Is there a version of complex analysis for p-adic numbers?

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In summary, analysis in Qp[i] has some similarities to complex analysis, such as the existence of derivatives and analytic functions. However, there are also notable differences, such as the lack of a fundamental theorem of calculus and the use of the Shnirelman integral. Additionally, the complex analysis proof of a polynomial having a root does not apply to Qp, as 1/p(x) may not be analytic despite being differentiable.
  • #1
The p-adic numbers Qp don't have a square root of -1, if p=3 mod 4.
So would differentiable functions from Qp -> Qp satisfy the
Cauchy-Riemann equations? I don't know why not.

To what extent would analysis in Qp have the familiar complex analysis
theorems? You couldn't prove that Qp is algebraically complete, I
wonder what would block the complex analysis proof of that, that 1/p(x)
would be a bounded entire function if it had no roots.

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  • #2
I looked into it some more -

Derivatives are used in the p-adics, although "strictly differentiable" is
a more useful concept than "differentiable". Strictly differentiable at a
point p means that as x and y approach p, then f(x)-f(y)/(x-y) approaches
f'(p), the derivative of f at p. If the function is strictly differentiable
and its derivative at a point is nonzero, it's locally injective, so you
get an inverse function theorem.

There's a definition of analytic functions for p-adics: a function f is analytic
in an open set D if f can be expressed as a power series around a point u in D.
Unlike with complex analysis, if a function is infinitely differentiable, that
doesn't mean it's analytic.

There are antiderivatives but no fundamental theorem of calculus in Qp, so the
antiderivatives aren't related to integration.

You can define an integral, for functions from the p-adic integers to Qp.

So, if a function is defined on a bounded subset of Qp, you could define
integration just by multiplying the argument by some power of p. Perhaps you
could use this to define an integral on an unbounded subset of Qp by a limit
process - but probably, the integral as defined by multiplying by one power of
p can't be made the same as the integral defined by multiplying by another
power of p, so that limit process might not work.

There IS a version of the Cauchy integral formula, residue theorem, and
maximum modulus principle for p-adics (see p. 129 of Koblitz "P-adic analysis:
a short course on recent work" online at ).
It uses the Shnirelman integral. The Shnirelman integral is ingenious: for
z in Cp, the metric completion of the algebraic closure of Qp, and f: Cp -> Cp,
and a point a in Cp, you define integral (f, a, z) as the limit n -> oo of
1/n (sum over e: e^n=1 of f(a+ez)). In the limit, you skip n's that are
divisible by p. So, it's taking the limit of the average of f on these
points sprinkled over a circle centered at a.

Just like in complex analysis, you can use this integral to show that
a bounded analytic function is a constant.

But that's as far as it goes. In
the complex analysis proof that a polynomial p(x) has a root in C, you show
that 1/p(x) is a bounded analytic function if p(x) has no roots. But for
p(x) in Qp[x], 1/p(x) isn't analytic just because it's differentiable.

Related to Is there a version of complex analysis for p-adic numbers?

What is P-adic analysis?

P-adic analysis is a branch of mathematics that deals with the properties and behavior of numbers in the context of the p-adic number system. It is based on the idea of extending the traditional real number system to include numbers that are expressed in terms of a prime number, p.

What is the significance of p in P-adic analysis?

The prime number p is the defining factor in P-adic analysis. It determines the structure and behavior of numbers in this system, and is used to construct the p-adic numbers. Different choices for p can lead to different properties and results in P-adic analysis.

What are the applications of P-adic analysis?

P-adic analysis has applications in various fields, including number theory, algebra, and theoretical physics. It has been used to solve problems in areas such as Diophantine equations, cryptography, and quantum mechanics.

How is P-adic analysis different from real analysis?

P-adic analysis differs from real analysis in several ways. In P-adic analysis, the numbers are not arranged in a linear order like in the real numbers. Instead, they are organized in a tree-like structure. Additionally, the metric used in P-adic analysis is different from the standard Euclidean metric in real analysis.

What are some open problems in P-adic analysis?

Some open problems in P-adic analysis include the existence and uniqueness of p-adic L-functions, the behavior of p-adic modular forms, and the relationship between p-adic and real algebraic numbers. The study of these and other open problems in P-adic analysis is an active area of research in mathematics.

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