A geometric construction of S-units in number fields

Ramesh Sreekantan Tata Institute of Fundamental Research

May 30, 2002

Abstract Special values of certain transcendental functions, like the trignometric functions, elliptic functions and modular functions are known to give units in number fields. In this note we show how some of these units are special cases of a general construction using the higher chow groups of curves. In general, we do not obtain units, but S-units for some set of primes determined by geometry.

1 Introduction

The values of certain special functions evaluated at special points often lie in certain special number fields. For example, the values of the function (1 − e^{z}) at rational multiples of 2πi lie in the cyclotomic fields, and the values of the j function on the upper half plane at imaginary quadratic points τ lie in the ring class fields of the imaginary quadratic extension. In some instances it is known that these values are actually units and those have other applications - for example in verifying Stark’s conjectures [Gr].

The aim of this note is to show that many of these examples can be viewed as special cases of a general geometric construction arising from the theory of higher chow groups of curves. This also suggests how to find other such examples as there are only isolated cases in which curves of genus > 1 have being used to construct units [Gr],[dS-Go].

In general, however, we do not get units, but S-units for some set of primes determined by the geometry of the curves. It seems to be a happy accident that the classical constructions actually give units - in those cases the varieties involved are rather special, like elliptic curves with complex multiplication, which happen to have potential good reduction.

2 The basic idea

The basic idea is to use the observation that certain higher chow groups of varieties depend only on the field of definition of the varieties. Therefore a geometric construction of elements of the these groups could lead to some interesting elements in the field of definition. We exploit this in the case of a higher chow group of curves.

Precisely, if C is a smooth projective curve over a number field K, then there is a push-forward map

π_{∗} : CH^{2}(C, 1) → K^{∗}

induced by the map π : C → Spec(K) which further induces a map

π_{∗} : CH^{2}(C, 1) → R^{∗}

where C is a model of C over Spec(R) where R is a subring. Usually R is the ring of integers O_{K} or O_{K}[S] for some set of primes S.

If K = K̄, then the map π_{∗} can be described as follows. An element of CH^{2}(C, 1) is represented as a formal sum

∑_{i}(x_{i}, u_{i})

where x_{i} are points on C and u_{i} are in K^{∗}. This map is simply

π_{∗}(∑_{i}(x_{i}, u_{i})) = ∏_{i} u_{i}

The point is that if there is some geometric way of constructing elements of the higher chow group, we could push the elements down to get some interesting elements of K^{∗}. Further, we could hope to understand the obstruction to the element being in R^{∗} from the geometry.

In general, there is no non obvious way of constructing elements of CH^{2}(C, 1), but there are some natural constructions of elements in CH^{2}(V, 1) for certain types of surfaces V on which C lies or maps to. We then pull-back those elements to get our elements. This procedure seems quite simplistic, but it appears to give some interesting results.

3 The group CH^{2}(V, 1)

In this section we describe the higher chow group CH^{2}(V, 1) where V is a surface over a number field K.

The group CH^{2}(V, 1) was introduced by Bloch in [Bl]. It has the following geometric description. An element of CH^{2}(V, 1) is represented by a sum

∑_{i}(C_{i}, f_{i})

where C_{i} are curves on V and f_{i} are functions on C_{i} such that

∑ div(f_{i}) = 0

A trivial example of such an is (C, a) where C is any curve on V and a is a constant.

Relations in this group are given by the Tame Symbols of functions. Suppose Z is a curve on V and f and g are two functions. The tame symbol is defined by

T_{Z}(f, g) = (−1)^{ord_{g}(Z)ord_{f} (Z)} \frac{f^{ord_{g}(Z)}}{g^{ord_{f} (Z)}}

and terms of the form

∑_{Z}(Z, T_{Z}(f, g))

are said to be 0 in the CH^{2}(V, 1).

3.1 The construction

In the subsequent sections we specialise to the case when V = X × X, where X is a certain type of curve over a number field K. We give a construction of elements of CH^{2}(X × X, 1) which is due in general to Beilinson [Be], though the first case was due to Bloch [Bl]. The idea is to remove some points from the closed curve so that one has some non-trivial functions on the open curve and then use them to construct non-trivial elements in the higher chow groups of the product without any points removed. It is these functions whose values give rise to interesting elements.

Let K be a field. Let X = X_{K} be a curve over K and S a set of K-rational points on X such that any divisor of degree 0 supported on S is torsion. Let Y = X\S

Then we have a localisation sequence,

0 → CH^{1}(X, 1) → CH^{1}(Y, 1) →^{∂} (⨁_{s∈S} ℤ)_{0} → 0  (3.1)

where (⨁_{s∈S} ℤ)_{0} is the hyperplane of elements of ℤ^{#(S)} which add up to 0.

From Bloch [Bl], the groups CH^{1}(X, 1) and CH^{1}(Y, 1) are O^{∗}(X) and O^{∗}(Y) respectively and the map ∂ is simply taking the divisor of the function in O^{∗}(Y).

For a divisor D of degree 0 supported on S, the cycle class is given by the extension class of the sequence

0 → CH^{1}(X, 1) → 𝔼 → ℤ → 0  (3.2)

(c.f [Ja]§9) where 𝔼 is the pull-back of D. Saying D is torsion is equivalent to saying that this splits rationally as an exact sequence of motivic cohomology groups.

Since any divisor of degree 0 supported on S is torsion, we have that the exact sequence 3.1 splits rationally as a sequence of motivic cohomology groups. Therefore there is a map

E : (⨁_{s∈S} ℤ)_{0} ⊗ ℚ → O^{∗}(Y) ⊗ ℚ  (3.3)

In other words, given a divisor D of degree 0 on S, there is a function on X whose divisor is D. We denote this function by E(D).

3.2 Main Lemma

More generally, we have the following:

Lemma 3.1. The sequence

0 → CH^{2}(X^{2}, 1)_{ℚ} → CH^{2}(X^{2}\S^{2}, 1)_{ℚ} → ((⊕_{s∈S}ℤ)^{2})_{0} ⊗ ℚ → 0  (3.4)

is a split exact sequence of motivic cohomology groups.

Proof. The idea is to use the splitting of the sequence 3.1 to split it in general. We have to produce an element of CH^{2}(X^{2}\S^{2}, 1)_{ℚ} for any 0-cycle of degree 0 on S^{2}.

Let (P_{1}, P_{2}) − (Q_{1}, Q_{2}) be a 0-cycle of degree 0 on S^{2}. It suffices to produce a lift of such an element. We do the usual trick

(P_{1}, P_{2}) − (Q_{1}, Q_{2}) = (P_{1}, P_{2}) − (Q_{1}, P_{2}) + (Q_{1}, P_{2}) − (Q_{1}, Q_{2})

For the pair (P_{1}, P_{2}) − (Q_{1}, P_{2}) we take the element

(X × (P_{2}), E(P_{1} − Q_{1}))

in CH^{2}(X^{2}\S^{2}, 1)_{ℚ}. Similarly we can choose and element for any pair which differ only in one place. Since we can write any 0 cycle as a sum of such 0 cycles, for any 0 cycle of degree 0 D supported on S^{2} we get an element in CH^{2}(X^{2}\S^{2}, 1)_{ℚ}. We denote this element by E^{2}(D).

A different representation of the 0 cycle in terms of the simple 0-cycles differs from this by an element which is trivial in CH^{2}(X^{2}\S^{2}, 1)_{ℚ} as it comes from the tame symbol.

3.3 Another construction

On the other hand, we have other ways of constructing elements in CH^{2}(X^{2}\S^{2}, 1)_{ℚ}. Suppose we have an embedding

Φ : X ↪ X^{2}

such that

• Φ : Y ↪ X^{2}\S^{2}

• Φ(S) ⊆ S^{2}

We have an induced proper pushforward

Φ_{∗} : CH^{1}(Y, 1) → CH^{2}(X^{2}\S^{2}, 1)

so if D is a divisor of degree 0 supported on S we get an element

Φ_{∗}(E(D)) ∈ CH^{2}(X^{2}\S^{2}, 1)_{ℚ}

The simplest example of such a Φ is the diagonal map,

Δ : Y → X^{2}\S^{2}
Δ(y) = (y, y)

Now, let D be a 0-cycle of degree 0 on X supported on S. We can look at the two elements Φ_{∗}(E(D)) and E^{2}(Φ(D)) of CH^{2}(X^{2}\S^{2}, 1). Since

∂(Φ)_{∗}(E(D)) = ∂(E^{2}(Φ(D)) = Φ(D)

the element

Ψ(D) = Φ_{∗}(E(D)) − E^{2}(Φ(D))

has no boundary and hence is an element of CH^{2}(X^{2}, 1) !

3.4 Eisenstein Series

The above construction asserts that given a divisor D of degree 0 supported on S, one can find a function E(D) whose divisor is D. A priori this is only determined up to an element of K^{∗}. However, for our purposes, we need it on the nose. In order to compute an explicit splitting we can use real analytic Eisenstein series as in [Sc3]. One can write down a series E_{D}(τ, s) whose value at s = 1 is log |E_{0}(D)| for some explicit function E_{0}(D) whose divisor is D.

However, in all the examples below there are some obvious functions whose divisor is are supported on S. In the case of modular curves, the Kronecker Limit Formula shows that the functions we write down and the Eisenstein series construction agree.

3.5 Decomposable Elements

If X = X_{K} is a variety over a field K, there is a map

CH^{1}(X_{K}, 1) ⊗ CH^{1}(X_{K}) → CH^{2}(X_{K}, 1)  (3.5)

induced by the product structure on higher chow groups. By a theorem of Bloch [Bl] the group CH^{1}(X_{K}, 1) is simply K^{∗}. Elements in CH^{2}(X_{K}, 1) which lie in the image of this map are said to be decomposable over K. The group is said to be decomposable if every element is decomposable over some extension L of K.

If X is a curve over K, the group CH^{2}(X, 1) is decomposable. This can also be stated as the map

⊕N_{L/K} : ⨁_{L⊇K} CH^{1}(X_{L}) ⊗ CH^{1}(C_{L}, 1) → CH^{2}(X, 1)  (3.6)

is surjective, where N_{L/K} denotes the norm map and L runs thorough all finite extensions of K.

Typically, our elements are not decomposable over K but are decomposable over some explicit extension L.

4 Units from all this

Suppose C is a codimension 1 cycle defined over a field K on the surface X. Then if ξ is an element of CH^{2}(X, 1) one can consider

U_{C}(ξ) := π_{∗}(i_{C}^{∗}(ξ))  (4.1)

where π : C → Spec(K) is the natural map and i_{C}^{∗} is the restriction. Then U_{C}(ξ) ∈ K^{∗}.

If ξ is decomposable over K itself, so it is of the form (D, u) for some cycle D and unit u in K^{∗}, then U_{C}(ξ) = u^{n}, where n is the interesection number of C and D.

To decide whether it is a unit or not, we have the following theorem, which determines the valuation at a prime P.

Theorem 4.1. If C is a curve on X and ξ is an element of CH^{2}(X, 1), then, for a prime P in Spec(K),

ord_{P} (U_{C}(ξ)) = #(∂_{P} (ξ) ∩ C_{P})  (4.2)

Proof. We have the following diagram with exact rows of localisation sequences which relate the chow groups of the models with chow groups of the general fibre and special fibres:

⟶ CH^{2}(X , 1) ⟶ CH^{2}(X, 1) ⟶^{⊕∂_{P}} ⨁_{P} CH^{1}(X_{P}) ⟶
↓^{i_{C}^{∗}}  ↓^{i_{C}^{∗}}  ↓^{⊕i_{C_{P}}^{∗}}
⟶ CH^{2}(C, 1) ⟶ CH^{2}(C, 1) ⟶^{⊕∂_{P}} ⨁_{P} CH^{1}(C_{P}) ⟶
↓^{π_{∗}}  ↓^{π_{∗}}  ↓^{π_{∗}}
⟶  R^{∗}  ⟶  K^{∗}  ⟶^{⊕ord_{P}}  ⨁_{P} ℤ

The sequences show that

ord_{P} (U_{C}(ξ)) = #(∂_{P} (ξ) ∩ C_{P})

In particular, U_{C}(ξ) is a unit at all those places where this is 0.

While in principle this allows one to compute the prime factorisation of the element, in practice it is not so easy. For example, to compute the boundary ∂_{P} one has to have a good understanding of the special fibre at the primes of bad reduction as well as understand the distribution of the horizontal sections coming from the divisor of the functions involved and then one has to be able to compute its intersection number with the special fibre of the curve. However, if the curve has everywhere good reduction then a suitable choice of the functions with the same divisor will give a unit.

The field K in which the elements lie is the compositum of the field of definition of the element ξ and the curve C. In general, the element i_{C}^{∗}(ξ) is not decomposable over K but is decomposable over some extension L of K, so U_{C}(ξ) is made up of functions evaluated at divisors made up of Galois conjugate points. As the points are Galois conjugate, one might expect a reciprocity law which relates the Galois action on the value of the function to the value of the function at Galois conjugate points. In some examples this is obvious or is a special case of the Shimura reciprocity law. In these cases, U_{C}(ξ) lies in N_{K}^{L}(O_{L}^{∗}[S]) ⊂ O_{K}^{∗}[S] for some extension L of K and some set of primes S of K. The componenents of U_{C}(ξ) are therefore also S-units.

5 Examples

The construction above requires two things, a curve X with a set S and a curve C on X × X to pullback the element ξ to. Examples of the curves X are Fermat curves, elliptic curves and modular curves which give rise to classical constructions. But essentially these are special cases of curves with a lot of automorphisms, whose graphs give rise to curves C and whose fixed points give examples of sets of the type S and our construction also works there.

5.1 Fermat Curves and Cyclotomic Units

Let p be an odd prime. We can apply the construction above to construct elements of CH^{2}(F_{p} × F_{p}, 1) where F_{p} is the p^{th} Fermat Curve

F_{p} : X^{p} + Y^{p} + Z^{p} = 0

considered as a curve over the cyclotomic field ℚ(ζ_{p}). Here the ‘cusps’, namely the 3p trivial solutions

a_{j} = [0, ϵζ^{j}, 1]
b_{j} = [ϵζ^{j}, 0, 1]
c_{j} = [1, ϵζ^{j}, 0]

where ϵ is a fixed 2p^{th} root of unity give a set of the type S, as shown by Rohrlich [Roh]. Using this in our construction, we recover some cyclotomic units. For example, if f is the function

f = \frac{Y − ϵZ}{Y − ϵX}  (5.1)

then one can see that it has divisor pa_{0} − pc_{0}. Using this function in our construction, with Φ being the diagonal, we can construct an element ξ of CH^{2}(F_{p} × F_{p}, 1) of the type

(F_{p} × a_{0}, f × a_{0}) + (Δ, f^{−1}) + (c_{0} × F_{p}, c_{0} × f)

There is an action of the group (ℤ/pℤ)^{2} on the Fermat curve by

ϕ_{abc}([X, Y, Z]) = [ζ^{a}X, ζ^{b}Y, ζ^{c}Z]

Consider the graph Γ_{abc} of ϕ_{abc} in F_{p} × F_{p}. We can pull back the element to CH^{2}(Γ_{abc}, 1). Γ_{abc} intersects the vertical and horizontal curves in the support at the points

(ϕ_{abc}^{−1}(a_{0}), a_{0}) = ((0, ζ^{−b}ϵ, ζ^{−c}), (0, ϵ, 1)) = (a_{c−b}, a_{0})

and

(c_{0}, ϕ_{abc}(c_{0})) = ((1, ϵ, 0), (ζ^{a}, ζ^{b}ϵ, 0)) = (c_{0}, c_{b−a})

With the diagonal, there are more cases as Case 1:If at least two of (a, b, c) are non-zero. In this case the graph does not intersect the diagonal so there are no points to consider. The resulting element of ℚ(ζ)^{∗} is

U_{Γ_{abc}}(ξ) = f(a_{c−b})f(c_{b−a}) = \frac{ζ^{c−b} − 1}{ζ^{c−b}} · \frac{ζ^{b−a}}{ζ^{b−a} − 1} = ζ^{2b−(a+c)} \frac{ζ^{c−b} − 1}{ζ^{b−a} − 1}  (5.2)

Case 2: If two of (a, b, c) are 0 and the third is not. In this case the graph does intersect the diagonal. There are two subcases.

• If b ≠ 0, so a = c = 0. In this case the fixed points on the diagonal are the points b_{r}, 0 ≤ r ≤ p−1. Therefore the resulting element is

U_{Γ_{0b0}}(ξ) = f(a_{−b})f(c_{b}) ∏_{r=0}^{p−1} \frac{1}{f(b_{r})} = \frac{ζ^{−b} − 1}{ζ^{−b}} · \frac{ζ^{b}}{ζ^{b} − 1} · ∏_{r=0}^{p−1} \frac{1}{ϵζ^{r}} = −ζ^{2b} \frac{ζ^{−b} − 1}{ζ^{b} − 1} = −ζ^{3b} (5.3)

• If b = 0. In this either a or c is also zero and the third is not. Suppose a = 0 and c ≠ 0. Then the fixed points on the diagonal are c_{r}, 0 ≤ r ≤ p − 1. The point (c_{0}, c_{0}) lies on two components, the diagonal and the horizontal, but the pole at one cancels the zero at the other. The resulting element of ℚ(ζ)^{∗} is

U_{Γ_{00c}}(ξ) = f(a_{c}) ∏_{r=1}^{p−1} f(c_{r}) = \frac{ζ^{c} − 1}{ζ^{c}} · ∏_{r=1}^{p−1} \frac{ζ^{r}}{ζ^{r} − 1} = \frac{ζ^{c} − 1}{ζ^{c}p}  (5.4)

By symmetry, if c = 0 and a is not, the resulting element is

U_{Γ_{a00}}(ξ) = ∏_{r=1}^{p−1} f(a_{r})f(c_{−a}) = \frac{pζ^{−a}}{1 − ζ^{−a}}  (5.5)

Both of these elements are clearly not units, but are ramified only at p. Since the Fermat curve F_{p} has bad reduction at p this is not surprising.

5.2 Elliptic Curves

If E is an elliptic curve defined over a number field K given by the the equation

E : Y^{2} = X(X − 1)(X − λ)  (5.6)

then, if P_{i}, 1 ≤ i ≤ 4 are the four 2-torsion points, there is a function f on E whose divisor is

div(f) = 2(P_{1}) − 2(P_{4})  (5.7)

and f(P_{2}) = 1, so f(P_{3}) = λ. This function is given by

f((X, Y)) = X  (5.8)

We can construct the element ξ of CH^{2}(E × E, 1) by the procedure above and then pull it back to CH^{2}(Δ^{−}, 1) where Δ^{−} is the anti-diagonal given by X → (X, −X).

The support of ξ is P_{1} × E,E × P_{4} and the diagonal Δ. Δ^{−} intersects P_{1} × E at (P_{1}, P_{1}) and E ×P_{4} at (P_{4}, P_{4}). It intersects the diagonal at the four 2-torsion points (P_{i}, P_{i}). Therefore the resulting element of K^{∗} is

U_{Δ^{−}}(ξ) = f(P_{1}) ∏_{i=1}^{4} \frac{1}{f(P_{i})}f(P_{4}) = \frac{1}{f(P_{2})f(P_{3})} = \frac{1}{λ}  (5.9)

If E is a CM elliptic curve so it has potential good reduction everywhere, U_{Δ^{−}}(ξ) is known to be a unit and is an example of an elliptic unit. This will always be a unit outside the primes dividing the discriminant. For primes dividing the discriminant, the situtation depends on the boundary of ξ - if P_{1} and P_{4} specialise to the same component of the special fibre, there is no boundary, but if they lie on different fibres, there can be a boundary. If P_{3} and P_{1} come together modulo a prime, then there will be a boundary, as when one resolves the singularity the points P_{1} and P_{4} will lie on different components. This can be translated in to saying λ ≡ 0 mod P. If P_{3} and P_{2} come together, there is no boundary, as both P_{1} and P_{4} will not be involved in any blowing up and so will lie on the same component. Indeed, this is the same as saying λ ≡ 1 mod P.

More generally, one can consider the following. If ϕ ∈ End(E) is an isogeny, then one can consider the restriction of ξ to the graph of ϕ,

Γ^{ϕ} := (z, ϕ(z)) ⊂ E × E {z ∈ E}

Then

• Γ^{ϕ} ∩ P_{1} × E = (P_{1}, ϕ(P_{1}))

• Γ^{ϕ} ∩ E × P_{4} = {(z, P_{4})| z ∈ ϕ^{−1}(P_{4})} = E[ϕ] + b, ϕ(b) = P_{4}. Here E[k] denotes the k-torsion.

• Γ^{ϕ} ∩ Δ = E[ϕ − 1]

Assuming P_{4} is the origin of the elliptic curve, the result depends on ϕ(P_{1}). If ϕ ≡ 1 mod 2 then Φ(P_{1}) = P_{4}, then the point (P_{1}, P_{4}) on P_{1} × E cancels with the point (P_{1}, P_{4}) lying in the ϕ torsion on E × P_{4}. The point (P_{4}, P_{4}) in the ϕ torsion cancels off with the (P_{4}, P_{4}) appearing with opposite sign in the ϕ − 1 torsion. Therefore, in this case the resulting element is

U_{Γ_{ϕ}}(ξ) = ∏_{z∈E[ϕ]′} f(z) ∏_{z∈E[ϕ−1]′′} \frac{1}{f(z)}  (5.10)

where the ′ indicates omitting the point (P_{4}, P_{4}) and the ′′ indictates omitting the two points (P_{4}, P_{4}) and (P_{1}, P_{4}).A similar calculation can be done in the case when ϕ(P_{1}) ≠ P_{4}. This element U_{Γ_{ϕ}}(ξ) is a unit at all places not dividing ϕ and ord_{ϕ}(U_{Γ_{ϕ}}(ξ)) = −2 as the function has a pole of order 2 at all primes dividing ϕ since the ϕ torsion collapses to a point.

If one normalises the function f differently, this construction gives the construction in [C-N,T], pg 68, which is an analogue of the statement in the p^{th}-cyclotomic fields that

∏_{a mod p}(1 − exp^{\frac{2πia}{p}}) = p

5.3 Hyperelliptic Curves

One can make a similar construction on the self product of a hyperelliptic curve C of genus g with S being the set of 2g + 2 ramification points. Let K be a field over which the curve C as well as the 2g + 2 points are defined.

We can assume that the curve is given by the equation

C : Y^{2} = X(X − 1)(X − λ_{1})...(X − λ_{2g+1})  (5.11)

If P_{i}, i ≤ i ≤ 2g + 2 are the points, with P_{2g+2} = ∞, then the function

f : (X, Y) → X

has the following properties:

• It is defined over K

• div(f) = 2(P_{1}) − 2(P_{2g+2})

• f(P_{2}) = 1

Using this function and the construction above, we get an element of CH^{2}(C × C, 1), supported on the curves C × P_{1}, P_{2g+2} × C and the diagonal. If ι is the hyperelliptic involution, one also has the curve C_{ι} = {(x, ι(x))|x ∈ C}, and one can pull back the element of CH^{2}(C × C, 1) to this curve.

The resulting element of K^{∗} is

U_{C_{ι}}(ξ) = f(P_{1})f(P_{2g+2}) ∏_{i=1}^{2g+2} \frac{1}{f(P_{i})} = ∏_{i=2}^{2g+1} \frac{1}{f(P_{i})}  (5.12)

As the function f can be expressed in terms of theta functions [Fa-Kr], this number also can, like the elliptic λ function.

There is a hyperelliptic discriminant Δ(C), the primes dividing which are the primes where the curve has bad reduction. This element will be a unit outside these primes. For these primes one has to compute the boundary and apply 4.2.

5.4 Curves of the form Y^{N} = XF (X)

Suppose C is a curve of the form

Y^{N} = XF (X)

where F is a polynomial of degree k. Let P_{i} be the points (a_{i}, 0) where a_{i} are the roots of F with P_{0}(0, 0) and P_{k+1} = (∞, 0) then one can consider the function

g(X, Y) = X

which has divisor N(P_{0}) − N(P_{k+1}). Using this we can construct, as before, an element ξ in CH^{2}(C × C, 1).

There is an automorphism of C given by

σ((X, Y)) = (X, ζ_{N}Y)

where ζ_{N} is an N^{th} root of unity. The graph of σ, C_{σ} gives a curve on C × C to which we can restrict the element ξ.

The points of interesection of C_{σ} with the diagonal, namely the fixed points, are precisely the points P_{i} for 1 ≤ i ≤ k and the resulting element

U_{C_{σ}}(ξ) = ∏_{i=1}^{k} \frac{1}{g(P_{i})} = ∏_{i=1}^{k} \frac{1}{a_{i}} = A_{0}/A_{k}

where F (X) = A_{k}X^{k} + .... + A_{0}. If A_{0}/A_{k} is a unit, then so are the g(P_{i}). This is of course obvious but can also be seen from our point of view as follows:

Whether it is a unit or not depends on the primes of bad reduction of C and the distribution of P_{0} and P_{k+1} on the special fibres. The bad fibres are those coming from the primes for which XF (X) has multiple roots mod p. This is equivalent to saying that either F (X) has a multiple root mod a prime p or a_{1} ≡ 0 mod p. The second can be stated as p divides A_{0}/A_{k} = ∏_{i} a_{i}. So if that is a unit, that case does not arise. Therefore, a prime of bad reduction is a prime dividing disc(F) or equivalently where at least two of the P_{i}, 1 ≤ i ≤ k come together mod p. In this case, the horizontal components P_{0} and P_{k+1} intersect the same component of the special fibre, so ∂_{p}(ξ) = 0.

All these are special cases of the following. If X is a curve with an action by a group G, then if g ∈ G and x and y are two fixed points, then x−y is torsion in the Jacobian of X, so there is a function f_{g} with div(f_{g}) = Nx − Ny. Using this, we can construct an element ξ_{g} in CH^{2}(C × C, 1), pull back to curves of the form Γ_{h}, the graph of an element h of G. The resulting element is composed of the function f_{g} evaluated at fixed points of h as well as hx and hy. In the examples above, one knew a little more about the fields of definition of C and the G action and the fixed points.

5.5 Modular Curves and Modular Units

Modular curves provide some interesting examples. The Manin-Drinfel’d theorem asserts precisely that any divisor of degree 0 on the cusps is torsion, hence the set of cusps is a set of the type S. Using this we can construct elements of CH^{2}(X_{0}(N) × X_{0}(N), 1). The functions involved are called modular units and can often can be described explicitly in terms of the Dedekind η function. If the cusps involved are defined over ℚ, the resulting element will also be defined over ℚ ( On X_{0}(N) with N squarefree, this is always the case ).

If the modular curve X_{0}(N) is hyperelliptic, which it is for some finite set of numbers N, the fixed points of the involution give another example of the set S. If N > 4 this set is distinct from the set of cusps and in fact can be described as a certain set of CM points. We can use this to construct other elements. The fact that it is modular allows us to make the functions quite explicit. In general, this element is not defined over ℚ as the functions involved are not.

On the self product of a modular curve, one way of finding interesting curves is to look at the Hecke correspondences, namely the graph of Hecke operators. The units that arise by restricting elements of CH^{2}(X_{0}(N) × X_{0}(N), 1) to these curves are the some of the classical modular units - units in ring class fields of imaginary quadratic fields.

In some circumstances, there are also certain curves which are not defined over ℚ, but defined over a cyclotomic fields. These are the components of the so called ‘Twisting correspondences’, and one can use them as well.

5.6 Hecke Operators

Flach [Fl] constructed elements of CH^{2}(X_{0}(N) × X_{0}(N), 1) supported on the graphs of T_{p} for p not dividing N and some vertical and horizontal components ( similar elements were also used by Mildenhall [Mi])

Let g be a modular form of level N and even weight k whose divisor is supported on the cusps. Following Flach [Fl] one can construct and element of CH^{2}(X_{0}(N) × X_{0}(N), 1) as follows. First, we think of g as a form on X_{0}(Np). Let w_{p} be the Atkin-Lerner involution on X_{0}(Np). Let

u_{g} = p^{−k/2} \frac{w_{p}(g)}{g}

This is a function on X_{0}(Np).If

div(g) = ∑_{c cusp of X_{0}(N)} n_{c}c

then

div(u_{g}) = ∑_{c cusp of X_{0}(N)} n_{c}(p − 1)(c − w_{p}(c))

where now w_{p}(c) is a cusp on X_{0}(Np). Using our construction, one can find functions u_{c} and a number l such that

ξ_{p} = ∑_{c cusp of X_{0}(N)}(X_{0}(N) × c, u_{c}^{−1}) + (c × X_{0}(N), u_{c}^{−1}) + (T_{p}, u_{g}^{l})

gives an element of CH^{2}(X_{0}(N) × X_{(}N), 1). By the computations of Flach, if g has weight k,

∂_{p}(ξ_{p}) = \frac{k}{2} l(Z_{1} − Z_{2})

where Z_{1} and Z_{2} are the components of the special fibre of X_{0}(Np) mod p, and there is no boundary at any other primes not dividing N. The field of definition of this element is the field of definition of g. Let K be this field.

There are several curves to which one can restrict this element. Restricting to the diagonal T_{1}, for example, one gets

U_{T_{1}}(ξ_{p}) = ∏_{z∈T_{1}∩T_{p}} u_{g}(z)^{l}

The interesection points T_{p} ∩ T_{1} are well understood: They are the so called Heegner points for certain discriminants D [G-K],[Hu]. Precisely, the set of points are points corresponding to elliptic curves with CM by an order of discriminant D = 4p−t^{2}, where t^{2} < 4p. Let H_{D} be the ring class field of dicriminant D and L_{D} the compositum of H_{D} and K.

The points are defined over ring class fields of H_{D} but this intersection is defined over ℚ, so the set is Galois invariant. The Shimura reciprocity law then says that Galois conjugates of u_{g}(z) for some point z is the values of u_{g} at at some galois conjugate point z′. Therefore, one has

U_{T_{1}}(ξ_{p}) = ∏_{D} N_{K}^{L_{D}}u_{g}^{l}(z_{D})

where z_{D} is a CM point of discriminant D. As U_{T_{1}}(ξ_{p}) is a unit outside p and N, so are the values u_{g}^{l}(z_{D}).

For different choices of g, one gets different units. And for different choices of curves, one gets S units in different fields. One choice of g is the classical Δ function. u_{g} is then essentially p^{−6} \frac{Δ(pz)}{Δ(z)}.

5.6.1 Hyperelliptic Modular Curves

There are a finite set of N for which X_{0}(N) is hyperelliptic [Ogg]. In almost all these examples, the involution is given by an Atkin-Lerner involution w = w_{N′} for some N′ dividing N. The fixed points are CM points with CM by an order in ℚ(\sqrt{N′}) where N′ is a factor of N [Ogg]. Such an order is either of discriminant −N′ or −4N′ so the points are defined over the ring class field of these orders, namely ℚ(\sqrt{N′}, j(O_{−4N′})) or a subfield of that. Let H_{N′} denote this field of definition.

Let P_{i}, 1 ≤ i ≤ 2g + 2 be the fixed points. To carry out the construction, we need an explicit function h whose divisor is, say, 2P_{1} −2P_{2}. To do this, we first use the theorem of Rovira [Ro] which gives function f whose divisor is of the form div(f) = P + w(P) − ∞ − w(∞) for some cusp P. This function can be explicitly expressed as a product of η functions.

To determine h explicitly, let λ_{i} = f(P_{i}). These points lie in H_{N′} as f is defined over ℚ. There is a unique matrix M in P SL_{2}(ℂ) which takes the points (λ_{1}, λ_{2}, λ_{3}) to (0, ∞, 1). It is explicitly given by

M = (\begin{matrix}α & β \\ γ & δ \end{matrix})

where

α = λ_{3} − λ_{2}
β = −λ_{1}(λ_{3} − λ_{2})
γ = λ_{3} − λ_{1}
δ = −λ_{2}(λ_{3} − λ_{1})

Let h be given by

h = M(f) = \frac{αf + β}{γf + δ}  (5.13)

Then h is w-invariant as f is so div(h) = 2P_{1} − 2P_{2} and h(P_{3}) = 1. Further, since α, β, γ, δ lie in H_{N′}, h is defined over H_{N′}.

Using this h we can construct an element ξ in CH^{2}(X_{0}(N) × X_{0}(N), 1) supported on the curves X_{0}(N) × P_{1}, P_{2} × X_{0}(N) and the diagonal. This will be defined over H_{N′}.

There are several curves we can restrict our element to. One is the anti-diagonal X_{0}(N)_{w} as in the example of hyperelliptic curves above. This results in the element

U_{X_{0}(N)_{w}}(ξ) = \frac{1}{h(P_{4})h(P_{5})h(P_{6})...h(P_{2g+2})}

The values h(P_{i}) are the cross-ratios of the λ_{i} with λ_{i}, λ_{2}, λ_{3} namely

h(P_{i}) = \frac{(λ_{i} − λ_{1})(λ_{3} − λ_{2})}{(λ_{i} − λ_{2})(λ_{3} − λ_{1})}  (5.14)

5.6.2 Twisting Correspondences

Supposed χ is a character mod r for some integer r. Then if f is a modular form for Γ_{0}(N) with fourier coefficients ∑ a_{n}q^{n}, one can consider its twist by this character, which has fourier expansion f_{χ}(z) = ∑ χ(n)a_{n}q^{n}. This is a modular form of level Γ_{0}(Nr^{2}). Shimura [Sh] showed that this correspondence of modular forms can be realised as a geometric correspondence between the modular curves X_{0}(N) and X_{0}(Nr^{2}), which is defined over the fixed field of χ. Explicitly, for τ on X_{0}(N),

R_{χ} = ∑_{a mod r} χ(a)ϵ_{a}(τ)  (5.15)

where ϵ_{a} is the matrix

ϵ_{a} = (\begin{matrix}1 & \frac{a}{r} \\ 0 & 1 \end{matrix})  (5.16)

In some situations, for example when N is not squarefree, the twisting operator preserves the curve X_{0}(N). In such a case, we can can restrict an element of CH^{2}(X_{0}(N) × X_{0}(N), 1) to the graph of ϵ_{a}, Γ_{a}. This is defined over the r^{th} cyclotomic field. The resulting element lies in the compositum of the field of definition of the element ξ and the cyclotomic field.We will work out one example in the case of X_{0}(50)

5.6.3 The modular curve X_{0}(50)

As an example of our construction, we will compute the resuting units for the modular curve X_{0}(50). This has the advantage of being both a hyperelliptic curve as well as one which is preserved by a twisting correspondence. X_{0}(50) has genus 2 with 12 cusps. The cusps are equivalent to 0, ∞,±1/5, ±2/5, ±3/10, ±1/10,1/2 and 1/25.

We first work out the case of the a component of a twisting correspondence. Let χ be the quadratic character mod 5. The operator R_{χ} preserves X_{0}(N). It is defined over ℚ(\sqrt{5}) and induces multiplication by \sqrt{5} on the abelian surface J_{0}(50). All the preliminary calculations are due to Birch [Bi].

I.Suppose g is a function with a divisor of the form nP − nQ for some points P and Q. Let ξ be the element of CH^{2}(X_{0}(50) × X_{0}(50), 1) constructed using this function. It is supported on P × X_{0}(50), X_{0}(50) × Q and the diagonal.

Let Γ_{a} be the graph of ϵ_{a}. The points of intersection of Γ_{a} with the diagonal copy of X_{0}(50) are the cusps ∞ and 1/25. Note that these do not depend on a. The points of intersection with the horizontal and vertical components are (P, ϵ_{a}(P)) and (ϵ_{−a}(Q), Q). If P and Q are cusps, then ϵ_{a}(P) and ϵ_{−a}(Q) are cusps as well, if they are CM points, then they are CM points as well. The resulting element U_{Γ_{a}}(ξ) is

U_{Γ_{a}}(ξ) = g(ϵ_{−a}(Q))g(ϵ_{a}(P))  (5.17)

If div(g) is supported on the cusps, for example, if

g(z) = \frac{η(2z)^{2}η(25z)}{η(50z)^{2}η(z)}

so div(g) = 3(1/2) − 3(∞), it turns out the resulting element is

g(ϵ_{a}(1/25))

ϵ_{a}(1/25) ∼ ±(a/10), where a = ±1 or ±3. Then calculations [At] show

g(a/10) = \sqrt{5}ζ^{−a}

where a = ±1 or ±3.

Remark 5.1. There is a version of the reciprocity law which determines the action of the Galois group of the r^{th}-cyclotomic field on these values, namely values of modular units at cusps. If (a/r) represents a cusp, b is in (ℤ/rℤ)^{∗} and g is a modular unit, defined over ℚ, whose divisor is supported outside (a/r), then

σ_{b}(g(\frac{a}{r})) = g(\frac{ab^{−1}}{r})

where σ_{b} is the element of the Galois group corresponding to b.

II. Let f be the function

f(z) = \frac{η(2z)η(25z)}{η(50z)η(z)}

which has divisor div(f) = (1/25) + (1/2) − (0) − (∞). This gives a hyperelliptic function on X_{0}(50). There are six ramification points corresponding to the points with CM by O_{−200}, the order of conductor 5 in ℤ[\sqrt{−2}] The numbers f(P_{i}) = λ_{i} can be computed as the roots of the polynomial

P (X) = (X^{3} − 2X^{2} − 2X + 1)^{2} − 20X^{2}

as the equation of X_{0}(50) is given by Y^{2} = P (X) [Ro].

As they are the fixed points of the Atkin-Lerner involution w_{50}, they lie in the ring class field ℚ(\sqrt{−2}, j(O_{−200})). There are 20 choices of the functions h depending on which 3 roots one sends to 0, 1, ∞. For each h, one can construct an element ξ in CH^{2}(X_{0}(50)×X_{0}(50), 1). This is defined only over H_{200} and one can restrict it either to the curves Γ_{a}, or the Hecke operators or the anti-diagonal.

Restricting to Γ_{a} gives an element

U_{Γ_{a}}(ξ) = h(ϵ_{−a}(Q))h(ϵ_{a}(P))  (5.18)

where P and Q are points with CM by O_{−200} and so ϵ_{a}(P) has CM by the order of conductor 25. This lies in the field generated by H_{200} and the 5^{th} cyclotomic field.

III.The points of intersection of the cycles in the support of ξ with a Hecke operator T_{p} are certain Heegner points. The diagonal intersects with T_{p} at those CM points where there is an isogeny of degree p, namely the orders where p splits. Similarly, the curve X_{0}(50) × P intersects T_{p} at those points z where the corresponding elliptic curves are isogenous to that at P by an isogeny of degree p and Q × X_{0}(50) intersects T_{p} at those points corresponding to elliptic curves which are isogenous to that at Q.

Finally one can also restrict to the anti-diagonal and different choices of h give 20 different numbers which are S-units for S the set of primes lying over 2 and 5. Whether it is a unit or not depends on the configuration of the points P_{i} of the support of h on the special fibres of bad reduction. If they lie in the same component for all the special fibres, then the resulting element is an integral unit.

5.7 Twists of Elliptic Curves

If E is an elliptic curve over ℚ and E_{χ} is a real quadratic twist, then the Beilinson conjectures predict that the rank of the integral motivic cohomology group CH^{2}(E × E_{χ}, 1)_{ℤ} is 1 . There are two ways of constructing elements of this group.

The first is to use Beilinson’s elements on a modular parameterisation. If X_{0}(N) is a common modular parametrisation of the two curves, then one can construct an element of CH^{2}(X_{0}(N)×X_{0}(N), 1) by the above construction using modular units. Then one can push it forward using the modular parametrisation to CH^{2}(E × E_{χ}, 1). Since E and E_{χ} are non-isogenous, this element actually lifts to the ’integral’ chow group CH^{2}(E × E_{χ}, 1)_{ℤ}. This procedure works in general to give elements of the higher chow groups of the product of two non-isogenous elliptic curves [Sc2]. Let ξ_{1} be the an element constructed in this manner.

In this case, however, since the curves become isogenous over a real quadratic extension, it turns out that there is another decomposable element of the form N_{ℚ}^{ℚ(χ)}(C_{χ}, u_{χ}) where C_{χ} is the graph of the isomorphism between E and E_{χ} which is only defined over ℚ(χ) and u_{χ} is a certain unit in this real quadratic field. Let ξ_{2} be this element.

The dimension of CH^{2}(E × E_{χ}, 1)_{ℤ} is conjectured to be 1 so the two elements should differ by a rational number. In this case by looking at regulator, one can see that it is true.

Suppose now D is a curve on X_{0}(N) × X_{0}(N) and D′ is its image in E × E_{χ}. Then U_{D′}(ξ_{1}) and U_{D′}(ξ_{2}) differ by a rational number, or, in other words,

U_{D′}(ξ_{1})^{m} = U_{D′}(ξ_{2})^{n}

Since the support of ξ_{2} is C_{χ} and C_{χ}^{σ},

U_{D′}(ξ_{2}) = u_{χ}^{a_{1}−a_{2}}

where a_{1} = #(D′ ∩ C_{χ}) and a_{2} = #(D′ ∩ C_{χ}^{σ}) as u_{χ}^{σ} = u_{χ}^{−1}.

On the other hand, the computation for ξ_{1} is more interesting. If D is a special curve like the composite of a Hecke correspondence with a twisting correspondence, the intersection of D with the support of the element of CH^{2}(X_{0}(N)×X_{0}(N), 1) is at certain CM points and the functions are certain modular units. Therefore, the resulting element is the pushforward of some modular units evaluated at CM points. On the other hand, since D it is not defined over ℚ the interesection with ξ_{2} gives a non-trivial power of u_{χ}. This gives a geometric form of Kroneckers solution to Pell’s equation [We] . In some sense there is nothing new, as the Limit Formula has been used implicitly.

6 Regulators and values of L-functions

Beilinson’s general conjectures [Be] relate the values of L-functions of algebraic verieties at special points with the determinants of generalized regulator maps.

In the case of a curve defined over a number field K, there is a regulator map

reg : CH^{2}(C, 1) ⊕ NS(C) → H_{D}^{3}(C_{/ℝ}, ℝ(2))

The target space, H_{D}^{3}(C_{/ℝ}, ℝ(2)) is a Real Deligne colohomology group which is a real vector space of dimension r_{1} + r_{2} and one has

H_{D}^{3}(C_{/ℝ}, ℝ(2)) ≃ ⊕_{σ|∞}H^{2}(C_{σ}(ℂ), ℝ) ≃ ℝ^{r_{1}+r_{2}}

where C_{σ} is the curve C being thought of as a complex manifold using the embedding σ For each archimedian place σ, this map restricted to CH^{2}(C, 1) is given by

reg(∏_{i}(x_{i}, u_{i})) = log |u_{i}|_{σ}ω  (6.1)

where ω is the cycle class of C. The conjectures assert that the determinant of this map should equal L^{∗}(H^{2}(C), s)_{s=1} up to a rational number, where ∗ represents taking the (r_{1} + r_{2} − 1)^{st} derivative. Since C is a curve over K, L(H^{2}(C), s) = ζ_{K}(s − 1), this is nothing but the usual Dirichlet class number formula.

However, our constructions give elements of the form ∏_{j,i}(x_{ij}, f_{j}(x_{ij})) for some functions f_{j} and points x_{ij} One might hope that there are some exact formulas for the value of the L-function in terms of the f_{j} in the spirit of the Kronecker Limit formula.

Acknowledgements: I would like to thank A.O.L. Atkin, S.Baba, A.Brown, S.Bloch for their comments and help and the Tata Institute of Fundamental Research for its hospitality while this work was done.

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