We prove a transcendence result of Hecke operator $T\left(\mathfrak{l}\right)$ over the ring generated by diamond operators over $\mathbb{Q}$ in a non-CM component of the cyclotomic ordinary big $p$-adic Hecke algebra, and we discuss its conjectural implication of a modulo-$p$ version to the vanishing of the cyclotomic $\mu$-invariant.

We study sums over primes of trace functions of $\ell$-adic sheaves. Using an extension of our earlier results on algebraic twists of modular forms to the case of Eisenstein series and bounds for Type II sums based on similar applications of the Riemann hypothesis over finite fields, we prove general estimates with power saving for such sums. We then derive various concrete applications.

We show that a surface group contained in a reductive real algebraic group can be deformed to become Zariski-dense, unless its Zariski closure acts transitively on a Hermitian symmetric space of tube type. This is a kind of converse to a rigidity result of Burger, Iozzi, and Wienhard.

It was proved recently that the correlation functions of a semisimple cohomological field theory satisfy the so-called local Eynard–Orantin topological recursion. We prove that in the settings of singularity theory, the local Eynard–Orantin recursion is equivalent to $N$ copies of Virasoro constraints for the total ancestor potential. The latter follow easily from some known properties of the period integrals in singularity theory. Our approach generalizes easily to an arbitrary semisimple cohomological field theory, which yields a simple proof of the local Eynard–Orantin recursion for an arbitrary semisimple cohomological field theory.