Modular elliptic curves over real abelian fields and the generalized Fermat equation x2l + y2m = zp

Let K be a real abelian field of odd class number in which 5 is unramified. Let S-5 be the set of places of K above 5. Suppose for every nonempty proper subset S subset of S-5 there is a totally positive unit u is an element of O-K such that Pi(q) (is an element of S) Norm Fq/F5 (u mod q) not equal (1) over bar. We prove that every semistable elliptic curve over K is modular, using a combination of several powerful modularity theorems and class field theory. We deduce that if K is a real abelian field of conductor n < 100, with 5 inverted iota n and n not equal 29, 87, 89, then every semistable elliptic curve E over K is modular. Let l, m; p be prime, with l, m >= 5 and p >= 3. To a putative nontrivial primitive solution of the generalized Fermat equation x(2l) + y(2m) = z(p) we associate a Frey elliptic curve defined over Q(zeta(p))(+), and study its mod l representation with the help of level lowering and our modularity result. We deduce the nonexistence of nontrivial primitive solutions if p <= 11, or if p = 13 and l, m not equal 7.