arXiv subjects

T. Shaska

Publications and source records attributed to T. Shaska.

38 records · Page 3Linked to original sources

arXiv2003

Curves of genus 2 with (n, n)-decomposable jacobians

Let $C$ be a curve of genus 2 and $ψ_1:C \lar E_1$ a map of degree $n$, from $C$ to an elliptic curve $E_1$, both curves defined over $\bC$. This map induces a degree $n$ map $ϕ_1:\bP^1 \lar \bP^1$ which we call a Frey-Kani covering. We determine all possible ramifications for $ϕ_1$. If $ψ_1:C \lar E_1$ is maximal then there exists a maximal map $ψ_2:C\lar E_2$, of degree $n$, to some elliptic curve $E_2$ such that there is an isogeny of degree $n^2$ from the Jacobian $J_C$ to $E_1 \times E_2$. We say that $J_C$ is $(n,n)$-decomposable. If the degree $n$ is odd the pair $(ψ_2, E_2)$ is canonically determined. For $n=3, 5$, and 7, we give arithmetic examples of curves whose Jacobians are $(n,n)$-decomposable.

math.AG↗

arXiv2002

Genus 2 curves with (3,3)-split Jacobian and large automorphism group

Let $\C$ be a genus 2 curve defined over $k$, $char (k) =0$. If $\C$ has a $(3,3)$-split Jacobian then we show that the automorphism group $Aut(\C)$ is isomorphic to one of the following: $\bZ_2, V_4, D_8$, or $D_{12}$. There are exactly six $\bC$-isomorphism classes of genus two curves $\C$ with $Aut(\C)$ isomorphic to $D_8$ (resp., $D_{12}$). %We compute their absolute invariants $i_1, i_2, i_3$. We show that exactly four (resp., three) of these classes with group $D_8$ (resp., $D_{12}$) have representatives defined over $\bQ$. We discuss some of these curves in detail and find their rational points.

math.AG↗