## background/motivation

let *E _{k}* denote the modular form of level one and weight

*k*with

*q*-expansion given by $E_k(q)=1- \frac{2k}{b_k}\sum_n \sigma_{k-1}(n)q^n$ where σ

_{i}is the divisor sum and

*b*is the

_{k}*k*-th bernoulli number. for

*k = p – 1*the the denominators of this series don't contain any powers of

*p*and hence by the

*q*-expansion principle

*E*defines a modular form over ℤ

_{p–1}*. that is,*

_{p}*E*∈

_{p–1}*H*

^{0}(

*X*

_{0}(1), ℤ

*⊗ ω*

_{p}^{⊗k}) where

*X*

_{0}(1) is the compactified modular curve for the full modular group, and ω is the pushforward of the canonical bundle on the universal elliptic curve over ℤ.

now i can use *p*-adic uniformisation to evaluate *E _{p–1}* via its

*q*-expansion at elliptic curves with

*j*-invariant satisfying |

*j*| > 1; in concrete terms this means formally inverting the power series for 1/j(

*q*), and then evaluating *E

_{p–1}(

*q*) on the result.

i don't suppose one should expect to get a well-defined value in ℂ_{p} for *E _{p–1}* evaluated at elliptic curves for which |

*j*| < 1 (as you would for an modular form over ℂ) since this would mean choosing a non-vanishing differential on the universal elliptic curve over ℤ

*. the*

_{p}*norm*|

*E*|, however,

_{p–1}*is*well defined, since any two choices of basis for the canonial bundle on a given elliptic curve can only differ by a unit.

of course the context i am interested in is when |*E _{p–1}*| is used to define the "overconvergent" region of the modular curve. this is (roughly) defined as the region of the $j$-line satisfying |

*E*|>

_{p–1}*r*for |

*r*|<1. this region will (always) include part of the |

*j*|<1 region. in order to understand what |

*E*|>

_{p–1}*r*"means" in concrete terms, i was hoping to numerically compare |

*E*| and |j| for some specific ℂ

_{4}_{p}-values of

*j*near 0 (for primes

*p*= 5, 7, 11 etc. for which

*j*= 0 is supersingular).

anyway, my question is:

how can i

explicitlyevaluate |E| at elliptic curves with |_{p–1}j| < 1?