Given an elliptic curve E and a positive integer N, we consider the problem of counting the number of primes p for which the reduction of E modulo p possesses exactly N points over 𝔽p. On average (over a family of elliptic curves), we show bounds that are significantly better than what is trivially obtained by the Hasse bound. Under some additional hypotheses, including a conjecture concerning the short-interval distribution of primes in arithmetic progressions, we obtain an asymptotic formula for the average.

We provide an unconditional proof of the André–Oort conjecture for the coarse moduli space 𝒜2,1 of principally polarized abelian surfaces, following the strategy outlined by Pila–Zannier.

We construct two bases for each cluster algebra coming from a triangulated surface without punctures. We work in the context of a coefficient system coming from a full-rank exchange matrix, such as principal coefficients.

Given two Lie 2-groups, we study the problem of integrating a weak morphism between the corresponding Lie 2-algebras to a weak morphism between the Lie 2-groups. To do so, we develop a theory of butterflies for 2-term L∞-algebras. In particular, we obtain a new description of the bicategory of 2-term L∞-algebras. An interesting observation here is that the role played by 1-connected Lie groups in Lie theory is now played by 2-connected Lie 2-groups. Using butterflies, we also give a functorial construction of 2-connected covers of Lie 2 -groups. Based on our results, we expect that a similar pattern generalizes to Lie n-groups and Lie n-algebras.

We use reduction maps to study the minimal model program. Our main result is that the existence of a good minimal model for a Kawamata log terminal pair (X,Δ) can be detected on a birational model of the base of the (KX+Δ)-trivial reduction map. We then interpret the main conjectures of the minimal model program as a natural statement about the existence of curves on X.

We give a necessary and sufficient condition in order for a hyperplane arrangement to be of Torelli type, namely that it is recovered as the set of unstable hyperplanes of its Dolgachev sheaf of logarithmic differentials. Decompositions and semistability of non-Torelli arrangements are investigated.