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Let $E_{a,b}$ be an elliptic curve defined by the equation $y^{2}=x^3+ax+b$ where $a,b \in \mathbb{Q}$. Suppose that for $a=a_{0}$ and $b=b_{0}$ the rank of $E_{a_{0},b_{0}}(\mathbb{Q})=1$.

question: is there an $\epsilon> 0$ such that for any $(\alpha,\beta) \in \mathbb{Q}^{2}$ and $|\alpha-a_{0}|+|\beta -b_{0}|< \epsilon$ then the rank of $E_{\alpha,\beta}(\mathbb{Q})=1$.

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    $\begingroup$ Arithmetic properties like the rank of the Mordell-Weil group are not continuous with respect to the real (or the $p$-adic) topology. A more interesting question would be: Given $\epsilon>0$ is there a curve of rank $1$? $\endgroup$ – Chris Wuthrich Nov 6 '17 at 21:28
  • $\begingroup$ @ChrisWuthrich is it a heuristic observation or a precise mathematical statement ? I'm talking about continuity. $\endgroup$ – M.O. Nov 6 '17 at 21:31
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    $\begingroup$ Heuristically, I believe that the answer is "no" for all $a_0$ and $b_0$. A counter-example can be constructed as follows. Take an elliptic surface over $\mathbb{Q}$ with rank $2$. By Silverman's specialisation theorem, the rank of each fibre is at least $2$ (but often larger) except for a finite number of fibres. At an exceptional fibre of rank $1$, you will get a counter-example. For instance $a_0=2$, $b_0=1$ is of rank $1$, but the family $E_t:y^2=x^3+(-t^2+t+2)\,x+1$ is of rank $2$ as it contains $(0,1)$ and $(t,t+1)$. For all $t$ close to zero the rank of $E_t$ will be $2$ or larger. $\endgroup$ – Chris Wuthrich Nov 6 '17 at 23:46
  • $\begingroup$ @ChrisWuthrich I'm satisfied with your comment. I think it would be better if you transform it to an answer, I will accept it. $\endgroup$ – M.O. Nov 7 '17 at 10:18
  • $\begingroup$ @ChrisWuthrich Can you do the converse : from an elliptic curve $E/\mathbb{Q}$ with a point $P$ of infinite order construct an elliptic curve $E_t/\mathbb{Q}(t)$ with a point $P(t)\in E_t,P = P(0)$ ? $\endgroup$ – reuns Nov 7 '17 at 15:36
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A counter-example, showing that the answer is "no" for some $(a_0,b_0)$ can be constructed as follows. Take an elliptic surface over $\mathbb{Q}$ with rank $2$. By Silverman's specialisation theorem, the rank of each fibre is at least $2$ (but often larger) except for a finite number of fibres. At an exceptional fibre of rank $1$, you will get a counter-example.

For instance $a_0=2$, $b_0=1$ the curve is of rank $1$, but the family $E_t:y^2=x^3+(−t^2+t+2)x+1$ is of rank at least $2$ over $\mathbb{Q}(t)$ as it contains $(0,1)$ and $(t,t+1)$. For all $t$ close to zero the rank of $E_t(\mathbb{Q})$ will be $2$ or larger.

I believe a similar argument should work in general and show that the answer is always "no".

Arithmetic properties like the rank of the Mordell-Weil group are not continuous in the real or $p$-adic topology. For instance, the number of prime factors of numerator and denominator of $\Delta$ will have an influence on the rank.

In the above example as $t=1/n$ approaches $0$, the rank will jump around $2$ and $3$ and sometimes larger values rather randomly: $$ \begin{array}{c|cccc} t=\tfrac{1}{n} & \tfrac{1}{2} & \tfrac{1}{3} & \tfrac{1}{4} &\tfrac{1}{5} & \tfrac{1}{6} & \tfrac{1}{7} & \tfrac{1}{8} & \tfrac{1}{9} & \tfrac{1}{10} & \tfrac{1}{11} & \tfrac{1}{12} & \tfrac{1}{13} & \tfrac{1}{14} &\tfrac{1}{15} & \tfrac{1}{16} & \tfrac{1}{16} & \tfrac{1}{17} & \tfrac{1}{18} & \tfrac{1}{19}\\ \text{rank} & 1 & 3 & 2 & 2 & 3 & 2 & 3 & 3 & 3 & 3 & 4 & 2 & 2 & 3 & 3 & 3 & 3 & 3 &2 \\ \end{array} $$

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