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I have several questions regarding the degrees of morphisms between algebraic curves.

  1. If we have algebraic curves $X$ and $Y$ defined over some perfect field $k$, can we determine the least degree of a morphism defined over $k$ between them (we can assume we know that a morphism between them exists)?

  2. Same question, but with $Y$ being an elliptic curve.

  3. Same question, but with $X$ being the modular curve $X_0(N)$ and $Y$ being an elliptic curve with conductor $M \mid N$ and $M<N$ (for $M=N$ it is the modular degree).

If determining the exact least degree is not possible, methods to obtain lower bounds would be appreciated. For example, we can look at the number of points modulo $\pi$, where $\pi$ is a prime of good reduction for both $X$ and $Y$. We can also use the Castelnuovo-Severi inequality. What are the other methods?

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  • $\begingroup$ Assume $X$ and $Y$ have genus at least two. Let $f:X\to Y$ be a non-constant morphism. Then $\deg f \leq (2g(X)-2)/(2g(Y)-2)$. This gives you an upper bound on the least possible degree. $\endgroup$
    – Ariyan Javanpeykar
    Commented Nov 4, 2022 at 6:17
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    $\begingroup$ (Same hypotheses on $X$ and $Y$.) The set of non-constant morphisms from $X$ to $Y$ is finite and "Effectively computable". So one could in practice compute the least minimal degree of a morphism from $X$ to $Y$. $\endgroup$
    – Ariyan Javanpeykar
    Commented Nov 4, 2022 at 6:18
  • $\begingroup$ There is a work "Paulin Roland - On the minimal degree of morphisms between algebraic curves". $\endgroup$
    – Watson
    Commented Nov 4, 2022 at 7:13
  • $\begingroup$ @AriyanJavanpeykar So there is an algorithm for finding the minimal degree? Where can I find it? $\endgroup$
    – Petar Orlic
    Commented Nov 5, 2022 at 18:00
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    $\begingroup$ @DamianRössler Actually that was the motivation for the question. I was studying the gonality of the modular curve X_0(N) and I am interested about results when $Y$ is an elliptic curve instead of $\mathbb{P}^1$. $\endgroup$
    – Petar Orlic
    Commented Nov 10, 2022 at 14:21

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