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lim sup didn't make sense there
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David Lampert
David Lampert
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Fix $E/K$ an elliptic curve over a number field $K$. For various towers of finite field extensions $K=K_0 \subset K_1 \subset K_2\subset\cdots$ how fast can $\operatorname{rank}(E(K_n))$ grow in terms of $|K_n:K|$?

If $E:y^2=f(x)$ then there exist $x_n \in \mathbb{Q}$ such that $\sqrt{f(x_n)} \notin K_{n-1} = K(\sqrt{f(x_1)},\sqrt{f(x_2)}, \ldots, \sqrt{f(x_{n-1})})$ and $P_n=(x_n,\sqrt{f(x_n)})$ is independent of $E(K_{n-1})$, and thus $\operatorname{rank}(E(K_n)) \ge \log_2(|K_n:K|) + \operatorname{rank}(E(K))$, and if $E/K$ has complex multiplication then $\operatorname{rank}(E(K_n)) \ge 2\log_2(|K_n:K|) + \operatorname{rank}(E(K))$.

What function of $|K_n:K|$ is an upper bound to $\limsup \operatorname{rank}(E(K_n))$$\operatorname{rank}(E(K_n))$ for all towers $\{K_n\}$? Is there an example with $\limsup \operatorname{rank}(E(K_n))/\log_2(|K_n:K|) \gt 1$ resp.${} \gt 2$ for fixed $E/K$ without resp. with complex multiplication?

Fix $E/K$ an elliptic curve over a number field $K$. For various towers of finite field extensions $K=K_0 \subset K_1 \subset K_2\subset\cdots$ how fast can $\operatorname{rank}(E(K_n))$ grow in terms of $|K_n:K|$?

If $E:y^2=f(x)$ then there exist $x_n \in \mathbb{Q}$ such that $\sqrt{f(x_n)} \notin K_{n-1} = K(\sqrt{f(x_1)},\sqrt{f(x_2)}, \ldots, \sqrt{f(x_{n-1})})$ and $P_n=(x_n,\sqrt{f(x_n)})$ is independent of $E(K_{n-1})$, and thus $\operatorname{rank}(E(K_n)) \ge \log_2(|K_n:K|) + \operatorname{rank}(E(K))$, and if $E/K$ has complex multiplication then $\operatorname{rank}(E(K_n)) \ge 2\log_2(|K_n:K|) + \operatorname{rank}(E(K))$.

What function of $|K_n:K|$ is an upper bound to $\limsup \operatorname{rank}(E(K_n))$ for all towers $\{K_n\}$? Is there an example with $\limsup \operatorname{rank}(E(K_n))/\log_2(|K_n:K|) \gt 1$ resp.${} \gt 2$ for fixed $E/K$ without resp. with complex multiplication?

Fix $E/K$ an elliptic curve over a number field $K$. For various towers of finite field extensions $K=K_0 \subset K_1 \subset K_2\subset\cdots$ how fast can $\operatorname{rank}(E(K_n))$ grow in terms of $|K_n:K|$?

If $E:y^2=f(x)$ then there exist $x_n \in \mathbb{Q}$ such that $\sqrt{f(x_n)} \notin K_{n-1} = K(\sqrt{f(x_1)},\sqrt{f(x_2)}, \ldots, \sqrt{f(x_{n-1})})$ and $P_n=(x_n,\sqrt{f(x_n)})$ is independent of $E(K_{n-1})$, and thus $\operatorname{rank}(E(K_n)) \ge \log_2(|K_n:K|) + \operatorname{rank}(E(K))$, and if $E/K$ has complex multiplication then $\operatorname{rank}(E(K_n)) \ge 2\log_2(|K_n:K|) + \operatorname{rank}(E(K))$.

What function of $|K_n:K|$ is an upper bound to $\operatorname{rank}(E(K_n))$ for all towers $\{K_n\}$? Is there an example with $\limsup \operatorname{rank}(E(K_n))/\log_2(|K_n:K|) \gt 1$ resp.${} \gt 2$ for fixed $E/K$ without resp. with complex multiplication?

badly needed MathJax usage corrections
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Michael Hardy
Michael Hardy
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Fix $E/K$ an elliptic curve over a number field $K$. For various towers of finite field extensions $K=K_0 \subset K_1 \subset K_2...$$K=K_0 \subset K_1 \subset K_2\subset\cdots$ how fast can $rank(E(K_n))$$\operatorname{rank}(E(K_n))$ grow in terms of $|K_n:K|$?

If $E:y^2=f(x)$ then there exist $x_n \in \mathbb{Q}$ such that $\sqrt{f(x_n)} \notin K_{n-1}=K(\sqrt{f(x_1)},\sqrt{f(x_2)},...,\sqrt{f(x_{n-1})})$$\sqrt{f(x_n)} \notin K_{n-1} = K(\sqrt{f(x_1)},\sqrt{f(x_2)}, \ldots, \sqrt{f(x_{n-1})})$ and $P_n=(x_n,\sqrt{f(x_n)})$ is independent of $E(K_{n-1})$, and thus $rank(E(K_n)) \ge log_2(|K_n:K|) + rank(E(K))$$\operatorname{rank}(E(K_n)) \ge \log_2(|K_n:K|) + \operatorname{rank}(E(K))$, and if $E/K$ has complex multiplication then $rank(E(K_n)) \ge 2log_2(|K_n:K|) + rank(E(K))$$\operatorname{rank}(E(K_n)) \ge 2\log_2(|K_n:K|) + \operatorname{rank}(E(K))$.

What function of $|K_n:K|$ is an upper bound to lim sup $rank(E(K_n))$$\limsup \operatorname{rank}(E(K_n))$ for all towers $\{K_n\}$? Is there an example with lim sup $rank(E(K_n))/log_2(|K_n:K|) \gt 1$$\limsup \operatorname{rank}(E(K_n))/\log_2(|K_n:K|) \gt 1$ resp. $\gt 2$${} \gt 2$ for fixed $E/K$ without resp. with complex multiplication?

Fix $E/K$ an elliptic curve over a number field $K$. For various towers of finite field extensions $K=K_0 \subset K_1 \subset K_2...$ how fast can $rank(E(K_n))$ grow in terms of $|K_n:K|$?

If $E:y^2=f(x)$ then there exist $x_n \in \mathbb{Q}$ such that $\sqrt{f(x_n)} \notin K_{n-1}=K(\sqrt{f(x_1)},\sqrt{f(x_2)},...,\sqrt{f(x_{n-1})})$ and $P_n=(x_n,\sqrt{f(x_n)})$ is independent of $E(K_{n-1})$, and thus $rank(E(K_n)) \ge log_2(|K_n:K|) + rank(E(K))$, and if $E/K$ has complex multiplication then $rank(E(K_n)) \ge 2log_2(|K_n:K|) + rank(E(K))$.

What function of $|K_n:K|$ is an upper bound to lim sup $rank(E(K_n))$ for all towers $\{K_n\}$? Is there an example with lim sup $rank(E(K_n))/log_2(|K_n:K|) \gt 1$ resp. $\gt 2$ for fixed $E/K$ without resp. with complex multiplication?

Fix $E/K$ an elliptic curve over a number field $K$. For various towers of finite field extensions $K=K_0 \subset K_1 \subset K_2\subset\cdots$ how fast can $\operatorname{rank}(E(K_n))$ grow in terms of $|K_n:K|$?

If $E:y^2=f(x)$ then there exist $x_n \in \mathbb{Q}$ such that $\sqrt{f(x_n)} \notin K_{n-1} = K(\sqrt{f(x_1)},\sqrt{f(x_2)}, \ldots, \sqrt{f(x_{n-1})})$ and $P_n=(x_n,\sqrt{f(x_n)})$ is independent of $E(K_{n-1})$, and thus $\operatorname{rank}(E(K_n)) \ge \log_2(|K_n:K|) + \operatorname{rank}(E(K))$, and if $E/K$ has complex multiplication then $\operatorname{rank}(E(K_n)) \ge 2\log_2(|K_n:K|) + \operatorname{rank}(E(K))$.

What function of $|K_n:K|$ is an upper bound to $\limsup \operatorname{rank}(E(K_n))$ for all towers $\{K_n\}$? Is there an example with $\limsup \operatorname{rank}(E(K_n))/\log_2(|K_n:K|) \gt 1$ resp.${} \gt 2$ for fixed $E/K$ without resp. with complex multiplication?

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David Lampert
David Lampert
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How fast can elliptic curve rank grow in towers of number fields?

Fix $E/K$ an elliptic curve over a number field $K$. For various towers of finite field extensions $K=K_0 \subset K_1 \subset K_2...$ how fast can $rank(E(K_n))$ grow in terms of $|K_n:K|$?

If $E:y^2=f(x)$ then there exist $x_n \in \mathbb{Q}$ such that $\sqrt{f(x_n)} \notin K_{n-1}=K(\sqrt{f(x_1)},\sqrt{f(x_2)},...,\sqrt{f(x_{n-1})})$ and $P_n=(x_n,\sqrt{f(x_n)})$ is independent of $E(K_{n-1})$, and thus $rank(E(K_n)) \ge log_2(|K_n:K|) + rank(E(K))$, and if $E/K$ has complex multiplication then $rank(E(K_n)) \ge 2log_2(|K_n:K|) + rank(E(K))$.

What function of $|K_n:K|$ is an upper bound to lim sup $rank(E(K_n))$ for all towers $\{K_n\}$? Is there an example with lim sup $rank(E(K_n))/log_2(|K_n:K|) \gt 1$ resp. $\gt 2$ for fixed $E/K$ without resp. with complex multiplication?