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Preprint Number 170

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170. Alexandra Shlapentokh
Using Indices of Points on an Elliptic Curve to Construct A Diophantine Model of Z and Define Z Using One Universal Quantifier in Very Large Subrings of Number Fields, Including Q

Submission date: 5 February 2009.


Let K be a number field and let E be an elliptic curve defined and of rank one over K. For a set \calW_K of primes of K, let O_{K,\calW_K={x\in K: ord_{p}x \geq 0, \forall p \not \in \calW_K}. Let P \in E(K) be a generator of E(K) modulo the torsion subgroup. Let (x_n(P),y_n(P)) be the affine coordinates of [n]P with respect to a fixed Weierstrass equation of E. We show that there exists a set \calW_K of primes of K of natural density one such that in O_{K,\calW_K} multiplication of indices (with respect to some fixed multiple of P) is existentially definable and therefore these indices can be used to construct a Diophantine model of Z. We also show that Z is definable over O_{K,\calW_K} using just one universal quantifier. Both, the construction of a Diophantine model using the indices and the first-order definition of Z can be lifted to the integral closure of O_{K,\calW_K} in any infinite extension K_{\infty} of K as long as E(K_{\infty}) is finitely generated and of rank one.

Mathematics Subject Classification: 11U05, 11G05

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