%Hi, here is a revised version of my write-up of my lectures in Tucson.
%Some mistakes are corrected. Are you ever going to do something with
%them?
%Felipe


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\centerline{\bf The Brill-Segre formula and the $abc$ conjecture}
\medskip
\centerline{\bf Jos\'e Felipe Voloch}
\bs
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
{\bf 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 $f_0,\ldots,f_n$ of $K$, not all zero, we define the height as
$$h(f_0,\ldots,f_n):=\sum_{P\in X}-\min\{v_P(f_0),\ldots,v_P(f_n)\},$$
where $v_P(f)$ is the order of $f$ at a point $P$ of $X$.

Let $f_0,\ldots,f_n \in K$ and be linearly independent over $k$.
Consider the morphism $\phi = (f_0,\ldots,f_n) : X  \to \P^n$. For
each $P \in X$ we have 
$$f(P)=((t_P^{e_P}f_0)(P),\ldots,(t_P^{e_P}f_n)(P))$$
where $e_P := -\min\{v_P(f_0),\ldots,v_P(f_n)\}$ and
$t_P$ is a local parameter of $X$ at $P$.
The set $\{ v_P(\sum_{i=0}^n a_if_i)+e_P \mid a_i\in k\}$
consists of $n+1$ integers, say $0\le j_0 < j_1 <\ldots <j_n \leq \deg \phi$,
where $\deg\phi = h(f_0,\ldots,f_n)$ is the degree of $\phi$.
This can be shown as follows. There is a descending
sequence $\P^n = V_0 \supseteq V_1 \supseteq V_2 \supseteq \cdots$ of
linear subspaces of $\P^n$ such that, for each $m \ge 0$
$$V_m(k) = \{(a_0:\ldots:a_n) \in \P^n(k) \mid
v_P(\sum_{i=1}^n a_if_i) + e_P \ge m)\}.$$
Indeed, the condition in question amounts to $m$ linear conditions on the
indeterminate constants $a_1,\ldots,a_n$. Geometrically, we view
$a_1,\ldots,a_n$ as the coordinates of the hyperplane $\sum a_iX_i = 0$
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 $\P^n = V_0 \supseteq V_1 \supseteq V_2 \supseteq \cdots$ must
stabilize and $\bigcap V_m = \emptyset$ since the $f_i$ are linearly independent
over $k$. Therefore there are exactly $n+1$ integers, say 
$0\le j_0 < j_1 <\ldots <j_n$ with $V_m=V_{m+1}$ if and only if $m \ne j_i$
for all $i$. In particular $\dim V_m=n-i$, for $j_{i-1}< m \le j_i$.

\proclaim Brill-Segre formula. Let $\phi: X  \to \P^n$
be a morphism. For each $P \in X$ define
$w_{\phi}(P) = \sum_{i=0}^n(j_i-i)$, then
$\sum_{P \in X}w_{\phi}(P) = n(n+1)(g-1)+(n+1)\deg \phi$.

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
$(j_0,\ldots,j_n)\neq(0,\ldots,n)$. These points are called 
the Weierstrass points 
for the morphism $\phi$ and $w_{\phi}(P)$ is called the weight of $P$.

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

\proclaim Theorem. Suppose  $u_1,\ldots,u_m \in K$ are  
linearly independent over $k$, satisfy $u_1+\ldots +u_m=1$ and
let $S=\{ P \in X \mid \exists i, v_P(u_i) \ne 0\}$ then:
$$h(u_1,\ldots,u_m) \le {m(m-1) \over 2}(2g-2+ \# S).$$

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.

\proclaim Lemma. If $v_P(f_0)\leq v_P(f_1)\leq \ldots \leq v_P(f_n)$,
then $j_i \geq v_P(f_i)+e_P$, for $i=0,...,n$.

{\it Proof}: The hypothesis implies that the linear space
$x_0=\cdots=x_{i-1}=0$ is contained in
$V_m$, where $m=v_P(f_i)+e_P$, so $\dim V_m \ge n-i$. 
However, $j_i$ is defined by $\dim V_{j_i}=n-i$ so $j_i \geq v_P(f_i)+e_P$,
as desired. 


{\it Proof of the theorem:} Consider the morphism $\phi:
X  \to \P^{m-1}$ given by $(u_1:\ldots:u_m)$. Given a point $P \in X$,
choose $j$ such that $v_P(u_j)=-e_P$. We can make a change
of coordinates that replaces $u_j$ by $\sum u_i=1$. After reordering
the new coordinates we can assume we are under the hypotheses of the lemma
and therefore 
$$\sum j_i \ge v_P(1)+e_P+ \sum_{i\ne j} (v_P(u_i)+e_P)= 
\sum_{i\ne j} v_P(u_i) + me_P = 
\sum v_P(u_i)+(m+1)e_P.$$ 
So
$w_{\phi}(P) = \sum j_i -m(m-1)/2 \ge \sum v_P(u_i)+(m+1)e_P-m(m-1)/2$.
Thus 
$$\sum_{P\in S} w_{\phi}(P) \ge \sum_{P\in S}(m+1)e_P-m(m-1)/2+\sum_iv_P(u_i)$$

$$=(m+1)h(u_1,\ldots,u_m)-m(m-1)/2|S|.$$

On the other hand, $\sum_{P\in S} w_{\phi}(P) \le \sum_{P\in X} w_{\phi}(P)=
m(m-1)(g-1) + mh(u_1,\ldots,u_m)$ by the Brill-Segre formula,
and these two inequalities immmmmmediately give the theorem.

We will now consider an arithmetic analogue of the above.
Let $a_1,\ldots,a_n$ be elements of a number field $K$.
Thinking of $K$ as the function field of a
non-existent curve, $(a_1:\ldots:a_n)$ 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 Pl\"ucker 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 $a_1,\ldots,a_n \in K$. As $K$ has no
constant field, we take as replacement the set of Teichm\"uller representatives
of the elements of of $k$, which we denote by $T(k)$.
Denote the valuation on $K$ by $v(.)$.

\proclaim Theorem ([TV]). Let $K$ be as above.
Given $a_1,\ldots,a_n \in K$, there exists
a positive integer $m$ such that for $\z_1,\ldots,\z_n \in T(k)$,
either $\sum \z_ia_i = 0$ or $v(\sum \z_ia_i) \le m$.

{\it Proof:} Without loss of generality, we can assume that $k$ is
algebraically closed and that $a_1,\ldots,a_n$ are in the ring of integers
of $k$, which we can identify with the ring of Witt vectors of infinite
length over $k$. For $z \in k$ we let $T(z)$ denote its Teichm\"uller
representative $T(z) = (z,0,0,\dots)$.
Since the Witt vectors of length $m$
form a ring scheme and $T$ is multiplicative, the condition $\sum \z_ia_i
\con 0\ (\mod p^m)$, for $\z_i = T(z_i)$, translates into a set of $m$
homogeneous polynomial equations
in $z_1,\ldots,z_n$ 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 $\z_1,\ldots,\z_n \in T(k)$ whose
residues $z_1,\ldots,z_n$ define a point in $V_{m_0}$ satisfies
$\sum \z_ia_i = 0$ and all others satisfy $v(\sum \z_ia_i) \le m_0-1$.
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 well-behaved 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 
$$M_w = {1 \over p^{(n-1)(n-2)/2}}
\sum_{(z_1:\ldots:z_n) \in V_{n-1}}\bigl(w(\sum a_iT(z_i))-(n-1)\bigr).$$
As a first approximation to the Brill-Segre formula one can ask
if the infinite series $\sum_w M_w\log Nw$ converges if $n \ge 3$. 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 $a^{p-1} \con 1 \ (\mod p^2)$ or $(\mod p^3)$, where $a$ is some
fixed integer. 





{\bf 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.

\bigskip
\centerline{\bf 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]  St\"ohr, 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
\smallskip
\noindent
e-mail: voloch@math.utexas.edu



\end