p-adic exponential function (${\exp }_p$)

The p-adic exponential function is $\exp (x)={\sum }_{n\ge 0}{x}^n/n!$ in a complete extension of ${\mathbb{Q}}_p$. It converges when $|x{|}_p<p^{-1/(p-1)}$, as the p-adic valuations of $n!$ determine the coefficient growth.

On ${\mathbb{Q}}_p$, this includes $p{\mathbb{Z}}_p$ for odd $p$, and $4{\mathbb{Z}}_2$ for $p=2$. In a sufficiently small neighborhood it satisfies $\exp (x+y)=\exp (x)\exp (y)$ and is inverse to the p-adic logarithm. Unlike the real exponential, its defining series is not convergent on the whole field, so its domain is a substantive part of its definition.

The 2-adic exponential is the power series $\exp (x)={\sum }_{n\ge 0}{x}^n/n!$ interpreted with the 2-adic absolute value. Over ${\mathbb{Q}}_2$, it converges for $v_2(x)>1$, equivalently $x\in 4{\mathbb{Z}}_2$. The general p-adic exponential function has convergence condition $|x{|}_p<p^{-1/(p-1)}$.

On $4{\mathbb{Z}}_2$, the 2-adic exponential and 2-adic logarithm give inverse group homomorphisms between the additive group $4{\mathbb{Z}}_2$ and the multiplicative group $1+4{\mathbb{Z}}_2$. The stronger domain restriction compared with odd primes comes from the powers of two dividing the factorial denominators.

On ${\mathbb{C}}_p$, the power series $\exp (x)={\sum }_{n\ge 0}{x}^n/n!$ has radius of convergenceIndeed, Legendre formula gives $v_p(n!)/n\to 1/(p-1)$, and the Cauchy-Hadamard theorem gives the stated radius.

The p-adic logarithm is defined by $\log (1+x)={\sum }_{n\ge 1}(-1{)}^{n+1}{x}^n/n$ when $|x{|}_p<1$ in a complete extension of ${\mathbb{Q}}_p$. The terms tend to zero in the p-adic absolute value, giving convergence even though the corresponding real series may behave differently.

On a sufficiently small group of principal units, the logarithm is a group homomorphism into the additive field and is inverse to the p-adic exponential function. For $p=2$, it converges on $1+2{\mathbb{Z}}_2$ but becomes injective only after restricting to $1+4{\mathbb{Z}}_2$, since $\log (-1)=0$. It converts multiplicative congruence questions into additive ones while retaining these domain restrictions.

The 2-adic logarithm is the power series $\log (1+x)={\sum }_{n\ge 1}(-1{)}^{n+1}{x}^n/n$, convergent for $x\in 2{\mathbb{Z}}_2$. Thus it is defined on the odd 2-adic units $1+2{\mathbb{Z}}_2$.

It is inverse to the 2-adic exponential after restriction to $1+4{\mathbb{Z}}_2$, with image $4{\mathbb{Z}}_2$. On the larger domain, $\log (-1)=0$, so it is not injective there. Indeed each odd unit is uniquely a sign times an element of $1+4{\mathbb{Z}}_2$, and the logarithm forgets the sign. Distinguishing the convergence domain from the inverse-function domain avoids a frequent error.

For a mixed-characteristic local field with residue characteristic $\ell$, the p-adic logarithm and p-adic exponential function are mutually inverse continuous group homomorphisms on these domains. Bounds on the valuations of the denominators show that higher terms have greater valuation than the linear term. Dividing the logarithm by ${\pi }^r$ identifies the multiplicative higher principal-unit group with the additive integer ring; in particular it is torsion-free.