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David Loeffler
David Loeffler
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Your belief is correct. A $\mathbb{Z}$-point has to reduce to an $\mathbb{F}_p$-point for all $p$, which kills examples with gcd > 1.

If you want to make this precise, try writing down an explicit description of X by patching affine pieces. All the essential ideas are already there in $\mathbb{A}^1 \backslash 0$$\mathbb{A}_{\mathbb{Z}}^1 \backslash 0$: this is the spectrum of $\mathbb{Z}[X, Y] / (XY - 1)$.

Your belief is correct. A $\mathbb{Z}$-point has to reduce to an $\mathbb{F}_p$-point for all $p$, which kills examples with gcd > 1.

If you want to make this precise, try writing down an explicit description of X by patching affine pieces. All the essential ideas are already there in $\mathbb{A}^1 \backslash 0$: this is the spectrum of $\mathbb{Z}[X, Y] / (XY - 1)$.

Your belief is correct. A $\mathbb{Z}$-point has to reduce to an $\mathbb{F}_p$-point for all $p$, which kills examples with gcd > 1.

If you want to make this precise, try writing down an explicit description of X by patching affine pieces. All the essential ideas are already there in $\mathbb{A}_{\mathbb{Z}}^1 \backslash 0$: this is the spectrum of $\mathbb{Z}[X, Y] / (XY - 1)$.

Source Link
David Loeffler
David Loeffler
  • 37k
  • 3
  • 89
  • 194

Your belief is correct. A $\mathbb{Z}$-point has to reduce to an $\mathbb{F}_p$-point for all $p$, which kills examples with gcd > 1.

If you want to make this precise, try writing down an explicit description of X by patching affine pieces. All the essential ideas are already there in $\mathbb{A}^1 \backslash 0$: this is the spectrum of $\mathbb{Z}[X, Y] / (XY - 1)$.