Abstract
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 elliptic curves with good ordinary or multiplicative reduction 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.
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