Galois representations attached to elliptic curves and Diophantine problems (Q2710116)

In this article, the author gives a nice and compact overview of a set of Diophantine conjectures and their interrelations, notably the ABC conjecture and the height conjecture for elliptic curves (formulated for general fields with divisor theory), and their implications for torsion structures on elliptic curves. NEWLINENEWLINENEWLINEIn the first part, the Riemann-Hurwitz genus formula is used to prove all of these conjectures in the function field case. The author points out that the ABC conjecture and the height conjecture for (semistable) elliptic curves can be proved from one another. NEWLINENEWLINENEWLINEThen the conjectures are given for fields with divisor theory. The main result here is the equivalence of (suitable versions of) the ABC conjecture and the height conjecture for elliptic curves, based on the same idea as in the function field case, which is to relate the equation \(a + b = c\) and the elliptic curve \(y^2 = x(x-a)(x+b)\). NEWLINENEWLINENEWLINEFor fields satisfying stronger conditions (for example number fields or global function fields), it is shown that the asymptotic Fermat conjecture (which is itself a consequence of the ABC conjecture) implies strong conjectural results on torsion structures of elliptic curves. NEWLINENEWLINENEWLINEOver \(\mathbb Q\), one can use the theory of modular forms and modular representations, culminating in the proof of the modularity conjecture for elliptic curves, to get at least some `real' results. For example, it is possible to deduce the ABC conjecture over \(\mathbb Q\) from a special case of the so-called degree conjecture. The idea is that it might be possible to use our extensive knowledge on elliptic curves to prove one of these results and thus deduce the ABC conjecture some day.NEWLINENEWLINEFor the entire collection see [Zbl 0959.00053].