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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 $\mathbb{F}{p^n}$ \mathbb{F}_{p^n}$ and the number of solutions over $\mathbb{F}{p}$ \mathbb{F}_p$ (for this, the coefficients of the equation are reduced mod $\mod p$). p$). Is there such a relation also at the bad primes?
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Given an elliptic curve $E$, 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 ${\mathbb{F}\mathbb{F}{p^n}}$ p^n}$ and the number of solutions over ${\mathbb{F}\mathbb{F}{p}}$. Is p}$ (for this, the same statement valid coefficients of the equation are reduced $\mod p$). Is there such a relation also at the bad primes?I can only find a proof at good primes, but then I have seen it stated for all primes, without proof. |
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How many points are there on an elliptic curve reduced at a bad prime?Given an elliptic curve $E$, 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 ${\mathbb{F}{p}}$. Is the same statement valid at the bad primes? I can only find a proof at good primes, but then I have seen it stated for all primes, without proof.
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