We use Hensel minimality, a non-Archimedean analog of o-minimality, to study several questions around transcendental number theory, unlikely intersections, and differential fields in a non-Archimedean setting. In particular, we focus on $p$-adic exponentiation and Tate uniformization on $\mathbb{C}_p$, which we show live in a Hensel minimal structure on $\mathbb{C}_p$. We start by constructing a large collection of derivations on Hensel minimal fields that respect definable functions, which we then apply to the $p$-adic Schanuel conjecture. We also study properties of local definability in analogy to work of Wilkie, and show that $p$-adic Schanuel implies a uniform version of itself. For Tate uniformization we show a strong closure property when blurring, and deduce that $\mathbb{C}_p$ with the blurred Tate uniformization is quasiminimal. Finally, we prove a result on $p$-adic density of likely intersections for powers of elliptic curves.
O-minimality is a framework coming from model theory in which one can carry out analytic arguments with guaranteed tameness. We remark that when we say "analytic arguments" we specifically mean arguments that are set in either real or complex analysis. The successful use of o-minimality in important arithmetic geometry problems is now well-established, see for example [Pil11, Tsi18, PST + 21] for applications to the André-Oort conjecture, [PT16, MPT19,BT19] for results in functions transcendence, and [DR18] for results regarding the Zilber-Pink conjecture. In part motivated by these achievements, various attempts have been made to come up with a similarly tame framework that supports non-Archimedean analytic arguments, such as P-minimality [HM97] or C-minimality [HM94,MS96]. We will specifically focus on the more recent notion of 1-h-minimality [CHRK22,CHRKV23], which applies more generally than the previous notions.
The purpose of this article is to showcase how 1-h-minimality can be used in analogy to various applications of o-minimality, to exhibit interesting examples of functions definable in 1-h-minimal structures, and to show how to sort some of the technical obstructions that arise. For this, we will study some non-Archimedean analogs of problems around transcendental number theory, unlikely intersections, and differential fields. Most analytic applications of o-minimality start by considering the expansion of the real field R by a collection of analytic functions F so that the resulting structure is o-minimal. Similarly, we will begin by considering expansions of C p by certain collections of analytic functions.
We first study the existence of non-trivial field derivations δ ∶ C p → C p that respect a collection of functions F such that the structure (C p , F) is 1-h-minimal.
Theorem 1.1 (Existence of non-trivial derivations, Theorem 3.1). Let T be a 1-hminimal theory and let K be a model of T . Then there exists a collection of commuting derivations ∆ = (∂ τ ) τ ∈B on K with the following properties:
(1) if U ⊂ K n is open and h ∶ U → K is a ∅-definable C 1 function, then for every a = (a 1 , . . . , a n ) and every ∂ τ we have
(2) We have ⋂ τ ker ∂ τ = acl(∅), where acl denotes the model-theoretic algebraic closure operator.
Once we have these derivations, we study the model-theoretic question of local definability, following the treatment of Wilkie [Wil08] for holomorphic functions. We note that a few obstructions need to be sorted, one having to do with choosing appropriate reducts of 1-h-minimal theories which preserve 1-h-minimality while having a countable language (in general, reducts do not preserve 1-h-minimality, see [CHRKV23, Remark 5.1.2]), but the main one being that there is no non-Archimedean analog of Gabrielov’s theorem which shows that the projections of analytic sets remain analytic.
After this, the rest of the paper focuses on two specific p-adically analytic functions, which we show are part of 1-h-minimal theories in Corollary 2.2 (even in a countable language). We work in C p , we denote the valuation ring by O Cp , the maximal ideal by M Cp and the valuation by v ∶ C p → Q. One of the most famous transcendental functions known to be part of an o-minimal expansion of R is the real exponential function, so the first function we will focus on is p-adic exponentiation. We denote by D p ⊂ C p the open ball of valuative radius 1/(p -1) around zero, and let exp ∶ D p → 1 + D p be the p-adic exponential map.
Using that p-adic exponentiation is part of a 1-h-minimal expansion of C p , we obtain some results concerning the p-adic version of Schanuel’s conjecture from transcendental number theory (see §3.2 for the statement of the conjecture). We first show that this conjecture implies a uniform version of itself (the analogous result about real exponentiation and Schanuel’s conjecture is due to Kirby and Zilber [KZ06]).
Theorem 1.2. Let V ⊆ G n a × G n m be an irreducible algebraic variety defined over Q with dim V < n. Then the p-adic Schanuel conjecture implies that there exist finitely many proper Q-linear subspaces L 1 , . . . , L m ⊂ G n a such that if (z, exp(z)) ∈ V (C p ), then z ∈ L i , for some i ∈ {1, . . . , n}.
Then we show two weak forms of the p-adic Schanuel conjecture using Ax’s theorem on functional transcendence of exponentiation (although, as we explain in §3.2, these two weak forms could have been already deduced from earlier work of Kirby [Kir10]).
Many applications of o-minimality to problems in arithmetic geometry (especially when it comes to problems of unlikely intersections) often involve periodic holomorphic functions, such as complex exponentiation, the exponential maps of abelian varieties, or automorphic functions of Shimura varieties. Some of the key applications of o-minimality in this area (like the o-minimal proofs of Ax-Schanuel type theorems) exploit the fact that these functions are not definable in an o-minimal structure (because of p
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