Grothendieck's section conjecture for the modular curves X0(N) and X1(N) over Q

Claims Theorem 8.2: the global section conjecture holds for every smooth proper geometrically connected curve of genus at least two over a number field whose finite-cover descent locus equals its rational points; by Stoll's criteria this includes curves mapping nonconstantly to an abelian variety with finitely many rational points and trivial divisible Sha, and the modular curves $X_0(N)$, $X_1(N)$ of genus at least two over $\mathbb Q$. It does NOT prove the section conjecture for arbitrary curves over number fields; the descent-locus equality remains a separate hypothesis in general.

Claims Theorem 8.2: the global section conjecture holds for every smooth proper geometrically connected curve of genus at least two over a number field whose finite-cover descent locus equals its rational points; by Stoll's criteria this includes curves mapping nonconstantly to an abelian variety with finitely many rational points and trivial divisible Sha, and the modular curves $X_0(N)$, $X_1(N)$ of genus at least two over $\mathbb Q$. It does NOT prove the section conjecture for arbitrary curves over number fields; the descent-locus equality remains a separate hypothesis in general.