Let $K$ be a number field and let $\mathcal O_K$ be the ring of integers. Following [this paper][1] of Cornelissen, Pheidas, and Zahidi, a key ingredient needed to show that Hilbert's tenth problem has a negative solution over $\mathcal O_K$ is an elliptic curve $E$ defined over $K$ with rank$(E(K))=1$.

Recently [Mazur and Rubin][2] have shown that such a curve exists assuming the Shafarevich-Tate conjecture for elliptic curves over number fields. They actually use a weaker, but still inaccessible hypothesis (See conjecture $ШT_2$).

If you wanted to eliminate the need for this hypothesis you would have to write a proof that simultaneously demonstrated that rank$(E(K))=1$ for infinitely many pairs $(K,E)$ where $E$ is an elliptic curve defined over $K.$ This raises (as opposed to begs) the easier question:

> Can you show unconditionally that rank$(E(\Bbb Q)) = 1$ for infinitely many elliptic curves $E$ over $\Bbb Q$?

It would appear that Byeon, Jeon, and Kim have done so in [this paper (probably need an institutional login)][3]. Vatsal obtains a weaker result [here][4] that still does the job. Unfortunately both of these results invoke the fact that the BSD rank conjecture is true for elliptic curves over $\Bbb Q$ with analytic rank 1. Which won't help at present working over number fields.

>Can anyone do the above **WITHOUT** invoking the proven part of the BSD rank conjecture or assuming any conjectures?


  [1]: http://www.emis.de/journals/JTNB/2005-3/article02.pdf
  [2]: http://arxiv.org/pdf/0904.3709v2
  [3]: http://www.reference-global.com/doi/pdf/10.1515/CRELLE.2009.060
  [4]: http://www.math.ubc.ca/~vatsal/research/der.PDF