In unpublished notes, Pila discussed some theory surrounding the modular function j and its derivatives. A focal point of these notes was the statement of two conjectures regarding j, j′ and j″: a Zilber–Pink-type statement incorporating j, j′ and j″, which was an extension of an apparently weaker conjecture of André–Oort type. In this paper, I first cover some background regarding j, j′ and j″, mostly covering the work already done by Pila. Then I use a seemingly novel adaptation of the o-minimal Pila–Zannier strategy to prove a weakened version of Pila's ‘Modular André–Oort with Derivatives’ conjecture. Under the assumption of a certain number-theoretic conjecture, the central theorem of the paper implies Pila's conjecture in full generality, as well as a more precise statement along the same lines.

We investigate the Hasse principle for complete intersections cut out by a quadric hypersurface and a cubic hypersurface defined over the rational numbers.

Let $X\subset\mathbb{R}^2$ be the graph of a Pfaffian function $f$ in the sense of Khovanskii. Suppose that $X$ is non-algebraic. This note gives an estimate for the number of rational points on $X$ of height less than or equal to $H$; the estimate is uniform in the order and degree of $f$.

A uniform version of the Shafarevich Conjecture for function fields (Theorem of Parshin–Arakelov) is proved, together with a uniform version of the geometric Mordell conjecture (Theorem of Manin). Such results are generalized to base varieties of arbitrary dimension (i.e. function fields of arbitrary transcendence degree).