3. Decidability

Problem 3.1.
[Chris Hall, Alexandra Shlapentokh] Given an elliptic curve $E$ given by $y^2 = x^3 + Ax + B$: Is there are algorithm that decides if $E(\mathbb Z) \setminus\{\infty\}$ is empty?
 Is there an algorithm for enumerating $E(\mathbb Z)$?
 Can these questions be answered if $E$ is replaced with an arbitrary curve?

Problem 3.2.
 Is the first order theory of $\mathbb C(t)$ decidable in signature with $1$ and $t$?
 Does it help if $\frac{\partial}{\partial t_i}$ is added to the signature?

Problem 3.3.
Is the existential theory of $\{\text{complex holomorphic functions over $\mathbb C$}\}$ in signature $0,1,+,\cdot, z$ decidable? 
Problem 3.4.
[Arno Fehm] Is $\exists \text{Th}(\mathbb F_p(t))$, in the language of rings without a name for $t$, decidable? 
Problem 3.5.
[Thanasis Pheidas] Let $p$ be a fixed prime. Define $$ x \mid_p y \Leftrightarrow \exists s\in \mathbb N, y = p^sx $$ Does $(\mathbb Z, +, \mid_p, 0, 1)$ have a decidable existential theory? 
Problem 3.6.
[Arno Fehm] Which fragments of $\text{Th}(\mathbb Q)$ are undecidable? Consider the number of quantifiers, not just alternations.
Cite this as: AimPL: Definability and decidability problems in number theory, available at http://aimpl.org/definedecide.