G. Wijesiri | Searcharxiv

arXiv2013

Degree 4 coverings of elliptic curves by genus 2 curves

Genus two curves covering elliptic curves have been the object of study of many articles. For a fixed degree $n$ the subloci of the moduli space $\mathcal M_2$ of curves having a degree $n$ elliptic subcover has been computed for $n=3, 5$ and discussed in detail for $n$ odd; see \cite{Sh1, SV2, Fr, FK}. When the degree of the cover is even the case in general has been treated in \cite{PRS}. In this paper we compute the sublocus of $\mathcal M_2$ of curves having a degree 4 elliptic subcover.

math.AG↗

arXiv2012

Codes over rings of size $p^2$ and lattices over imaginary quadratic fields

Let $\ell>0$ be a square-free integer congruent to 3 mod 4 and $Ø_K$ the ring of integers of the imaginary quadratic field $K=Q(\sqrt{-\ell})$. Codes $C$ over rings $Ø_K / p Ø_K$ determine lattices $Λ_\ell (C) $ over $K$. If $ p \nmid \ell$ then the ring $\R:=Ø_K / p Ø_K$ is isomorphic to $\F_{p^2}$ or $\F_p \times \F_p$. Given a code $C$ over $\R$, theta functions on the corresponding lattices are defined. These theta series $θ_{Λ_{\ell}(C)}$ can be written in terms of the complete weight enumerator of $C$. We show that for any two $\ell < \ell^\prime$ the first $\frac {\ell + 1} 4$ terms of their corresponding theta functions are the same. Moreover, we conjecture that for $\ell > \frac {p(n+1)(n+2)} 2$ there is a unique complete weight enumerator corresponding to a given theta function. We verify the conjecture for primes $p< 7$ and $\ell \leq 59$.

math.AG↗