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Will Sawin
Will Sawin
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  1. It shouldn't be too hard to find some necessary conditions and some sufficient conditions, e.g., complex conjugation acts on dessins by reflection, so a dessin defined over Q should certainly have a mirror symmetry.

  2. One can certainly give an explicit dessin for all the modular curves, since they all have a map to $X(1) \cong \mathbb P^1$ ramified over only $3$ points (the elliptic points and the cusp), the precondition for a dessin. One can compute it by looking at the group action of $\Gamma/\Gamma(n)$$\Gamma/\Gamma(N)$.

  3. The existence of a map which factors through the map to $\mathbb P^1$ is an obvious sufficient condition. It is fairly painless to check but seems very unlikely to be strong enough. I would expect that the problem is extremely hard in general.

  1. It shouldn't be too hard to find some necessary conditions and some sufficient conditions, e.g., complex conjugation acts on dessins by reflection, so a dessin defined over Q should certainly have a mirror symmetry.

  2. One can certainly give an explicit dessin for all the modular curves, since they all have a map to $X(1) \cong \mathbb P^1$ ramified over only $3$ points (the elliptic points and the cusp), the precondition for a dessin. One can compute it by looking at the group action of $\Gamma/\Gamma(n)$.

  3. The existence of a map which factors through the map to $\mathbb P^1$ is an obvious sufficient condition. It is fairly painless to check but seems very unlikely to be strong enough. I would expect that the problem is extremely hard in general.

  1. It shouldn't be too hard to find some necessary conditions and some sufficient conditions, e.g., complex conjugation acts on dessins by reflection, so a dessin defined over Q should certainly have a mirror symmetry.

  2. One can certainly give an explicit dessin for all the modular curves, since they all have a map to $X(1) \cong \mathbb P^1$ ramified over only $3$ points (the elliptic points and the cusp), the precondition for a dessin. One can compute it by looking at the group action of $\Gamma/\Gamma(N)$.

  3. The existence of a map which factors through the map to $\mathbb P^1$ is an obvious sufficient condition. It is fairly painless to check but seems very unlikely to be strong enough. I would expect that the problem is extremely hard in general.

Source Link
Will Sawin
Will Sawin
  • 148.5k
  • 9
  • 324
  • 563

  1. It shouldn't be too hard to find some necessary conditions and some sufficient conditions, e.g., complex conjugation acts on dessins by reflection, so a dessin defined over Q should certainly have a mirror symmetry.

  2. One can certainly give an explicit dessin for all the modular curves, since they all have a map to $X(1) \cong \mathbb P^1$ ramified over only $3$ points (the elliptic points and the cusp), the precondition for a dessin. One can compute it by looking at the group action of $\Gamma/\Gamma(n)$.

  3. The existence of a map which factors through the map to $\mathbb P^1$ is an obvious sufficient condition. It is fairly painless to check but seems very unlikely to be strong enough. I would expect that the problem is extremely hard in general.