p-adic analysis

In mathematics, p-adic analysis is a branch of number theory that deals with the mathematical analysis of functions of p-adic numbers.

The theory of complex-valued numerical functions on the p-adic numbers is part of the theory of locally compact groups. The usual meaning taken for p-adic analysis is the theory of p-adic-valued functions on spaces of interest.

Applications of p-adic analysis have mainly been in

Ostrowski's theorem, due to Alexander Ostrowski (1916), states that every non-trivial absolute value on the rational numbers Q is equivalent to either the usual real absolute value or a p-adic absolute value.^[1]

Mahler's theorem

Mahler's theorem, introduced by Kurt Mahler,^[2] expresses continuous p-adic functions in terms of polynomials.

In any field of characteristic 0, one has the following result. Let

${\displaystyle (\Delta f)(x)=f(x+1)-f(x)}$

be the forward

${\displaystyle f(x)=\sum _{k=0}^{\infty }(\Delta ^{k}f)(0){x \choose k},}$

where

${\displaystyle {x \choose k}={\frac {x(x-1)(x-2)\cdots (x-k+1)}{k!}}}$

is the kth binomial coefficient polynomial.

Over the field of real numbers, the assumption that the function f is a polynomial can be weakened, but it cannot be weakened all the way down to mere continuity.

Mahler proved the following result:

Mahler's theorem: If f is a continuous p-adic-valued function on the p-adic integers then the same identity holds.

Hensel's lemma

Hensel's lemma, also known as Hensel's lifting lemma, named after

To state the result, let ${\displaystyle f(x)}$ be a polynomial with integer (or p-adic integer) coefficients, and let m,k be positive integers such that m ≤ k. If r is an integer such that

${\displaystyle f(r)\equiv 0{\pmod {p^{k}}}}$ and ${\displaystyle f'(r)\not \equiv 0{\pmod {p}}}$

then there exists an integer s such that

${\displaystyle f(s)\equiv 0{\pmod {p^{k+m}}}}$ and ${\displaystyle r\equiv s{\pmod {p^{k}}}.}$

Furthermore, this s is unique modulo p^k+m, and can be computed explicitly as

${\displaystyle s=r+tp^{k}}$ where ${\displaystyle t=-{\frac {f(r)}{p^{k}}}\cdot (f'(r)^{-1}).}$

Applications

P-adic quantum mechanics

p-adic quantum mechanics is a relatively recent approach to understanding the nature of fundamental physics. It is the application of p-adic analysis to quantum mechanics. There are now hundreds of research articles on the subject,^[3][4] along with international journals.

There are two main approaches to the subject.^[5][6] The first considers particles in a p-adic potential well, and the goal is to find solutions with smoothly varying complex-valued wavefunctions. Here the solutions are to have a certain amount of familiarity from ordinary life. The second considers particles in p-adic potential wells, and the goal is to find p-adic valued wavefunctions. In this case, the physical interpretation is more difficult. Yet the math often exhibits striking characteristics, therefore people continue to explore it. The situation was summed up in 2005 by one scientist as follows: "I simply cannot think of all this as a sequence of amusing accidents and dismiss it as a 'toy model'. I think more work on this is both needed and worthwhile."^[7]

Local–global principle

• p-adic number
• p-adic exponential function
• p-adic Teichmüller theory
• Locally compact space
• Real analysis
• Complex analysis
• Hypercomplex analysis
• Harmonic analysis

References