2016/10/22 by Monsky, Paul
#FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.1610.07058
Let F be the element ∑n odd, n>0x^n2 of Z/2[[x]]. Set G=F(x5), D=F(x)+F(x25). For k>0, (k,10)=1, define Dk as follows. D1=D, D3=D8/G, D7=D2G, D9=D4G; furthermore Dk+10=G2Dk. Using modular forms of level Γ0(5) we show that the space W spanned by the Dk is stabilized by the formal Hecke operators Tp:Z/2[[x]]→ Z/2[[x]], p≠ 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 T3 and T7 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 Γ0(5), and our Hecke algebra result parallels findings in level 1 (by J.-L. Nicolas and J.-P. Serre) and in level Γ0(3) by us.