Abstract. I give a model-theoretic setting for the modular j function and its derivatives. These structures, here called j-fields, provide an adequate setting for interpreting the Ax-Schanuel theorem for j (Theorem 1.3 of [16]). Following the ideas of [3] and [7] for exponential fields, I prove a generic transcendence property for the j function.

Assuming a modular version of Schanuel's conjecture and the modular Zilber-Pink conjecture, we show that the existence of generic solutions of certain families of equations involving the modular j function can be reduced to the problem of finding a Zariski dense set of solutions. By imposing some conditions on the field of definition of the variety, we are also able to obtain unconditional versions of this result.

This is a short note on various results about the combinatorial properties of line arrangements in terms of the Chern numbers of the corresponding log surfaces. This resembles the study of the geography of surfaces of general type. We prove some new results about the distribution of Chern slopes, we prove a connection between their accumulation points and the accumulation points of linear H-constants on the plane, and we present two open problems in relation to geography over Q and over C.

We give three equivalent characterisations of the natural closure operator on a field equipped with functions replicating the algebraic behaviour of the modular j-function and its derivatives, following similar work on exponential fields. As an application of these results, we give unconditional cases of a theorem of Eterović-Herrero on the Existential Closedness problem for the complex j-function which previously assumed the modular version of Schanuel's conjecture.

We prove the Existential Closedness conjecture for the differential equation of the j-function and its derivatives. It states that in a differentially closed field certain equations involving the differential equation of the j-function have solutions. Its consequences include a complete axiomatisation of j-reducts of differentially closed fields, a dichotomy result for strongly minimal sets in those reducts, and a functional analogue of the Modular Zilber-Pink with Derivatives conjecture.