## Welcome to MathOverflow

MathOverflow is a question and answer site for professional mathematicians. It's built and run by you as part of the Stack Exchange network of Q&A sites. With your help, we're working together to build a library of detailed answers to every question about research level mathematics.

We're a little bit different from other sites. Here's how:

This site is all about getting answers. It's not a discussion forum. There's no chit-chat.

Just questions...

up vote

Good answers are voted up and rise to the top.

The best answers show up first so that they are always easy to find.

accept

Accepting doesn't mean it's the best answer, it just means that it worked for the person who asked.

# Order of reduction of infinite order rational point on an Elliptic Curve

Let $E/$ℚ be an elliptic curve and $P$ ∈ $E($ℚ$)$ a rational point of infinite order. Does the reduction of $P$ mod $p$ generate a maximal cyclic subgroup of $E(\mathbb{F}$$p$$)$ for almost all primes $p$?

However, it is true that $$f_p=\text{order of \tilde P in \tilde E(\mathbb F_p)}$$ cannot be "too small, too often." For example, for every $\epsilon>0$, the series $$\sum_{p~\text{prime}} \frac{\log p}{p\cdot f_p^\epsilon}$$ converges. (More precisely, the series is ${}\le 3\epsilon^{-1}+O(1)$ as $\epsilon\to0$.)

No. There is a positive density of primes that split in ℚ$(Q, E[2])$ (where $2Q=P$) and excluding the finitely many primes for which reduction of $E[2]$ isn't injective. For such primes any maximal cyclic subgroup of $E(\mathbb{F}$$p$$)$ has even order so reduction of $P$ can't be a generator (since reduction of $P$ is 2-divisible).

## Get answers to practical, detailed questions

Focus on questions about an actual problem you have faced. Include details about what you have tried and exactly what you are trying to do.

• Specific issues with research level mathematics
• Real problems or questions that you’ve encountered

Not all questions work well in our format. Avoid questions that are primarily opinion-based, or that are likely to generate discussion rather than answers.

Questions that need improvement may be closed until someone fixes them.

• Anything not directly related to research level mathematics
• Questions that are primarily opinion-based
• Questions with too many possible answers or that would require an extremely long answer

## Tags make it easy to find interesting questions

All questions are tagged with their subject areas. Each can have up to 5 tags, since a question might be related to several subjects.

Click any tag to see a list of questions with that tag, or go to the tag list to browse for topics that interest you.

# Order of reduction of infinite order rational point on an Elliptic Curve

Let $E/$ℚ be an elliptic curve and $P$ ∈ $E($ℚ$)$ a rational point of infinite order. Does the reduction of $P$ mod $p$ generate a maximal cyclic subgroup of $E(\mathbb{F}$$p$$)$ for almost all primes $p$?

## You earn reputation when people vote on your posts

+5 question voted up
+2 edit approved

As you earn reputation, you'll unlock new privileges like the ability to vote, comment, and even edit other people's posts.

Reputation Privilege
15 Vote up
125 Vote down (costs 1 rep on answers)

At the highest levels, you'll have access to special moderation tools. You'll be able to work alongside our community moderators to keep the site focused and helpful.

2000 Edit other people's posts Vote to close, reopen, or migrate questions Access to moderation tools
see all privileges

## Improve posts by editing or commenting

Our goal is to have the best answers to every question, so if you see questions or answers that can be improved, you can edit them.

Use edits to fix mistakes, improve formatting, or clarify the meaning of a post.

You can always comment on your own questions and answers. Once you earn 50 reputation, you can comment on anybody's post.

Remember: we're all here to learn, so be friendly and helpful!

However, it is true that $$f_p=\text{order of \tilde P in \tilde E(\mathbb F_p)}$$ cannot be "too small, too often." For example, for every $\epsilon>0$, the series $$\sum_{p~\text{prime}} \frac{\log p}{p\cdot f_p^\epsilon}$$ converges. (More precisely, the series is ${}\le 3\epsilon^{-1}+O(1)$ as $\epsilon\to0$.)

That particular result is a special case of one of the main results of: Murty, Rosen, Silverman, Variations on a theme of Romanoff, Internat. J. Math. 7 (1996), 373-391. Thank you for the kind words about my books. There are also many papers by Murty and others on the elliptic analogue of Artin's conjecture. If you google "elliptic artin conjecture", you'll find some references that will lead you to the literature on the subject. - Joe Silverman Nov 13 at 17:36

## Unlock badges for special achievements

Badges are special achievements you earn for participating on the site. They come in three levels: bronze, silver, and gold.

 Informed Read the entire tour page
 Student First question with score of 1 or more Editor First edit Good Answer Answer score of 25 or more Civic Duty Vote 300 or more times Famous Question Question with 10,000 views