The Brill-Segre formula and the abc conjecture

José Felipe Voloch

This is a write-up of lectures presented at the first Arizona Winter School in Arithmetic Geometry on the abc conjecture. There isn’t anything new in these notes, except perhaps the point of view. Most of the results are in [V] and [TV].

The Brill-Segre formula counts the number of osculation points for a morphism of a curve to n-dimensional space and generalizes the Hurwitz formula (n = 1) and the Plucker formula (n = 2). The Brill-Segre formula implies the abc theorem for function fields for arbitrarily many summands. Smirnov has suggested a conjectural analogue of Hurwitz formula for number fields which implies the abc conjecture. We had hoped to be able to formulate a corresponding number field analogue of the Brill-Segre formula, but had to stop short of that goal and discuss only local aspects of such an analogue.

Let X be an irreducible, nonsingular, projective algebraic curve of genus g defined over an algebraically closed field k of characteristic zero (see the papers of J. Wang [W1,2] for the case of positive characteristic). Let K be the function field of X. For elements f_{0}, …, f_{n} of K, not all zero, we define the height as

h(f_{0}, …, f_{n}) := ∑_{P ∈X} − min{v_{P} (f_{0}), …, v_{P} (f_{n})},

where v_{P} (f) is the order of f at a point P of X.

Let f_{0}, …, f_{n} ∈ K and be linearly independent over k. Consider the morphism ϕ = (f_{0}, …, f_{n}) : X → P^{n}. For each P ∈ X we have

f(P) = ((t_{P}^{e_{P}} f_{0})(P), …, (t_{P}^{e_{P}} f_{n})(P))

where e_{P} := − min{v_{P} (f_{0}), …, v_{P} (f_{n})} and t_{P} is a local parameter of X at P. The set {v_{P} (∑_{i=0}^{n} a_{i}f_{i})+e_{P} | a_{i} ∈ k} consists of n+1 integers, say 0 ≤ j_{0} < j_{1} < … < j_{n} ≤ deg ϕ, where deg ϕ = h(f_{0}, …, f_{n}) is the degree of ϕ. This can be shown as follows. There is a descending sequence P^{n} = V_{0} ⊇ V_{1} ⊇ V_{2} ⊇ ⋯ of linear subspaces of P^{n} such that, for each m ≥ 0

V_{m}(k) = {(a_{0} : …: a_{n}) ∈ P^{n}(k) | v_{P} (∑_{i=1}^{n} a_{i}f_{i}) + e_{P} ≥ m)}.

Indeed, the condition in question amounts to m linear conditions on the indeterminate constants a_{1}, …, a_{n}. Geometrically, we view a_{1}, …, a_{n} as the coordinates of the hyperplane ∑ a_{i}X_{i} = 0 and interpret V_{m} as the space of hyperplanes in P^{n} which meet our curve with multiplicity at least m at P. At each stage V_{m} = V_{m+1} or V_{m+1} is of codimension one in V_{m}. Also, it is clear that V_{m} is empty for m large, since the descending sequence P^{n} = V_{0} ⊇ V_{1} ⊇ V_{2} ⊇ ⋯ must stabilize and ⋂ V_{m} = ∅ since the f_{i} are linearly independent over k. Therefore there are exactly n + 1 integers, say 0 ≤ j_{0} < j_{1} < … < j_{n} with V_{m} = V_{m+1} if and only if m ≠ j_{i} for all i. In particular dim V_{m} = n − i, for j_{i−1} < m ≤ j_{i}.

Brill-Segre formula. Let ϕ : X → P^{n} be a morphism. For each P ∈ X define w_{ϕ}(P) = ∑_{i=0}^{n}(j_{i} − i), then ∑_{P ∈X} w_{ϕ}(P) = n(n + 1)(g − 1) + (n + 1) deg ϕ.

We will not give a proof of this result here since it is readily available in the literature, see, e.g., [FK],[GH],[SV], etc.

It follows from the Brill-Segre formula that there exists only finitely many points in X where (j_{0}, …, j_{n}) ≠ (0, …, n). These points are called the Weierstrass points for the morphism ϕ and w_{ϕ}(P) is called the weight of P.

We will now use the Brill-Segre formula to prove (see [BM] or [V]):

Theorem. Suppose u_{1}, …, u_{m} ∈ K are linearly independent over k, satisfy u_{1}+…+u_{m} = 1 and let S = {P ∈ X | ∃i, v_{P} (u_{i}) ≠ 0} then:

h(u_{1}, …, u_{m}) ≤ \frac{m(m − 1)}{2}(2g − 2 + #S).

This is a generalization of Mason’s abc theorem which corresponds to m = 2. Before embarking on the proof (which will follow [V]) we need a lemma.

Lemma. If v_{P} (f_{0}) ≤ v_{P} (f_{1}) ≤ … ≤ v_{P} (f_{n}), then j_{i} ≥ v_{P} (f_{i}) + e_{P}, for i = 0, ..., n.

Proof: The hypothesis implies that the linear space x_{0} = ⋯ = x_{i−1} = 0 is contained in V_{m}, where m = v_{P} (f_{i})+e_{P}, so dim V_{m} ≥ n−i. However, j_{i} is defined by dim V_{j_{i}} = n−i so j_{i} ≥ v_{P} (f_{i}) + e_{P}, as desired.

Proof of the theorem: Consider the morphism ϕ : X → P^{m−1} given by (u_{1} : …: u_{m}). Given a point P ∈ X, choose j such that v_{P} (u_{j}) = −e_{P}. We can make a change of coordinates that replaces u_{j} by ∑ u_{i} = 1. After reordering the new coordinates we can assume we are under the hypotheses of the lemma and therefore

∑ j_{i} ≥ v_{P} (1) + e_{P} + ∑_{i≠j}(v_{P} (u_{i}) + e_{P}) = ∑_{i≠j} v_{P} (u_{i}) + me_{P} = ∑ v_{P} (u_{i}) + (m + 1)e_{P} .

So w_{ϕ}(P) = ∑ j_{i} − m(m − 1)/2 ≥ ∑ v_{P} (u_{i}) + (m + 1)e_{P} − m(m − 1)/2. Thus

∑_{P ∈S} w_{ϕ}(P) ≥ ∑_{P ∈S}(m + 1)e_{P} − m(m − 1)/2 + ∑_{i} v_{P} (u_{i})

= (m + 1)h(u_{1}, …, u_{m}) − m(m − 1)/2|S|.

On the other hand, ∑_{P ∈S} w_{ϕ}(P) ≤ ∑_{P ∈X} w_{ϕ}(P) = m(m−1)(g−1)+mh(u_{1}, …, u_{m}) by the Brill-Segre formula, and these two inequalities immmmmmediately give the theorem.

We will now consider an arithmetic analogue of the above. Let a_{1}, …, a_{n} be elements of a number field K. Thinking of K as the function field of a non-existent curve, (a_{1} : …: a_{n}) can be viewed as a map of the curve to P^{n−1}. This point of view has been dubbed ”geometry over the field of one element” (see [M] or [Sm] and also [I], which is perhaps in the same spirit). We can consider the local and global aspects of the situation. In [Sm], Smirnov studies the case n = 2, where the local theory is trivial, and makes some global conjectures in the spirit of a number field analogue of the Hurwitz formula. He asks about a possible higher dimensional generalization of his conjecture, which should be an analogue of the Plücker formulas in dimension two which corresponds to n = 3, and their higher dimensional extensions, that is, the Brill-Segre formula. The results obtained in [TV] provide a possible local theory towards such a generalization and the next step will be to formulate such a higher dimensional global conjecture.

Consider an unramified local field K of characteristic zero with perfect residue field k of characteristic p > 0, and a_{1}, …, a_{n} ∈ K. As K has no constant field, we take as replacement the set of Teichmüller representatives of the elements of of k, which we denote by T (k). Denote the valuation on K by v(.).

Theorem ([TV]). Let K be as above. Given a_{1}, …, a_{n} ∈ K, there exists a positive integer m such that for ζ_{1}, …, ζ_{n} ∈ T (k), either ∑ ζ_{i}a_{i} = 0 or v(∑ ζ_{i}a_{i}) ≤ m.

Proof: Without loss of generality, we can assume that k is algebraically closed and that a_{1}, …, a_{n} are in the ring of integers of k, which we can identify with the ring of Witt vectors of infinite length over k. For z ∈ k we let T (z) denote its Teichmüller representative T (z) = (z, 0, 0, …). Since the Witt vectors of length m form a ring scheme and T is multiplicative, the condition ∑ ζ_{i}a_{i} ≡ 0 (mod p^{m}), for ζ_{i} = T (z_{i}), translates into a set of m homogeneous polynomial equations in z_{1}, …, z_{n} and therefore defines a closed subscheme V_{m} of P^{n−1} over k. Moreover, the V_{m} form a decreasing sequence, so must become constant. If V_{m_{0}} = V_{m}, for m > m_{0}, then any ζ_{1}, …, ζ_{n} ∈ T (k) whose residues z_{1}, …, z_{n} define a point in V_{m_{0}} satisfies ∑ ζ_{i}a_{i} = 0 and all others satisfy v(∑ ζ_{i}a_{i}) ≤ m_{0} − 1. This completes the proof.

The V_{m} in the above proof are the analogues of the V_{m} in the function field case. In the function field case they were linear spaces, thus reduced, irreducible and equidimensional. In the arithmetic case they are just schemes which may not be reduced, irreducible or equidimensional. See [TV] for examples and for results that ensure that the V_{m} are wellbehaved for small m.

Assume now we are dealing with the global situation. To begin with define the weight of a place w of K. A weak version is the following

M_{w} = \frac{1}{p^{(n−1)(n−2)/2}} ∑_{(z_{1}:...:z_{n})∈V_{n−1}}(w(∑ a_{i}T (z_{i})) − (n − 1)).

As a first approximation to the Brill-Segre formula one can ask if the infinite series ∑_{w} M_{w} log Nw converges if n ≥ 3. There is no good reason to assume it is a finite sum, but the convergence seems reasonable. A formula, or at least an estimate, for the sum of this series would then be the required conjecture. One would like to add an archimedian term, as in [Sm], also. At this point, further theoretical work and numerical experimentation seem advisable before hazarding a shape for this formula. Such a formula should imply the generalized abc-conjecture for number fields with arbitrarily many summands, so it lies quite deep. Also, it should be mentioned that in [SV] we obtain a variant of the Brill-Segre formula “twisted by Frobenius” that leads to a proof of the Riemann hypothesis for function fields and one may speculate whether there is something similar in the number field case. Like in [Sm], such a conjecture would have implications to some classical arithmetic questions such as whether there are infinitely many primes satisfying a^{p−1} ≡ 1 (mod p^{2}) or (mod p^{3}), where a is some fixed integer.

Acknowledgements: The NSF GIG grant provided support for my attendance at the meeting and for the meeting itself. I would also like to acknowledge financial support from the NSA (grant MDA904-97-1-0037). I’d like to thank Jeff Lagarias for lending me the notes he took during my lectures which were very useful in the preparation of these notes and Jaap Top who corrected some mistakes in an earlier version of these notes.

References. [BM]W. D. Brownawell and D. W. Masser, Vanishing sums in function fields, Math.

Proc. Cambridge Philos. Soc. 100 (1986), no. 3, 427–434] [FK] H. Farkas and I. Kra, Riemann surfaces, Springer GTM.

[GH] P. Griffiths and J. Harris, Principles of Algebraic Geometry, Wiley.

[M] Mason, R. C. Diophantine equations over function fields. London Mathematical Society Lecture Note Series, 96. Cambridge University Press, Cambridge-New York, 1984.

[Si] Silverman, J. H. The S-unit equation over function fields. Math. Proc. Cambridge Philos. Soc. 95 (1984), no. 1, 3–4

[Sm] Smirnov, A. L. Hurwitz inequalities for number fields. (Russian) Algebra i Analiz 4 (1992), no. 2, 186–209; translation in St. Petersburg Math. J. 4 (1993), no. 2, 357–37

[SV] Stöhr, K-O. and Voloch, J.F. Weierstrass Points and Curve over Function Fields Proc. London Math. Soc.(3) vol 52 1986 1–19

[TV] Tate, J. and Voloch, J. F., Linear forms in p-adic roots of unity. Internat. Math. Res. Notices

[V] Voloch, J. F., Diagonal equations over function fields. Bol. Soc. Brasil. Mat. 16 (1985), no. 2, 29–39

[W] Wang, J, T.-Y., The Truncated Second Main Theorem of Function Fields, J. Number Theory 58 (1996) 139-157.

Dept. of Mathematics, Univ. of Texas, Austin, TX 78712, USA e-mail: voloch@math.utexas.edu