We give the first explicit examples beyond the Chabauty–Coleman method where Kim’s nonabelian Chabauty program determines the set of rational points of a curve defined over $\mathbb{Q}$ or a quadratic number field. We accomplish this by studying the role of $p$-adic heights in explicit non-Abelian Chabauty.

Let $E$ and $E\text{'}$ be two elliptic curves over a number field. We prove that the reductions of $E$ and $E\text{'}$ at a finite place $\mathfrak{p}$ are geometrically isogenous for infinitely many $\mathfrak{p}$, and we draw consequences for the existence of supersingular primes. This result is an analogue for distributions of Frobenius traces of known results on the density of Noether–Lefschetz loci in Hodge theory. The proof relies on dynamical properties of the Hecke correspondences on the modular curve.

We establish an energy quantization result for sequences of Willmore surfaces when the underlying sequence of Riemann surfaces is degenerating in the moduli space. We notably exhibit a new residue which quantifies the possible loss of energy in collar regions. Thanks to this residue, we also establish the compactness (modulo the action of the Möbius group of conformal transformations of ${\mathbb{R}}^3\cup \{\infty \}$) of the space of Willmore immersions of any arbitrary closed $2$-dimensional oriented manifold into ${\mathbb{R}}^3$ with uniformly bounded conformal class and energy below $12\pi$.

We prove that any two $C^4$ critical circle maps with the same irrational rotation number and the same odd criticality are conjugate to each other by a $C^1$ circle diffeomorphism. The conjugacy is $C^{1+\alpha }$ for a full Lebesgue measure set of rotation numbers.