Function-fields page

Some documents:

‘Elementary equivalence and function fields’, Bull. London Math. Soc. 31 (1999), no. 4, 431–440.
Supplementary note
Slides
On the model theory of function fields (doctoral dissertation), University of Maryland, May 1997, written with the advice of Chris Laskowski.
‘Undecidability of C(T₀, T₁)’
Elliptic curves and modular forms

Commentary:

In the paper “Elementary equivalence and function fields”, together with the supplementary note, is proved the following:

Theorem. Suppose C₀ and C₁ are curves over an algebraically closed field k (of any characteristic). If one of the curves is not an elliptic curve with complex multiplication, then the following are equivalent:

The function fields k(C₀) and k(C₁) of the curves agree on all ∀∃ sentences in the signature of k-algebras.
The function fields k(C₀) and k(C₁) are isomorphic over k.

However, if both C₀ and C₁ are elliptic curves with complex multiplication, and the characteristic of k is 0, then the following are equivalent:

The function fields k(C₀) and k(C₁) of the curves agree on all ∀∃ sentences in the signature of k-algebras.
The endomorphism rings End(C₀) and End(C₁) are isomorphic.

(It is known that over the complex numbers there are just thirteen elliptic curves with complex multiplication that are uniquely determined by their endomorphism rings.)

The paper is cited by:

Vidaux, Xavier. Multiplication complexe et équivalence élémentaire dans le langage des corps. (French) [Complex multiplication and elementary equivalents in the language of fields] J. Symbolic Logic 67 (2002), no. 2, 635--648. MR1905159 (2003g:03063)
Pop, Florian. Elementary equivalence versus isomorphism. Invent. Math. 150 (2002), no. 2, 385--408. MR1933588 (2003i:12016)

The slides were for a general seminar talk, given at METU in September 2000, concerning the results of the paper.

The paper is based on my Ph.D. dissertation

The dissertation also contains, an (unpublished) elaboration of the proof in the article of K.H. Kim and F.W. Roush, `Diophantine Undecidability of C(t₁, t₂)', in Journal of Algebra 150, 35–44 (1992).

I prepared some notes, Elliptic curves and modular forms, which constitute mainly a sketch of the proof of the Uniformization Theorem, that is, the parametrization of the elliptic curves over the complex numbers, by the complex numbers (4 pages).