On the universal p-adic sigma and Weierstrass zeta functions

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.(c) 2023 Elsevier Inc. All rights reserved.