Given an elliptic curve $E$ defined over $\mathbb{Z}$, and a prime $p$, I know that Hasse's theorem gives, when $p$ is a good prime, a relation between the number of solutions over $\mathbb{F}_{p^n}$ and the number of solutions over $\mathbb{F}_p$ (for this, the coefficients of the equation are reduced mod $p$).

Is there such a relation also at the bad primes?

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    $\begingroup$ For bad primes it is much easier as there are essentially three cases to consider: additive reduction, split multiplicative reduction and non-split multiplicative reduction. It is a good exercise to try to count the number of points in each case yourself. $\endgroup$ – Daniel Loughran May 31 '12 at 7:38
  • $\begingroup$ Related: mathoverflow.net/questions/239513 $\endgroup$ – Watson Nov 7 '18 at 16:04

If the reduction is additive, there are $p+1$ points including one singular point. If it is split multiplicative it is $p$ and if non-split multiplicative, then it is $p+2$. See Washington "Elliptic curves, Number Theory and Cryptography ", section 2.10 on page 59.

... and see François comment below for $n>1$.

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    $\begingroup$ The way to remember it is that when you remove the singularity, the rest has a group structure, and it is isomorphic to the additive group (order $p$) in the case of additive reduction, to the multiplicative group of the base field (order $p-1$) or to the kernel of the norm map from the quadratic extension of the base (order $p+1$) in the two cases of multiplicative reduction. $\endgroup$ – Chandan Singh Dalawat May 31 '12 at 8:05
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    $\begingroup$ So when one considers the reduction of $E$ as a curve defined over $\mathbf{F}_{p^n}$, then the number of points is $p^n+1$, $p^n$ in the case of additive resp. split multiplicative reduction. In the non-split multiplicative case it will depend whether $n$ is odd or even. $\endgroup$ – François Brunault May 31 '12 at 8:14
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    $\begingroup$ The question was specifically when $n>1$, so Francois answered that. As a consequence of all this, the relation $|E(\mathbb{F}_p)| = p+1-a_p$ remains true for primes of bad reduction, where $a_p$ are coefficients of the Hasse-Weil L-function. It fails however when $p$ is replaced by $q=p^n$, except for primes where the reduction is additive. $\endgroup$ – Anonymous May 31 '12 at 8:55
  • $\begingroup$ Sorry I did not see the $n$ in the unedited version. $\endgroup$ – Chris Wuthrich May 31 '12 at 9:45
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    $\begingroup$ Careful. This answer is correct when the elliptic curve is given in Weierstrass form but not in general. For instance, $xy(x-y)=p$ defines an elliptic curve whose reduction modulo $p$ has $3p+1$ points. $\endgroup$ – Felipe Voloch May 31 '12 at 17:25

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