galois representations, modular curves and

galois representations, modular curves and

GALOIS REPRESENTATIONS, MODULAR CURVES AND
RUNGE'S METHOD
YURI BILU (U. BORDEAUX)
2o Encuentro Chileno de Teoría de Números
16 de Abril 2015, Valparaiso, Chile
Let E/Q be an elliptic curve without complex multiplication. Serre (1972) proved that for all primes p > p0 (E), the Galois
representation obtained by the action of the absolute Galois group on
the p-torsion points of E is surjective. He asked if the constant p0 (E)
above can be made independent of E. In particular, is it true that p0 = 37
would do?
The problem reduces to classifying rational points on modular curves
+
+
X0 (p), Xsp
(p) and Xns
(p) associated to three types of the maximal subgroups of the linear group GL2 (Fp ):
- the Borel subgroups;
- the normalizers of split Cartan subgroups;
- the normalizers of non-split Cartan subgroups.
Abstract.
For X0 (p) the problem was completely solved by Mazur (1978). For
+
Xsp
(p) the problem was solved for all p distinct from 13 in a joint work
with P. Parent and M. Rebolledo (2013). An important ingredient in
our work is so-called Runge's method, which was successfully used in
recent times for attacking several dicult Diophantine problems. I will
explain how Runge's method works and will briey say how it applies
in our work.
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