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Joe Silverman
Joe Silverman
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You need to use Siegel's theorem (which is quite deep and relies on Roth's theorem or some such). This is in Chapter IX of Arithmetic of Elliptic Curves, specifically Theoream IX.3.1. You'll need to unsort the definitions a bit, since itsit's stated for number fields, but in your notation, one has $$ \lim_{n\to\infty} \frac{2\log d_n}{n^2} = \lim_{n\to\infty} \frac{\log a_n}{n^2} = \hat h(P). $$$$ \lim_{n\to\infty} \frac{2\log d_n}{n^2} = \lim_{n\to\infty} \frac{\log |x_n|}{n^2} = \hat h(P). $$ Hmmm.. Actually, the definitions are unsorted for you in Example IX.3.3, where you'll find the following formula (using your notation) in the middle of page 279 (of the 2nd edition): $$ \lim_{n\to\infty} \frac{\log|a_n|}{\log d_n^2} = 1. $$$$ \lim_{n\to\infty} \frac{\log|x_n|}{\log d_n^2} = 1. $$

BTW, the sequence $(d_n)_{n\ge1}$ is called the Elliptic Divisibility Sequence associated to the curve $E$ and point $P$. The fact that $\log d_n$ grows like a multiple of $n^2$ is an essential fact used to prove that elliptic divisibility sequences satisfy the Zsigmondy property: for all but finitely many $n$, there is a prime $p$ such that $p\mid d_n$ and $p\nmid d_m$ for all $m < n$.

You need to use Siegel's theorem (which is quite deep and relies on Roth's theorem or some such). This is in Chapter IX of Arithmetic of Elliptic Curves, specifically Theoream IX.3.1. You'll need to unsort the definitions a bit, since its stated for number fields, but in your notation, one has $$ \lim_{n\to\infty} \frac{2\log d_n}{n^2} = \lim_{n\to\infty} \frac{\log a_n}{n^2} = \hat h(P). $$ Hmmm.. Actually, the definitions are unsorted for you in Example IX.3.3, where you'll find the following formula (using your notation) in the middle of page 279 (of the 2nd edition): $$ \lim_{n\to\infty} \frac{\log|a_n|}{\log d_n^2} = 1. $$

BTW, the sequence $(d_n)_{n\ge1}$ is called the Elliptic Divisibility Sequence associated to the curve $E$ and point $P$. The fact that $\log d_n$ grows like a multiple of $n^2$ is an essential fact used to prove that elliptic divisibility sequences satisfy the Zsigmondy property: for all but finitely many $n$, there is a prime $p$ such that $p\mid d_n$ and $p\nmid d_m$ for all $m < n$.

You need to use Siegel's theorem (which is quite deep and relies on Roth's theorem or some such). This is in Chapter IX of Arithmetic of Elliptic Curves, specifically Theoream IX.3.1. You'll need to unsort the definitions a bit, since it's stated for number fields, but in your notation, one has $$ \lim_{n\to\infty} \frac{2\log d_n}{n^2} = \lim_{n\to\infty} \frac{\log |x_n|}{n^2} = \hat h(P). $$ Hmmm.. Actually, the definitions are unsorted for you in Example IX.3.3, where you'll find the following formula (using your notation) in the middle of page 279 (of the 2nd edition): $$ \lim_{n\to\infty} \frac{\log|x_n|}{\log d_n^2} = 1. $$

BTW, the sequence $(d_n)_{n\ge1}$ is called the Elliptic Divisibility Sequence associated to the curve $E$ and point $P$. The fact that $\log d_n$ grows like a multiple of $n^2$ is an essential fact used to prove that elliptic divisibility sequences satisfy the Zsigmondy property: for all but finitely many $n$, there is a prime $p$ such that $p\mid d_n$ and $p\nmid d_m$ for all $m < n$.

Source Link
Joe Silverman
Joe Silverman
  • 47.4k
  • 2
  • 149
  • 241

You need to use Siegel's theorem (which is quite deep and relies on Roth's theorem or some such). This is in Chapter IX of Arithmetic of Elliptic Curves, specifically Theoream IX.3.1. You'll need to unsort the definitions a bit, since its stated for number fields, but in your notation, one has $$ \lim_{n\to\infty} \frac{2\log d_n}{n^2} = \lim_{n\to\infty} \frac{\log a_n}{n^2} = \hat h(P). $$ Hmmm.. Actually, the definitions are unsorted for you in Example IX.3.3, where you'll find the following formula (using your notation) in the middle of page 279 (of the 2nd edition): $$ \lim_{n\to\infty} \frac{\log|a_n|}{\log d_n^2} = 1. $$

BTW, the sequence $(d_n)_{n\ge1}$ is called the Elliptic Divisibility Sequence associated to the curve $E$ and point $P$. The fact that $\log d_n$ grows like a multiple of $n^2$ is an essential fact used to prove that elliptic divisibility sequences satisfy the Zsigmondy property: for all but finitely many $n$, there is a prime $p$ such that $p\mid d_n$ and $p\nmid d_m$ for all $m < n$.