2016/12/06 by Paul Monsky, Monsky, Paul
Mathematics · #FOS: Mathematics #Number Theory (math.NT) #math.NT
paper · pdf · doi:10.48550/arxiv.1612.01599
14 pages
arxiv created 2016/12/06 · arxiv updated 2016/12/07
In arXiv:1603.03910 [math.NT] we introduced some Cn in Z/2[t] defined by a linear recurrence and showed that each Cn, n≡ 0 \bmod4, is a sum of Ck, 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 Γ0(3) that are annihilated by the operator U3+I, has a basis mi,j "adapted to T7 and T13" in the sense of Nicolas and Serre. (And so the "completed shallow Hecke algebra" attached to K is a power series ring in T7 and T13.) This note derives analogous results in level Γ0(5). Now U3+I is replaced by U5+I, and the operators T7 and T13 by T3 and T7. In place of level Γ0(3) results from 1508.07523, we use level Γ0(5) results from arXiv:1603.07085 [math.NT]. A linear recurrence again plays the key role. Now Cn+6 = Cn+5 + (t6+t5+t2+t)Cn+tn(t2+t), C0=0, C1=C2=1, C3=t, C4=t2, C5=t4+t2+t, and we prove that each Cn, n≡ 0 or 2\bmod6 is a sum of Ck, k<n.