A characteristic 2 recurrence related to $U_5$, with a Hecke algebra application

Abstract

In arXiv:1603.03910 [math.NT] we introduced some $C_n$ in $Z/2[t]$ defined by a linear recurrence and showed that each $C_n$, $n\equiv 0\mathrm{mod}4$, is a sum of $C_k$, $k<n$. Combining this with results from arXiv:1508.07523 [math.NT] we proved that the space $K$, consisting of those odd mod~2 modular forms of level ${\Gamma }_0(3)$ that are annihilated by the operator $U_3+I$, has a basis $m_{i,j}$ "adapted to $T_7$ and $T_{13}$" in the sense of Nicolas and Serre. (And so the "completed shallow Hecke algebra" attached to $K$ is a power series ring in $T_7$ and $T_{13}$.) This note derives analogous results in level ${\Gamma }_0(5)$. Now $U_3+I$ is replaced by $U_5+I$, and the operators $T_7$ and $T_{13}$ by $T_3$ and $T_7$. In place of level ${\Gamma }_0(3)$ results from 1508.07523, we use level ${\Gamma }_0(5)$ results from arXiv:1603.07085 [math.NT]. A linear recurrence again plays the key role. Now $C_{n+6}=C_{n+5}+(t^6+t^5+t^2+t)C_n+t^n(t^2+t)$, $C_0=0$, $C_1=C_2=1$, $C_3=t$, $C_4=t^2$, $C_5=t^4+t^2+t$, and we prove that each $C_n$, $n\equiv 0$ or $2\mathrm{mod}6$ is a sum of $C_k$, $k<n$.

14 pages