In 1985 Joseph Oesterle (left) and David Masser (right) formulated the conjecture that for three relative prime integers satisfying $A+B=C$, the product of the prime divisors of $ABC$ is rarely much smaller than $C$.
More precisely, if $A,B,C \in \mathbb{Z}$ are such that $A+B=C$ and $\gcd(A,B,C)=1$, then their conjecture states that for each $\epsilon > 0$ there is a constant $M_{\epsilon}$ such that for all triples $(A,B,C)$ satisfying the conditions we have
$$\max( |A|, |B|, |C|) \leq M_{\epsilon}\left(\underset{p | ABC}{\prod} p\right)^{1+\epsilon}$$
The ABC-conjecture has several consequences, some obvious ones such as proving Fermat’s last theorem for large exponents, some less obvious such as Falting’s theorem. However, many people consider a proof the ABC-conjecture to be beyond the range of the available methods.
Since 2006 the ABC@Home project tries to find triples $(A,B,C)$ of large ‘quality’ meaning that the ratio
$$\operatorname{q}(A,B,C) = \frac{\log(C)}{\log(\operatorname{rad}(ABC))}$$
is as large as possible. To date, the champion-triple is $2+3^{10}109 = 23^5$ (discovered by Eric Reyssat) with a quality of $1.6299$.
If we write $u = \tfrac{A}{C}$ and $v=\tfrac{B}{C}$ then the ABC-conjecture can be recast as the statement that there is a constant $M_{\epsilon}$ such that when $u,v \in \mathbb{Q}^*$ satisfy $u+v=1$ we have
$$\max\left(\operatorname{ht}(u),\operatorname{ht}(v)\right) \leq M_{\epsilon} + (1+\epsilon)\left(\sum_{p | ABC} \log(p)\right)$$
where $A$ and $B$ are the numerators of $u$ and $v$ and $C$ is their common denominator, and where the ‘height’ $\operatorname{ht}(u)$ of a rational number $u=\tfrac{A}{C}$ with $(A,B)=1$ is $\max\left(\log|A|, \log|C|\right)$.
The latter formulation can be extended to the case of function fields of curves. So, let $K \in \mathsf{1Fields}$ with a perfect field of constants $k$ and suppose $u,v \in K^*$ are non-constants satisfying $u+v=1$. We need a substitute for the notion of height.
If $L$ is the maximal separable extension of $k(u)$ in $K$, then we call the dimension $[L : k(u)]$ the separability degree of $u$ and denote it with $\deg_s(u)$. Clearly, $\deg_s(u) \leq \deg(u) = [K : k(u)]$.
If $R$ is the integral closure of $k[u]$ in $K$, then there are maximal ideals $P_i$ in the Dedekind domain $R$ such that
$$(u) = P_1^{e_1} \cdots P_r^{e_r}$$
Because the local ring in $P_i$ is a discrete valuation ring in $K$ it determines a point in the curve $C$ with $K=k(C)$ (see here) also denoted $P_i$. But then, the zero-divisor of $u$ is $\operatorname{div}_0(u) = A = \sum_i e_i [P_i]$ with degree $\deg(A) = \sum_i e_i \deg(P_i)$.
Similarly, in the integral closure $S$ of $k[\tfrac{1}{u}]$ we have a decomposition
$$(\tfrac{1}{u}) = Q_1^{f_1} \cdots Q_s^{f_s}$$
and the pole-divisor of $u$ is $\operatorname{div}_{\infty}(u) = C = \sum_j f_j [Q_j]$ with degree $\deg(C) = \sum_j f_j \deg(Q_j)$. With these conventions, the ABC-conjecture for function fields can now be formulated as the following claim:
Let $K \in \mathsf{1Fields}/k$ and $u,v \in K^*$ with $u+v=1$, then
$$\deg_s(u) = \deg_s(v) \leq 2 g_K – 2 + \sum_{P \in \operatorname{Supp}(A+B+C)} \deg(P)$$
where $A=\operatorname{div}_0(u)$, $B=\operatorname{div}_0(v)$, $C=\operatorname{div}_{\infty}(u)=\operatorname{div}_{\infty}(v)$ and $g_K$ is the genus of $C$. Observe that there is no $\epsilon$ in this function field ABC-conjecture.
Perhaps surprisingly, the function-field ABC-conjecture can be proved fairly easily from the Riemann-Hurwitz genus formula. Details are in the book Number Theory in Function Fields by Michael Rosen (theorem 7.17) or in an upcoming prep-notes post.
Can we view the abc conjecture as saying something about the curve in P^2 defined by x + y = z? (analogous to Fermat hypersurfaces and FLT)? Clearly it is not as trivial as rational points. The ‘max(A,B,C)’ makes me think of tropical geometry. Perhaps there is a change of base sort of thing happening…
Just some idle thoughts…