[Paper Review] Elliptic curves and Hilbert's tenth problem for algebraic function fields over real and p-adic fields

Laurent Moret-Bailly|arXiv (Cornell University)|Sep 7, 2004

Algebraic Geometry and Number Theory4 citations

TL;DR

This paper establishes the Diophantine undecidability of function fields over real and p-adic fields by extending Denef’s method using twisted elliptic curves with Mordell-Weil rank one over rational function fields. It proves that Hilbert’s tenth problem is undecidable over semilocal rings and function fields in these settings, using geometric constructions and valuation-theoretic arguments to define a Diophantine copy of Z within the rings in question.

ABSTRACT

Let k be a field of characteristic zero, V a smooth, positive-dimensional, quasiprojective variety over k, and D a nonempty effective divisor on V. Let K be the function field of V, and A the semilocal ring of D in K. In this paper, we prove the Diophantine undecidability of: (1) A, in all cases; (2) K, when k is (formally) real and V has a real point; (3) K, when k is a subfield of a p-adic field, for some odd prime p. To achieve this, we use Denef's method: from an elliptic curve E over Q, without complex multiplication, one constructs a quadratic twist E' of E over Q(t), which has Mordell-Weil rank one. Most of the paper is devoted to proving (using a theorem of R. Noot) that one can choose f in K, vanishing at D, such that the group E'(K) deduced from the field extension K/Q(f)=Q(t) is equal to E'(Q(t)). Then we mimic the arguments of Denef (for the real case) and of Kim and Roush (for the p-adic case).

Motivation & Objective

• To establish the undecidability of the positive-existential theory of function fields over real and p-adic fields, extending known results to broader classes of fields.
• To prove that the ring of integers in a function field over a real or p-adic field is Diophantine-undefinable, implying no algorithm can decide the solvability of Diophantine equations.
• To extend Denef’s method—originally for real function fields—to p-adic settings, using quadratic twists of elliptic curves with rank one over rational function fields.
• To construct a Diophantine subset of the function field isomorphic to Z, leveraging valuation theory and isotropy of quadratic forms over p-adic fields.
• To show that the semilocal ring at a divisor on a smooth quasiprojective variety over such fields also has undecidable positive-existential theory.

Proposed method

• Construct a quadratic twist Ẽ of a rational elliptic curve E over Q(t) with Mordell-Weil rank one, ensuring the twist is defined over a function field K via a rational function f vanishing at a divisor Q.
• Use R. Noot’s theorem to ensure the Mordell-Weil group Ẽ(K) is isomorphic to Ẽ(Q(t)), preserving the rank-one structure over the function field.
• Define a Diophantine subset Λ ⊂ K² with a ring structure isomorphic to Z, using explicit polynomial equations and valuation conditions.
• Apply valuation-theoretic techniques: define subsets Y₀, Y₁ ⊂ Q(f) based on v₀ and v∞ valuations, and prove Y₁ is relatively Diophantine in Y via isotropy of quadratic forms.
• Use the isotropy of quadratic forms φ₀ and φ₁ over K, depending on parameters c₃, c₅ ∈ C (a Diophantine subset dense in Qₚ), to define the ring multiplication via Diophantine conditions.
• Leverage results from Kim and Roush on isotropy and Newton polygon conditions to ensure that for r ∈ Y₁, both forms are isotropic, while for r ∈ Y₀, at least one is anisotropic.

Experimental results

Research questions

• RQ1 Can Denef’s method for proving undecidability over real function fields be extended to p-adic function fields?
• RQ2 Under what conditions is the Mordell-Weil group of a twisted elliptic curve over a function field isomorphic to that over the base rational function field?
• RQ3 Is there a Diophantine subset of a function field over a real or p-adic field that is isomorphic to Z as a ring?
• RQ4 Can the ring structure of such a Diophantine subset be defined using only existential formulas in the language of rings?
• RQ5 What role do valuation conditions and isotropy of quadratic forms play in defining the arithmetic of function fields over non-algebraically closed fields?

Key findings

• The positive-existential theory of the semilocal ring A at a nonempty divisor Q on a smooth quasiprojective variety V over a field k of characteristic zero is undecidable.
• For function fields K of curves over a formally real field k, the positive-existential theory is undecidable, extending Denef’s result to non-rational function fields.
• When k is a subfield of a finite extension of Qₚ for an odd prime p, the positive-existential theory of K is undecidable, completing the p-adic case.
• A Diophantine subset Λ ⊂ K² is constructed with a ring structure isomorphic to Z, using a twisted elliptic curve and valuation-theoretic conditions.
• The multiplication in Λ is Diophantine, relying on the isotropy of quadratic forms φ₀ and φ₁ over K, which is ensured via Newton polygon conditions and p-adic density of parameters.
• The construction is effective in the sense that the required Diophantine sets can be explicitly defined using rational functions and valuation conditions, though full effectivity is not guaranteed in all cases.