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The Tate conjecture is known for function fields (Zarhin). To check whether the Tate modules are isomorphic you need to check that the image of Frobenius match for a finite set of places that can be b …

A reference for all quotients of the Fermat curve is Lang's book "Introduction to Algebraic and Abelian Functions". There, you'll find your maps (up to twist) and several others. If you have a map $X …

If $\iota$ denotes the hyperelliptic involution, then the condition $n[P-\infty] = [Q-\infty]$ is equivalent to $nP+\iota(Q)$ linearly equivalent to $(n+1)\infty$. In a few pathological cases, where t …

The degree of ${\hat \psi}_n$ is the intersection number of the image of $\psi_n$ with the divisor on the Kummer surface defined by $\kappa_4 = 0$. Maybe the issues you are having are because the mode …

Since you wrote $\mathbb{C}^*$, I'll assume you are in characteristic zero. Then $r=2g+2$ and the group you want (modulo constants) has rank $r-1$ but is bigger than the group coming from $\mathbb{P}^ …

Joking aside... Let $x: X \to \mathbb{P}^1$ be of degree $2$ and $\sigma: X \to X$ be an automorphism. Consider $f: X \to \mathbb{P}^1\times\mathbb{P}^1, f=(x,x\circ\sigma)$. If $f$ is injective, then …