It is widely believed that Weierstrass ignored Eisenstein’s theory of elliptic functions and developed an alternative treatment, which is now standard, because of a convergence issue. In particular, the Eisenstein series of weight two does not converge absolutely while Eisenstein’s theory assigned a value to this series.
It is now well-known that the quantity which Eisentsein assigned to this series is not only correct, but it has interesting interpretations and attracted much attention. It has been proved by Damerell in 1970 that this quantity is an algebraic number if the underlying elliptic curve has complex multiplication.
In 1976, N. Katz interpreted Damerell’s theorem in terms of DeRham cohomology; that allowed for a p-adic approach to this algebraic number. This p-adic version of Damerell’s theorem was instrumental in Katz’s theory of p-adic modular forms and p-adic L-functions of CM-fields. The approach, by design, works for those primes which split in the CM-field.
In this talk, we offer a modification of Katz’ p-adic approach to the weight two Eisenstein series which works uniformly well for all primes of good reduction, both inert and splitting in the CM-field.