1. Let X be a smooth curve over the perfect field k. Suppose f is a non-constant element of the function field k(X), and let G_{f}(k) be the subgroup of k^{∗} defined above. Then f induces a unique homomorphism

(2)  ϕ_{f} : Pic(X) → k^{∗}/G_{f}(k)

satisfying the following property: if ∑ m_{j}Q_{j} is a divisor on X whose support is disjoint from the poles and zeros of f, then the class [∑ m_{j}Q_{j}] of this divisor in Pic(X) is mapped, by ϕ_{f}, to the coset represented by

∏ Norm_{k(Q_{j})/k}(f(Q_{j}))^{m_{j}}

in the group on the right-hand side.

2. Proof and Discussion of Theorem 1

Throughout this section X denotes a smooth curve over a perfect field k, and \overline{k} the separable closure of k.

2.1. Preliminaries. It is worth recalling at the outset the relationship between the points of X, and the elements of X(\overline{k}). We started out by saying that a closed point P of X corresponds to discrete valuation rings O_{P} of k(X) containing k. Such a point would simultaneously correspond to an orbit of elements of X(\overline{k}), say \{P_{1}, …, P_{d}\}, under the action of Gal (\overline{k}/k). When convenient, we may identify the two by writing P = {P_{1}, …, P_{d}}. Note that the size d of the orbit corresponding to P equals its degree |P| = [k(P) : k]. The points P_{1}, …, P_{d} in fact correspond to the distinct embeddings of the residue field k(P) into \overline{k}. If g ∈ O_{P} then

Norm_{k(P)/k}(g(P)) = ∏ g(P_{i})

where as stated before, g(P) is the image of g in k(P) := O_{P}/𝔪_{P}, and the g(P_{i}) have the usual meaning.

We now come to the definition of the Picard group, Pic(X), which we reproduce here since there is some discrepancy in the literature. Recall that we have defined X^{c} to be the set of closed points of X. The divisor group of X, denoted Div(X), is the free group on the points of X^{c}. The subgroup of principal divisors is denoted by Princ(X), and we let Pic(X) := Div(X)/Princ(X). If X\overline{k} := X ×_{k} \overline{k}, then there is a natural injection

Pic(X) ↪ H^{0}(Gal (\overline{k}/k) , Pic(X\overline{k})),

that is not always an isomorphism, though it often is. In particular, this natural injection is known to be an isomorphism in the following cases (see [5, Section 3]):

• when k is a local field and X possesses a k-rational divisor of degree 1.

• when k is a number field and, for every prime υ of k, the corresponding

curve X_{υ} = X ×_{k} k_{υ} possesses a k_{υ}-rational divisor of degree 1. It follows in these cases that the degree 0 part of the Picard group, Pic^{0}(X), can be identified with J_{X}(k), where J_{X} is the Jacobian of the curve X. Although it is useful to be aware of this, we do not make any such assumption in this paper.

We now come to discuss the notation and tools needed for the proof of Theorem 1. By the support of a divisor ∑ m_{j}Q_{j} we mean the set {Q_{j} : m_{j} ≠ 0}. We say that a divisor is coprime to a set of points S if its support is disjoint from S. The support of a non-constant function h ∈ k(X) is the support of its divisor. If h ∈ k(X) is a non-constant function, and ∑ m_{j}Q_{j} is a divisor coprime to the support of h then we define

h (∑ m_{j}Q_{j}) := ∏ Norm_{k(Q_{j})/k}(h(Q_{j}))^{m_{j}}.

Recall the identification made above between closed points of X and orbits of elements of X(\overline{k}) under the action of Gal (\overline{k}/k). One immediately sees that this definition of h(∑ m_{j}P_{j}) is in harmony with the usual definition found elsewhere (for example [7, page 37] or [8, page 43]). Before proving Theorem 1 we need to recall Weil’s reciprocity.

Weil’s Reciprocity. Suppose X is a smooth curve over a perfect field k, and h_{1}, h_{2} ∈ k(X) are non-constant functions having disjoint supports. Then

h_{1}(div(h_{2})) = h_{2}(div(h_{1})).

See [7, page 37], or [8, page 43].

2.2. Proof of Theorem 1. Let S be the support of f. We let Div(X)_{S} be the subgroup of Div(X) of divisors coprime to S. Define a map

(3)  Div(X)_{S} → k^{∗}/G_{f}(k)

sending ∑ m_{j}Q_{j} to the coset represented by f(∑ m_{j}Q_{j}). Clearly this map is a homomorphism. Now let Princ(X)_{S} be the subgroup of principal divisors coprime to S; thus

Princ(X)_{S} := Div(X)_{S} ∩ Princ(X).

We first show that Princ(X)_{S} is contained in the kernel of the homomorphism (3). Thus suppose g ∈ k(X) is a non-constant function such that div(g) ∈ Princ(X)_{S}. Clearly f, g have disjoint support. The map in (3), sends div(g) to the coset represented by f(div(g)) in k^{∗}/G_{f}(k), and hence to show that div(g) is in the kernel it is sufficient to show that f(div(g)) ∈ G_{f}(k). We observe that

f(div(g)) = g(div(f))  (By Weil’s reciprocity)

= g ( ∑_{P∈X^{c}} ord_{P}(f)P)

= ∏_{P∈X^{c}} Norm_{k(P)/k}(g(P))^{ord_{P} (f)}

However, Norm_{k(P)/k}(g(P)) is in Norm_{k(P)/k}(k(P)^{∗}), and it immediately follows that f(div(g)) is in G_{f}(k) := ∏_{P∈X} Norm_{k(P)/k}(k(P)^{∗})^{ord_{P} (f)}, and so div(g) is in the kernel of the map (3). We thus obtain an induced homomorphism

Div(X)_{S}/Princ(X)_{S} → k^{∗}/G_{f}(k),

sending the class of a divisor ∑ m_{j}Q_{j} that is coprime to S to the coset on the right-hand side represented by f(∑ m_{j}Q_{j}) = ∏ Norm_{k(Q_{j})/k}(f(Q_{j}))^{m_{j}}. The proof of Theorem 1 is complete upon observing that the obvious injection

Div(X)_{S}/Princ(X)_{S} ↪ Pic(X)

is indeed an isomorphism. This follows from the fact, proven by Lang, that any divisor class containing a k-rational divisor also contains a k-rational divisor whose support is disjoint from a given finite set^{1} (see [3, page 166]).

2.3. A discussion of Theorem 1. It is appropriate to make some remarks regarding the proof of Theorem 1.

(1) The proof of Theorem 1 is similar to Schaefer’s proof of his [6, Lemma 2.1]; the main difference is the replacement of q-th powers by norms.

(2) Moving the divisor in its class so as to avoid the ‘bad set’ is a standard device in algebraic geometry; compare the above proof to the construction of the intersection pairing Pic(X) × Pic(X) → ℤ for a surface (see [2, page 357]), and to the construction of the pairing Pic(X) × Br(X) → Br(k) for a curve (see [4]). As far as we are aware, Schaefer was the first to apply this to the construction of descent maps. The older approach used patching arguments, and extending these to work in our situation would have been infinitely troublesome, if not outright impossible!

(3) We have taken the domain of our map ϕ_{k} to be Pic(X) where as in descent maps the domain is usually J_{X}(k) (after making suitable assumptions to identify this with Pic^{0}(X); see page 3).

2.4. An Example. It is a good idea to give an example to show how Theorem 1 extends descent maps even when we restrict the domain to Pic^{0}(X). Let X be the elliptic curve (over ℚ) given by the Weierstrass equation

X : y^{2} = x^{3} + ax + b.

1This is in fact an easy consequence of the weak approximation theorem for function fields. To see this suppose that ∑ m_{j}Q_{j} is a divisor and that we want to find an equivalent divisor avoiding the finite set S. By the weak approximation theorem (see [9, page 11]) there exists a function h ∈ k(X) such that ord_{Q_{j}} (h) = m_{j} for all j and ord_{Q}(h) = 0 for any Q ∈ S that is not one of the Q_{j}. Then −div(h) + ∑ m_{j}Q_{j} avoids S and is equivalent to ∑ m_{j}Q_{j}.

We denote the point at infinity by O, and take f = x. Theorem 1 defines a homomorphism ϕ_{x} : Pic(X) → ℚ^{∗}/G_{x}(ℚ). Now the map X(ℚ) → Pic^{0}(X) given by P ↦ [P − O] is an isomorphism (where X(ℚ) has the usual group law). Thus composing with ϕ_{x} we obtain a homomorphism

ψ : X(ℚ) → ℚ^{∗}/G_{x}(ℚ)

given by

ψ(P) = x(P)G_{x}(ℚ)

for P ≠ O and x(P) ≠ 0 (it is not hard to show that ϕ_{x}([O]) = 1 · G_{x}(ℚ)). It remains to compute G_{x}(ℚ), and there are three cases: Case 1: b = 0. The divisor of x is just 2(0, 0)−2O. Both points in the support of the divisor have residue field ℚ. Hence G_{x}(ℚ) = ℚ^{∗}^{2}, and the map ψ : X(ℚ) → ℚ^{∗}/ℚ^{∗}^{2} is the usual one arising from the descent via 2-isogeny. Case 2: b ≠ 0, b ∈ ℚ^{∗}^{2}, say b = c^{2}. The divisor of x is (0, c) + (0, −c) − 2O. The residue fields are all ℚ again, but this time G_{x}(ℚ) = ℚ^{∗} and so the map ψ = 1. Case3: b ≠ 0, b ∉ ℚ^{∗}^{2}. The divisor of x is (0, \sqrt{b}) + (0, −\sqrt{b}) − 2O. We find that G_{x}(ℚ) = Norm(ℚ(\sqrt{b})^{∗})ℚ^{∗}^{−2} = Norm(ℚ(\sqrt{b})^{∗}). Hence we obtain a homomorphism

ψ : X(ℚ) → ℚ^{∗}/ Norm(ℚ(\sqrt{b})^{∗})

given by

ψ(P) = \{ \begin{matrix}x(P) Norm(ℚ(\sqrt{b})^{∗}) & if P ≠ O, \\ 1 · Norm(ℚ(\sqrt{b})^{∗}) & if P = O. \end{matrix}

Acknowledgment. I would like to thank J. Cremona, B. Poonen, E. Schaefer, A. Skorobogatov, and J. Top for useful discussions during the course of writing this paper. I am deeply indebted to Professor J.-L. Colliot-Thélène for detailed criticisms of a previous version of this paper, and for drawing to my attention [1].

References

[1] J.-L. Colliot-Thélène and J.-J. Sansuc La descente sur les variétés rationnelles, II, Duke J. Math. 54 (1987), 375–492.

[2] R. Hartshorne, Algebraic Geometry, Springer-Verlag, 1977.

[3] S. Lang, Abelian varieties, Interscience Publishers, New York, 1959.

[4] S. Lichtenbaum, Duality theorems for curves over P -adic Fields, Invent. math. 7 (1969), 120–136.

[5] B. Poonen, E.F. Schaefer, Explicit descent for the Jacobians of cyclic covers of the projective line, J. reine angew. Math. 488 (1997), 141–188.

[6] E.F. Schaefer, Computing a Selmer group of a Jacobian using functions on the curve, Math. Ann. 310 (1998), 447-471.

[7] J.-P. Serre, Algebraic groups and class fields, Springer-Verlag, New York, 1988.

[8] J. H. Silverman, The Arithmetic of Elliptic Curves, GTM 106, Springer-Verlag, 1986.

[9] H. Stichtenoth, Algebraic Function Fields and Codes, Springer-Verlag, 1993.

Department of Mathematics, Faculty of Science, Sultan Qaboos University, P.O. Box 36, Al-Khod 123, Oman

   E-mail address: siksek@squ.edu.om