  For primes $p>3$ we produce a new derivation of the universal $p$-adic sigma
function and $p$-adic Weierstrass zeta functions of Mazur and Tate for ordinary
elliptic curves by a method that highlights congruences among coefficients in
Laurent expansions of elliptic functions, and works simultaneously for
generalized elliptic curves defined by Weierstrass equations.
