Modular curves and the refined distance conjecture

We test the refined distance conjecture in the vector multiplet moduli space of 4D $$ \mathcal{N} $$ N = 2 compactifications of the type IIA string that admit a dual heterotic description. In the weakly coupled regime of the heterotic string, the moduli space geometry is governed by the perturbative heterotic dualities, which allows for exact computations. This is reflected in the type IIA frame through the existence of a K3 fibration. We identify the degree d = 2N of the K3 fiber as a parameter that could potentially lead to large distances, which is substantiated by studying several explicit models. The moduli space geometry degenerates into the modular curve for the congruence subgroup Γ0(N)+. In order to probe the large N regime, we initiate the study of Calabi-Yau threefolds fibered by general degree d > 8 K3 surfaces by suggesting a construction as complete intersections in Grassmann bundles.

Computing models for quotients of modular curves

We describe an algorithm for computing a $${\mathbb {Q}}$$ Q -rational model for the quotient of a modular curve by an automorphism group, under mild assumptions on the curve and the automorphisms, by determining q-expansions for a basis of the corresponding space of cusp forms. We also give a moduli interpretation for general morphisms between modular curves.

Quadratic points on non-split Cartan modular curves

In this paper, we study quadratic points on the non-split Cartan modular curves [Formula: see text], for [Formula: see text] and [Formula: see text]. Recently, Siksek proved that all quadratic points on [Formula: see text] arise as pullbacks of rational points on [Formula: see text]. Using similar techniques for [Formula: see text], and employing a version of Chabauty for symmetric powers of curves for [Formula: see text], we show that the same holds for [Formula: see text] and [Formula: see text]. As a consequence, we prove that certain classes of elliptic curves over quadratic fields are modular.

CM values of higher automorphic Green functions for orthogonal groups

Gross and Zagier conjectured that the CM values (of certain Hecke translates) of the automorphic Green function $$G_s(z_1,z_2)$$ G s ( z 1 , z 2 ) for the elliptic modular group at positive integral spectral parameter s are given by logarithms of algebraic numbers in suitable class fields. We prove a partial average version of this conjecture, where we sum in the first variable $$z_1$$ z 1 over all CM points of a fixed discriminant $$d_1$$ d 1 (twisted by a genus character), and allow in the second variable the evaluation at individual CM points of discriminant $$d_2$$ d 2 . This result is deduced from more general statements for automorphic Green functions on Shimura varieties associated with the group $${\text {GSpin}}(n,2)$$ GSpin ( n , 2 ) . We also use our approach to prove a Gross–Kohnen–Zagier theorem for higher Heegner divisors on Kuga–Sato varieties over modular curves.