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In this article, we show that in each of four standard families of hyperelliptic curves, there is a density-$1$ subset of members with the property that their Jacobians have adelic Galois representation with image as large as possible. This result constitutes an explicit application of a general theorem on arbitrary rational families of abelian varieties to the case of families of Jacobians of hyperelliptic curves. Furthermore, we provide explicit examples of hyperelliptic curves of genus $2$ and $3$ over $mathbb Q$ whose Jacobians have such maximal adelic Galois representations.
In this paper we study the Coleman-Oort conjecture for superelliptic curves, i.e., curves defined by affine equations $y^n=F(x)$ with $F$ a separable polynomial. We prove that up to isomorphism there are at most finitely many superelliptic curves of
Let $K$ be a field of characteristic different from $2$, $bar{K}$ its algebraic closure. Let $n ge 3$ be an odd prime such that $2$ is a primitive root modulo $n$. Let $f(x)$ and $h(x)$ be degree $n$ polynomials with coefficients in $K$ and without r
Let $C$ be a hyperelliptic curve of genus $g>1$ over an algebraically closed field $K$ of characteristic zero and $O$ one of the $(2g+2)$ Weierstrass points in $C(K)$. Let $J$ be the jacobian of $C$, which is a $g$-dimensional abelian variety over $K
Let $F$ be any field. We give a short and elementary proof that any finite subgroup $G$ of $PGL(2,F)$ occurs as a Galois group over the function field $F(x)$. We also develop a theory of descent to subfields of $F$. This enables us to realize the aut
By constructing suitable Borcherds forms on Shimura curves and using Schofers formula for norms of values of Borcherds forms at CM-points, we determine all the equations of hyperelliptic Shimura curves $X_0^D(N)$. As a byproduct, we also address the