Ax-Schanuel and strong minimality for the j-function

Abstract

Let K := (K; +, •, D, 0, 1) be a differentially closed field of characteristic 0 with field of constants C.In the first part of the paper we explore the connection between Ax-Schanuel type theorems (predimension inequalities) for a differential equation E(x, y) and the geometry of the fibres U s := {y : E(s, y) ∧ y / ∈ C} where s is a non-constant element. We show that certain types of predimension inequalities imply strong minimality and geometric triviality of U s . Moreover, the induced structure on the Cartesian powers of U s is given by special subvarieties. In particular, since the j-function satisfies an Ax-Schanuel inequality of the required form (due to Pila and Tsimerman), applying our results to the j-function we recover a theorem of Freitag and Scanlon stating that the differential equation of j defines a strongly minimal set with trivial geometry.In the second part of the paper we study strongly minimal sets in the j-reducts of differentially closed fields. Let E j (x, y) be the (two-variable) differential equation of the j-function. We prove a Zilber style classification result for strongly minimal sets in the reduct K := (K; +, •, E j ). More precisely, we show that in K all strongly minimal sets are geometrically trivial or non-orthogonal to C. Our proof is based on the Ax-Schanuel theorem and a matching Existential Closedness statement which asserts that systems of equations in terms of E j have solutions in K unless having a solution contradicts Ax-Schanuel.

A differential approach to the Ax-Schanuel, I

“…When dim(Y ) = 1, this corresponds to the assumption that one only considers non-constant solutions. (2) Using Lemma 5.1 (2) for any υ ∈ Y , we have that tr.deg. C( t) C( t) υ = k. Hence it also follows that tr.deg.…”

“…, y k ∈ K υ are algebraically independent over K then they are algebraically independent over F . (2) We say that tp(υ/K) has U -rank 1 (or is minimal) if and only if υ ∈ K alg but every forking extension of tp(υ/K) is algebraic, that is has only finitely many realizations.…”

“…Over the past several years, in a series of works Aslanyan, and later Aslanyan, Kirby and Eterović [1,2,5,6,4] develop the connection between Ax-Schanuel type transcendence statements and the existential closedness of certain reducts of differentially closed fields related to equations satisfied by the j-function. This series of work builds on the earlier program of Kirby, Zilber and others mainly around the exponential function, see e.g.…”

“…Intermediate differential algebraic results from Ax's work are utilized in the program studying existential closedness results around the exponential function. A different approach was required in [1,2,5,6,4] for studying the j-function, in part since the intermediate results of the functional transcendence results of [31] take place in the o-minimal rather than abstract differential setting. For instance, the motivation for a differential algebraic proof of Ax-Schanuel results and this issue is pointed out specifically following Theorem 1.3 of [5] and in Section 4.4 of [4].…”

Generic differential equations are strongly minimal

“…This result ultimately relies on point-counting and o-minimality via the Pila–Wilkie theorem as applied in [Pil11, Pil13]; the argument there is very specific to the third order nonlinear differential equation satisfied by the -function. Later, Aslanyan [Asl20] produced another proof, ultimately relying on similar (stronger) inputs of [PT16]. Casale et al.…”