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$E$ is an elliptic curve with non-split multiplicative reduction at prime $p$. I'm trying to find the number of points $E$ over $F_{p^n}$.

I know that when I remove the singularity, the rest is a group isomorphic to the kernel of the norm map from the quadratic extension of the base, t.i. includes $p^n+1$ elements (if I'm not mistaken). Plus one singular point and the answer is $p^n+2$.

But I found that this number of points depends whether n is odd or even. Why it is so?

Kartan12
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  • Relevant MO thread: https://mathoverflow.net/questions/98466/how-many-points-are-there-on-an-elliptic-curve-reduced-at-a-bad-prime – Viktor Vaughn Jun 14 '22 at 22:14
  • Thanks! But there isn't answer why the number depends on evenness of n – Kartan12 Jun 15 '22 at 06:49
  • From zeta-function I found that for odd n the answer is p^n+2, and for even - p^n. The second one seems strange. Why it is so? – Kartan12 Jun 15 '22 at 07:37

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