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Fix a conductor and a prime $p$. Then

1) Do the elliptic curves in the same isogeny class after reduction modulo $p$ have the same number of points over the finite field $\mathbb{F}_{p} ?$

2) Do the elliptic curves belonging to two different isogeny classes corresponding to the fixed conductor, after reduction modulo $p$ have the same number of points over the finite field $\mathbb{F}_{p} ?$

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3) Yes. They have the same characteristic polynomial of Frobenius acting on the Tate module, hence the same number of points.

4) Depends on how you want to reduce mod p. Certainly if they are isogenous with the same conductor, then they will have the same reduction type: split / non-split multiplicative reduction or additive reduction, corresponding to p-1, p+1, or p points. (Or 1 more, counting the singular point.)

However another reasonable way to count points over the reduction includes the number of components in the component group of the Neron model. This is not preserved by isogeny.

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    $\begingroup$ I'm a little confused by your answer to the second question, which seems to assume that the reduction is bad. From the way the question is written I would instead have assumed, if anything, that the reduction is good (i.e., $p$ doesn't divide the conductor). But it's a little vague... $\endgroup$ – Pete L. Clark Dec 4 '13 at 6:44
  • $\begingroup$ My question is for both good and bad reduction. $\endgroup$ – Suman Dec 4 '13 at 7:34
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If in (2) you are asking whether the conductor entirely determines the number of points on all reductions, the answer is most assuredly not. If that were the case then there would only be one cusp form over $\mathbf{Q}$ for each $\Gamma_0(N)$ where $N$ is your conductor. But there are counterexamples all over the place for this.

For your specific example, take $p =3 $ and $N=37$. The 37a isogeny class is supersingular while the 37b isogeny class is not. Therefore the number of points mod $3$ on say 37.a1 and 37.b1 (LMFDB labels) will be different.

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