On the Mordell--Weil lattice of $y^2=x^3+bx+t^{3^n+1}$ in characteristic $3$

Abstract

We study the elliptic curves given by $y^2=x^3+bx+t^{3^n+1}$ over global function fields of characteristic $3$; in particular we perform an explicit computation of the $L$-function by relating it to the zeta function of a certain superelliptic curve $u^3+bu=v^{3^n+1}$. In this way, using the Néron-Tate height on the Mordell--Weil group, we obtain lattices in dimension $2\cdot 3^n$ for every $n\ge 1$, which improve on the currently best known sphere packing densities in dimensions 162 (case $n=4$) and 486 (case $n=5$). For $n=3$, the construction has the same packing density as the best currently known sphere packing in dimension $54$, and for $n=1$ it has the same density as the lattice $E_6$ in dimension $6$.

arXiv commentv2: Minor revisions. Comments are welcome. 23 pages

References in corpus (1)

• On the arithmetic of a family of twisted constant elliptic curves