| Let p, l be different odd prime numbers, and let kappa ∈ XQp := Homcont( Zxp,Q xp ) be a continuous character from Zxp to Qxp . We construct a modular symbol Ylk for the congruence subgroup Gamma0(pl) taking values in the space of rigid differentials on an annulus W∈ P1Cp . We show that the coefficients of that modular symbol vary continuously with kappa ∈ XQp . When k = 0, we compute the coefficients of Yl0 at (0) -- (infinity) ∈ Div0 ( P1&parl0;Qp&parr0; ) to be c0=&parl0;1-1p&parr0; ˙Logl, c1=0 andcn=1-l- n˙zp n+1˙bn, forn≥2, where bn are the Bernoulli numbers, Log is the p-adic Iwasawa logarithm, and zp (s) is the Kubota-Leopoldt p-adic L-function.;This implies that in the case p = 3, l = 11, the symbol Yl0 matches with an Eisenstein symbol phieis conjectured to exists by V. Pasol and G. Stevens, based on numerical evidence from a computer program written by R. Pollack. The importance and novelty of the symbol phieis stem from the fact that it specializes to a critical slope 1 Eisenstein symbol in SymbG033 Q3 . |
