This is a glossary of arithmetic and diophantine geometry in
mathematics, areas growing out of the traditional study of
Diophantine equations to encompass large parts of
number theory and
algebraic geometry
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrica ...
. Much of the theory is in the form of proposed
conjecture
In mathematics, a conjecture is a conclusion or a proposition that is proffered on a tentative basis without proof. Some conjectures, such as the Riemann hypothesis (still a conjecture) or Fermat's Last Theorem (a conjecture until proven in 1 ...
s, which can be related at various levels of generality.
Diophantine geometry in general is the study of
algebraic varieties
Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. ...
''V'' over fields ''K'' that are finitely generated over their
prime fields—including as of special interest
number fields and
finite fields—and over
local fields. Of those, only the
complex numbers are
algebraically closed
In mathematics, a field is algebraically closed if every non-constant polynomial in (the univariate polynomial ring with coefficients in ) has a root in .
Examples
As an example, the field of real numbers is not algebraically closed, because ...
; over any other ''K'' the existence of points of ''V'' with coordinates in ''K'' is something to be proved and studied as an extra topic, even knowing the geometry of ''V''.
Arithmetic geometry
In mathematics, arithmetic geometry is roughly the application of techniques from algebraic geometry to problems in number theory. Arithmetic geometry is centered around Diophantine geometry, the study of rational points of algebraic varieties.
...
can be more generally defined as the study of
schemes of finite type over the
spectrum
A spectrum (plural ''spectra'' or ''spectrums'') is a condition that is not limited to a specific set of values but can vary, without gaps, across a continuum. The word was first used scientifically in optics to describe the rainbow of colors ...
of the
ring of integers. Arithmetic geometry has also been defined as the application of the techniques of algebraic geometry to problems in
number theory.
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See also
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Arithmetic topology
*
Arithmetic dynamics Arithmetic dynamics is a field that amalgamates two areas of mathematics, dynamical systems and number theory. Classically, discrete dynamics refers to the study of the iteration of self-maps of the complex plane or real line. Arithmetic dynamics is ...
References
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Further reading
*Dino Lorenzini (1996)
An invitation to arithmetic geometry AMS Bookstore,
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Diophantine geometry
Algebraic geometry
Geometry
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