A Hecke algebra attached to mod 2 modular forms of level 5

Abstract

Let $F$ be the element ${\sum }_{nodd,n>0}{x}^{n^2}$ of $Z/2[[x]]$. Set $G=F(x^5)$, $D=F(x)+F(x^{25})$. For $k>0$, $(k,10)=1$, define $D_k$ as follows. $D_1=D$, $D_3=D^8/G$, $D_7=D^{2}G$, $D_9=D^{4}G$; furthermore $D_{k+10}=G^2{D}_k$. Using modular forms of level ${\Gamma }_0(5)$ we show that the space $W$ spanned by the $D_k$ is stabilized by the formal Hecke operators $T_p:Z/2[[x]]\to Z/2[[x]]$, $p\ne 2$ or $5$. And we determine the structure of the (completed) shallow Hecke algebra attached to $W$. This algebra proves to be a power series ring in $T_3$ and $T_7$ with an element of square $0$ adjoined. As Hecke module, $W$ identifies with a certain subquotient of the space of mod~2 modular forms of level ${\Gamma }_0(5)$, and our Hecke algebra result parallels findings in level 1 (by J.-L. Nicolas and J.-P. Serre) and in level ${\Gamma }_0(3)$ by us.

25 pages

References in corpus (2)

• A Hecke algebra attached to mod 2 modular forms of level 3
• Variations on a Lemma of Nicolas and Serre

Cited by in corpus (2)

• Variations on a Lemma of Nicolas and Serre
• Newforms mod p in squarefree level, with applications to Monsky's Hecke-stable filtration