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Project III topics 2024-25
Subjects: Algebraic number theory.
Topic: p-adic numbers and diophantine equations.
p-adic numbers appear in a very similar way to real numbers as
the completion of the field of rational numbers. In the case of real numbers
we use the usual metric given by the modulus of distance between two given rational numbers.
In the case of p-adic numbers we must use the so-called p-adic distance:
two rational numbers are close if their difference is divisible by a large
power of prime number p. The theory of p-adic numbers
is based on elementary number theory and at the same time contains many
results which are similar to the corresponding properties of usual real numbers.
1. p-adic numbers and diophantine equations.
A typical diophantine problem is solving in rational (or integral) numbers
equations of the form F(X1,...,Xn)=0,
where F is a polynomial of a total degree m in variables
X1,...,Xn with integral coefficients. The case
of equations of degree m=1 is easy.
The case m>1 contains, for example, the Fermat Equation
and leads to many beautiful results and open problems in
modern Algebraic Number Theory.
One of basic ideas in solving diophantine problems is the study
of solutions of a given diophantine equation
in bigger domains: in the field of real numbers and in
all p-adic fields. If our equation has no solutions in one of these bigger domains then
it has no integer solutions as well. The inverse statement is not always true, but in the case
of equations of total degree 2 it is true by the Minkowski Theorem.
The project will be concerned with the study of properties of p-adic
numbers, the proof of Minkowsky Theorem and its counterexamples
for equations of degree m>2.
Prerequisites:
Algebra and Number Theory II
Basic resources:
Number theory, by Z.I. Borevich, I.R. Shafarevich, ch.I, sections 3-7
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2. p-adic analysis.
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As explained above, the p-adic numbers are a perfect analogue of
the usual real numbers. Therefore, these numbers can be used for developing all
basic concepts of calculus: functions, limits, derivatives, integrals, infinite series and so on.
For example, in the field of 3-adic numbers the series 1+3+9+27+81+... converges and its sum
equals(!) minus 1/2. One can introduce an analogue of the usual exponential function
exp(x), which satisfies many usual properties, but the corresponding infinite Taylor series
does not converge for all x. There is very interesting theory of p-adic
integration.
The project will be concerned with the study of basic properties of p-adic numbers and
functions and the construction of p-adic integration.
Prerequisites:
Algebra and Number Theory II
Basic resources:
p-adic analysis: a short course on recent work, by N. Koblitz, ch.I and IV