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Branch of number theory
In mathematics, p-adic analysis is a branch of number theory that studies functions of p-adic numbers. Along with the more classical fields of real and
P-adic_analysis
Number system extending the rational numbers
p, the p-adic numbers form an extension of the rational numbers that is distinct from the real numbers, though with some similar properties; p-adic numbers
P-adic_number
Mathematical function
In mathematics, particularly p-adic analysis, the p-adic exponential function is a p-adic analogue of the usual exponential function on the complex numbers
P-adic_exponential_function
Study of objects of arithmetic interest over infinite towers of number fields
{\displaystyle \Gamma } isomorphic to the additive group of p-adic integers for some prime p. (These were called Γ {\displaystyle \Gamma } -extensions in
Iwasawa_theory
on 2012-03-11, retrieved 2011-05-12 Koblitz, Neal (1984), p-adic Numbers, p-adic Analysis, and Zeta-Functions, Graduate Texts in Mathematics, vol. 58
P-adic_distribution
Theorem in p-adic analysis
Mahler's theorem, introduced by Kurt Mahler (1958), expresses any continuous p-adic function as an infinite series of certain special polynomials. This series
Mahler's_theorem
Research program
a constant field, and the harmonic oscillator. p-adic analysis Volovich, I. V. (1987-06-01). "p-adic space-time and string theory". Theoretical and Mathematical
P-adic_quantum_mechanics
French mathematician (born 1962)
recherche at the CNRS (IMJ-PRG) known for his work in number theory and p-adic analysis. Colmez studied at École Normale Supérieure and obtained his doctorate
Pierre_Colmez
notion of differentiability of functions that is particularly suited to p-adic analysis. In short, the definition is made more restrictive by allowing both
Strict_differentiability
Mathematical function
In mathematics, the p-adic gamma function Γp is a function of a p-adic variable analogous to the gamma function. It was first explicitly defined by Morita
P-adic_gamma_function
Natural number
Knuth & Patashnik 1994, p. 111. Kennedy 1974, pp. 389. Peano 1889, p. 1. Peano 1908, p. 27. Halmos 1974, p. 32. Hodges 2009, p. 14. Hext 1990. Graham,
1
In mathematics, a p-adic zeta function, or more generally a p-adic L-function, is a function analogous to the Riemann zeta function, or more general L-functions
P-adic_L-function
Relates the topology of a complete non-archimedean field to its algebraic extensions
In number theory, more specifically in p-adic analysis, Krasner's lemma is a basic result relating the topology of a complete non-archimedean field to
Krasner's_lemma
American mathematician
1998) was an American mathematician, known for his application of p-adic analysis to local zeta functions, and in particular for a proof of the first
Bernard_Dwork
Measure of divisibility by a prime number
the p-adic valuation or p-adic order of an integer n is the exponent of the highest power of the prime number p that divides n. It is denoted ν p ( n
P-adic_valuation
German mathematician (1903–1988)
fields of transcendental number theory, diophantine approximation, p-adic analysis, and the geometry of numbers. Mahler was a student at the universities
Kurt_Mahler
fields were originally introduced in p-adic analysis since the fields Q p {\displaystyle \mathbb {Q} _{p}} of p-adic numbers are locally compact topological
Locally_compact_field
French mathematician (1936–1993)
research in the areas of number theory and p-adic analysis. The Amice transform and Amice theorem for p-adic functions are named after her. She headed
Yvette_Amice
Theorem on polynomial roots modulo prime powers
power of p tends to infinity, it follows that a root or a factorization modulo p can be lifted to a root or a factorization over the p-adic integers.
Hensel's_lemma
Branch of algebraic geometry
varieties. p-adic Hodge theory gives tools to examine when cohomological properties of varieties over the complex numbers extend to those over p-adic fields
Arithmetic_geometry
Mathematics of real numbers and real functions
York: Wiley, ISBN 978-0-471-31716-6. Koblitz, Neal (1984), p-adic Numbers, p-adic Analysis, and Zeta-Functions, Graduate Texts in Mathematics, vol. 58
Real_analysis
Israeli mathematician and professor
– Israel Institute of Technology. He is known for his work in p-adic analysis, p-adic quantum mechanics, and non-additive geometry, including the field
Shai_Haran
Mathematical integration method
of p-adic analysis, the Volkenborn integral is a method of integration for p-adic functions. Let : f : Z p → C p {\displaystyle f:\mathbb {Z} _{p}\to
Volkenborn_integral
Number
(part 1 ed.). Addison-Wesley Publishing Company, Inc. p. 18. Foerster 1980, p. 283. Foerster 1980, p. 3. "Subtracting Integers Rules: Definition and Rules
0
mathematical analysis (3., [Nachdr.] ed.). New York: McGraw-Hill. pp. 47, 52–54. ISBN 978-0-07-054235-8. Koblitz, Neal. (1984). P-adic Numbers, p-adic Analysis, and
Complete_field
forms, computational complexity theory, algebraic combinatorics and p-adic analysis. Sergei Evdokimov was born in Leningrad (now Saint Petersburg, Russia)
Sergei_Evdokimov
Branch of mathematics
monogenic or Clifford analytic functions. p-adic analysis, the study of analysis within the context of p-adic numbers, which differs in some interesting
Mathematical_analysis
French mathematician (1924–1987)
mathematician who worked on the theory of Lie groups in the context of p-adic analysis. Born in Paris, Lazard studied at the University of Paris–Sorbonne
Michel_Lazard
Mathematical theory
In mathematics, p-adic Hodge theory is a theory that provides a way to classify and study p-adic Galois representations of characteristic 0 local fields
P-adic_Hodge_theory
specifically in p-adic analysis, the Artin–Hasse exponential, introduced by Emil Artin and Helmut Hasse in 1928, is the power series given by E p ( x ) = exp
Artin–Hasse_exponential
American mathematician (1933–2013)
director of undergraduate studies for 30 years. His research areas were p-adic analysis and representation theory. Sally was born in the Roslindale neighborhood
Paul_Sally
Test for the divergence of an infinite series
test is often checked first due to its ease of use. In the case of p-adic analysis the term test is a necessary and sufficient condition for convergence
Nth-term_test
The zeros of a linear recurrence relation mostly form a regularly repeating pattern
with values in any field of characteristic zero. Its known proofs use p-adic analysis and are non-constructive. Let K {\displaystyle K} be a field of characteristic
Skolem–Mahler–Lech_theorem
Sum of an (infinite) geometric progression
11996214. ISSN 0025-570X. Robert, Alain M. (2000). A Course in p {\displaystyle p} -adic Analysis. Graduate Texts in Mathematics. Vol. 198. New York, USA: Springer-Verlag
Geometric_series
American mathematician and cryptographer
of Waterloo people Gross–Koblitz formula — (1984) [1977]. p-adic Numbers, p-adic Analysis, and Zeta-Functions. Graduate Texts in Mathematics. Vol. 58
Neal_Koblitz
theory p-adic analysis a branch of number theory that deals with the analysis of functions of p-adic numbers. p-adic dynamics an application of p-adic analysis
Glossary of areas of mathematics
Glossary_of_areas_of_mathematics
Algorithm for finding zeros of functions
used cubic approximations. In p-adic analysis, the standard method to show a polynomial equation in one variable has a p-adic root is Hensel's lemma, which
Newton's_method
Russian mathematician (1923–2012)
physics, quantum field theory, numerical analysis, generalized functions, several complex variables, p-adic analysis, multidimensional Tauberian theorems
Vasily_Vladimirov
Study of discrete mathematical structures
objects include transcendental numbers, diophantine approximation, p-adic analysis and function fields. Algebraic structures occur as both discrete examples
Discrete_mathematics
Special character in number theory
ISBN 978-0-387-49922-2, MR 2312337 Koblitz, Neal (1984), p-adic Numbers, p-adic Analysis, and Zeta-Functions, Graduate Texts in Mathematics, vol. 58
Teichmüller_character
Canadian-American mathematician (born 1974)
representation theory of quadratic forms, to interpolation problems and p-adic analysis, to the study of ideal class groups of algebraic number fields, and
Manjul_Bhargava
Function which measures the "size" of elements in a field or integral domain
cases. Koblitz, Neal (1984). P-adic numbers, p-adic analysis, and zeta-functions (2nd ed.). New York: Springer-Verlag. p. 1. ISBN 978-0-387-96017-3. Retrieved
Absolute_value_(algebra)
All numbers between two given numbers
numerical analysis, including adaptive mesh refinement, multigrid methods and wavelet analysis. Another way to represent such a structure is p-adic analysis (for
Interval_(mathematics)
Theory in number theory
alternative proofs of partial cases of the Grothendieck conjecture without using p-adic Hodge theory. Combinatorial anabelian geometry helps to study various aspects
Anabelian_geometry
Mathematical property of algebraic structures
Verslag Afd. Natuurk. (52): 74–84. MR 0015678. Neal Koblitz, "p-adic Numbers, p-adic Analysis, and Zeta-Functions", Springer-Verlag,1977. Shell, Niel, Topological
Archimedean_property
Product of numbers from 1 to n
{\displaystyle p} -adic valuation of a factorial". A Course in p {\displaystyle p} -adic Analysis. Graduate Texts in Mathematics. Vol. 198. New York: Springer-Verlag
Factorial
Type of metric space
Similar ideas can be found in domain theory. p-adic analysis makes heavy use of the ultrametric nature of the p-adic metric. In condensed matter physics, the
Ultrametric_space
Function in algebra
1967, p. 2. Emil Artin Geometric Algebra, pages 47 to 49, via Internet Archive Robert, Alain M. (2000), A Course in p-adic Analysis, Springer, p. 129,
Valuation_(algebra)
Undergraduate math course at Harvard University
weeks of point-set topology and special topics (for instance, in 1994, p-adic analysis was taught by Wilfried Schmid), students would take a quiz. As of 2012
Math_55
Algebraic structure with addition, multiplication, and division
algebraic function fields, algebraic number fields, finite fields, and p-adic fields are commonly used and studied in mathematics, particularly in number
Field_(mathematics)
Particular kind of algebraic structure
also be defined over fields of p {\displaystyle p} -adic numbers. This is part of p {\displaystyle p} -adic analysis. The prototypical example of a Banach
Banach_algebra
Type of topological space in mathematics
p-adic numbers is locally compact, because it is homeomorphic to the Cantor set minus one point. Thus locally compact spaces are as useful in p-adic analysis
Locally_compact_space
Result in number theory showing congruences involving Bernoulli numbers
to define the p-adic zeta function. The simplest form of Kummer's congruence states that B h h ≡ B k k ( mod p ) whenever h ≡ k ( mod p − 1 ) {\displaystyle
Kummer's_congruence
Mahler's compactness theorem (geometry of numbers) Mahler's theorem (p-adic analysis) Maier's theorem (analytic number theory) Mann's theorem (number theory)
List_of_theorems
Chebotarev's density theorem Totally real field Local field p-adic number p-adic analysis Adele ring Idele group Idele class group Adelic algebraic group
List of algebraic number theory topics
List_of_algebraic_number_theory_topics
On all absolute values of rational numbers
\mathbb {Q} } is equivalent to either the usual real absolute value or a p-adic absolute value. An absolute value on the rational numbers is a function
Ostrowski's_theorem
Belgian mathematician (1936–2008)
July 2008) was a Belgian mathematician known as a pioneer of p-adic functional analysis, and particularly for her work on locally convex topological vector
Nicole_De_Grande-De_Kimpe
Operation in differential geometry
analytic functions between real or complex domains, to p-adic analysis, and to other areas of analysis. Let C ∞ ( R n , R m ) {\displaystyle C^{\infty }({\mathbb
Jet_(mathematics)
Finite extension of the rationals
tools such as intermediate value theorem at the archimedean places and p-adic analysis at the nonarchimedean places) can be used. This implication does not
Algebraic_number_field
Mathematics of varieties with integer coordinates
geometry Arakelov geometry Hindry & Silverman 2000, p. vii, Preface. Hindry & Silverman 2000, p. viii, Preface. "Mordell : Review: Serge Lang, Diophantine
Diophantine_geometry
Academic fields of study or professions
Non-standard analysis Ordinary differential equations p-adic analysis Partial differential equations Real analysis Calculus (outline) Probability theory Ergodic
Outline of academic disciplines
Outline_of_academic_disciplines
Complement of an open subset
for many examples, including the Cantor set and spaces arising in p-adic analysis. In algebraic number theory, topological groups over non-Archimedean
Closed_set
Series of mathematics textbooks
Richard H. Crowell, Ralph H. Fox (1977, ISBN 978-0-387-90272-2) p-adic Numbers, p-adic Analysis, and Zeta-Functions, Neal Koblitz (1984, 2nd ed., ISBN 978-0-387-96017-3)
Graduate_Texts_in_Mathematics
Number whose square ends in the same digits
base, i)) Arithmetic dynamics Kaprekar number p-adic number p-adic analysis Zero-divisor See Gérard Michon's article at "spherical number"
Automorphic_number
Infinite series that diverges
p-adic Numbers, p-adic Analysis, and Zeta-Functions. Graduate Texts in Mathematics, vol. 58. Springer-Verlag. pp. chapter I, exercise 16, p. 20. ISBN 0-387-96017-1
1_+_2_+_4_+_8_+_⋯
Russian–American mathematician
Chicago Press, 1992. Russian edition, Faktorial Press, Moscow, 2002. p-adic Analysis Compared with Real, Student Mathematical Library, vol. 37, American
Svetlana_Katok
Fundamental construction of differential calculus
arbitrary algebraic varieties, instead of just smooth manifolds. In p-adic analysis, the usual definition of derivative is not quite strong enough, and
Generalizations of the derivative
Generalizations_of_the_derivative
Award in mathematical logic
Shoenfield, The Priority Method. 1993 Angus Macintyre, Logic of Real and p-adic Analysis: Achievements and Challenges. 1994 Donald A. Martin, L(R): A Survey
Gödel_Lecture
Fraction with denominator a power of two
Fractional and integral parts of p {\displaystyle p} -adic numbers", A Course in p {\displaystyle p} -adic Analysis, Graduate Texts in Mathematics, vol
Dyadic_rational
International specialist organization
Annual Gödel Lecture 1993 1993 Angus Macintyre, Logic of Real and p-adic Analysis: Achievements and Challenges The Third Annual Gödel Lecture 1992 1992
Association for Symbolic Logic
Association_for_Symbolic_Logic
German mathematician
internazionale per la ricerca matematica; Congress on "p-adic Analysis" (1990). P-adic analysis : proceedings of the international conference held in Trento
Siegfried_Bosch
Linear representation in mathematics
Harish-Chandra (1973), "Harmonic analysis on reductive p-adic groups", in Moore, Calvin C. (ed.), Harmonic analysis on homogeneous spaces (Proc. Sympos
Steinberg_representation
space Metric topology Manhattan distance Ultrametric space P-adic numbers, p-adic analysis Open ball Bounded subset Pointwise convergence Metrization
List of general topology topics
List_of_general_topology_topics
Type of zeta function
1017/is010004028jkt103. Sources François Bruhat (1963). Lectures on some aspects of p-adic analysis. Tata Institute of Fundamental Research. Serre, Jean-Pierre (1969–1970)
Arithmetic_zeta_function
American mathematician
theory, with specific interests in p-adic analysis and arithmetic geometry. In particular, he developed a theory of p-adic integration analogous to the classical
Robert_F._Coleman
consequences. Dwork's method Bernard Dwork used distinctive methods of p-adic analysis, p-adic algebraic differential equations, Koszul complexes and other techniques
Glossary of arithmetic and diophantine geometry
Glossary_of_arithmetic_and_diophantine_geometry
Infinite sum that is considered independently from any notion of convergence
seen as the (x)-adic completion of the polynomial ring R [ x ] , {\displaystyle R[x],} in the same way as the p-adic integers are the p-adic completion of
Formal_power_series
American mathematician
1977). Kottwitz works in the Langlands program, including harmonic analysis on p-adic Lie groups and automorphic forms and the general linear groups and
Robert_Kottwitz
Analogue of a complex analytic space over a nonarchimedean field
on uniformizing p-adic elliptic curves with bad reduction using the multiplicative group. In contrast to the classical theory of p-adic analytic manifolds
Rigid_analytic_space
French mathematician investigating the areas of number theory and p-adic analysis. Jeanne Baret (1740–1887). French explorer, the first woman to have
List of women's names for the Eiffel Tower
List_of_women's_names_for_the_Eiffel_Tower
Swiss mathematician
of p-adic analysis of one variable (except the rationality of the zeta function of an algebraic variety over a finite field and the theory of p-adic differential
Alain_M._Robert
Mathematical concept
analogous theory to the classical theory of complex modular forms and the p-adic theory of modular forms. Modular forms are analytic functions, so they admit
Modular_forms_modulo_p
American mathematician
research interests include algebraic number theory, Diophantine analysis, p-adic analysis, geometry of numbers, and the theory of continued fractions. He
Edward_Burger
for indigenous students Yvette Amice (1936–1993), French expert on p-adic analysis who became president of the French mathematical society Divsha Amirà
List_of_women_in_mathematics
Field of mathematics
also called p-adic or nonarchimedean dynamics, is an analogue of complex dynamics in which one replaces the complex numbers C by a p-adic field such as
Arithmetic_dynamics
Function named after Harish Chandra
c-function for p-adic Lie groups. Macdonald (1968, 1971) and Langlands (1971) found an analogous product formula for the c-function of a p-adic Lie group.
Harish-Chandra's_c-function
Alternative decimal expansion of 1
p {\displaystyle p} -adic numbers are an alternative number system of interest in number theory. Like the real numbers, the p {\displaystyle p} -adic
0.999...
Indian mathematician (born 1961)
Number Theory, International Journal of Number Theory, and P-adic Numbers, Ultrametric Analysis and Applications. Thakur is a founding member of-and for
Dinesh_Thakur_(mathematician)
Topological space that is maximally disconnected
homeomorphic to the set of p-adic integers. Another example, playing a key role in algebraic number theory, is the field Qp of p-adic numbers. A topological
Totally_disconnected_space
Infinite series that diverges
summation methods to the series, as well as the limit of the series using the 2-adic metric. Gottfried Leibniz considered the divergent alternating series 1 −
1_−_2_+_4_−_8_+_⋯
Solving integer equations from all modular solutions
then this also yields a real solution and a p-adic solution, as the rationals embed in the reals and p-adics: a global solution yields local solutions at
Hasse_principle
Branch of algebra that studies commutative rings
integers, including the ordinary integers Z {\displaystyle \mathbb {Z} } ; and p-adic integers. Commutative algebra is the main technical tool of algebraic geometry
Commutative_algebra
Uzbek mathematician and professor
including quantum probability, ergodic theory, dynamical systems, and p-adic analysis. In quantum probability, he applied methods from statistical mechanics
Farrukh_Mukhamedov
Algebra based on a vector space with a quadratic form
(it is not a subalgebra). This Z2-grading plays an important role in the analysis and application of Clifford algebras. The automorphism α is called the
Clifford_algebra
Mathematical term
ISBN 978-3-540-42533-5. Robert, Alain M. (2000-05-31). A Course in p-adic Analysis. Springer Science & Business Media. p. 129. ISBN 978-0-387-98669-2. v t e
Spherically_complete_field
Application of Fourier analysis to non-abelian topological groups
interesting examples include many Lie groups, and also algebraic groups over p-adic fields. These examples are of interest and frequently applied in mathematical
Noncommutative harmonic analysis
Noncommutative_harmonic_analysis
Retrieved 15 August 2023. Analysis of Tobacco Market in Sri-Lanka (PDF) (in British English and Indian English). Colombo: ADIC Sri Lanka – Alcohol & Drug
List_of_cigarette_brands
Mathematical space with a notion of distance
graphs may be viewed as metric spaces. In abstract algebra, the field of p-adic numbers is the completion of the field of rational numbers with respect
Metric_space
Used to count, measure, and label
algebraic structures are explicitly referred to as numbers (such as the p-adic numbers and hypercomplex numbers) while others are not, but this is more
Number
French mathematician (1912–1985)
emeritus. Krasner did research on p-adic analysis. In 1944 he introduced the concept of ultrametric spaces, to which p-adic numbers belong. In 1951, alongside
Marc_Krasner
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