Serre's uniformity problem in the split Cartan case

Abstract

We prove that there exists an integer p0 such that X split (p)(Q) is made of cusps and CM-points for any prime p > p0. Equivalently, for any non-CM elliptic curve E over Q and any prime p > p0 the image of Gal(Q/Q) by the representation induced by the Galois action on the p-division points of E is not contained in the normalizer of a split Cartan subgroup. This gives a partial answer to an old question of Serre.

On uniform large Galois images for modular abelian varieties

“…Moreover, a > 0 (otherwise, the equations X p i = δ i X i+1 would show A j is isomorphic overF p to Spec(F p [X]/(X p r − cX)) for some c = 0 inF p ; the A j would therefore beétale over O K , which is a contradiction). So v(δ i ) > 0 and Raynaud's equations X p i = δ i X i+1 do give our claim (2). But this is not compatible with what we know about A[P] × OKFp .…”

“…The exceptional cases (a) are relatively easy to rule out for elliptic curves over arbitrary number fields (see [17, p. 36]). For elliptic curves over Q, it is known that such absolute upper bounds also exist in cases (b) [18], and (c) when the Cartan group is split [2], but we still do not know if there are rational non-CM elliptic curves with normalizer of non-split Cartan structure modulo arbitrarily large p.…”

Surjectivity of Galois representations associated with quadratic $${\mathbb {Q}}$$ Q -curves

“…This proves (4). From this equation, we know that for every γ ∈ SL 2 (Z), the q-expansion of g |γ (that is, the image of g by the usual right action of SL 2 (Z) on functions on H) is a formal series in q …”

“…The proof mechanism improves on [7] and is akin to that of [4]. More precisely, let K be an imaginary quadratic field.…”

“…The sum h of the first two terms of h give the good candidate for elliptic curves over Q (see [4]). As in Borel case, the generalisation to Q-curves relies in the consideration of h + h • w d .…”