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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, but
P-adic_L-function
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
Theorem in algebraic number theory relating p-adic L-functions and ideal class groups
main conjecture of Iwasawa theory is a deep relationship between p-adic L-functions and ideal class groups of cyclotomic fields, proved by Kenkichi Iwasawa
Main conjecture of Iwasawa theory
Main_conjecture_of_Iwasawa_theory
Study of objects of arithmetic interest over infinite towers of number fields
In each case, there is a main conjecture linking the tower to a p-adic L-function. In 2002, Christopher Skinner and Eric Urban claimed a proof of a
Iwasawa_theory
mathematics, a p-adic distribution is an analogue of ordinary distributions (i.e. generalized functions) that takes values in a ring of p-adic numbers. If
P-adic_distribution
Meromorphic function on the complex plane
generalisation of that phenomenon. In that case results have been obtained for p-adic L-functions, which describe certain Galois modules. The statistics of the zero
L-function
Highest power of p dividing a given 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
1960s meant that Hasse–Weil L-functions could be regarded as Artin L-functions for the Galois representations on l-adic cohomology groups. Bad reduction
Glossary of arithmetic and diophantine geometry
Glossary_of_arithmetic_and_diophantine_geometry
American mathematician (born 1943)
American mathematician, working in arithmetic geometry, particularly on p-adic methods, monodromy and moduli problems, and number theory. He is currently
Nick_Katz
Motivic zeta function of a motive Multiple zeta function, or Mordell–Tornheim zeta function of several variables p-adic zeta function of a p-adic number Prime
List_of_zeta_functions
Conjecture on zeros of the zeta function
a p-adic L-function with the eigenvalues of an operator, so can be thought of as an analogue of the Hilbert–Pólya conjecture for p-adic L-functions. Several
Riemann_hypothesis
Type of Dirichlet series associated to number field extensions
In mathematics, Artin L-functions are a type of Dirichlet series defined for finite extensions of number fields, encoding informations about linear representations
Artin_L-function
χℓ form a strictly compatible system of ℓ-adic representations. The p-adic cyclotomic character is the p-adic Tate module of the multiplicative group
Cyclotomic_character
Function in algebra
the p-adic completions of Q . {\displaystyle \mathbb {Q} .} Let v be a valuation of K and let L be a field extension of K. An extension of v (to L) is
Valuation_(algebra)
Branch of algebraic geometry
system of polynomial equations over number fields, finite fields, p-adic fields, or function fields, i.e. fields that are not algebraically closed excluding
Arithmetic_geometry
American mathematician and novelist
with a thesis on p-adic L-functions attached to elliptic curves, a Ph.D. in mathematics in 1990 with a thesis on p-Adic L-functions and Galois groups
Leila_Schneps
both analytic functions. The work of Lubotzky and Mann, combined with Michel Lazard's solution to Hilbert's fifth problem over the p-adic numbers, shows
Pro-p_group
A p-adic language is defined as the set of strings L η ( p ) = { n 1 n 2 n 3 … | 0 ≤ n k < p and 0. n 1 n 2 n 3 … > η } {\displaystyle L_{\eta }(p
Probabilistic_automaton
Gives the rank of the group of units in the ring of algebraic integers of a number field
(PDF) on 2008-05-10. Neukirch et al. (2008) p. 626–627 Iwasawa, Kenkichi (1972). Lectures on p-adic L-functions. Annals of Mathematics Studies. Vol. 74.
Dirichlet's_unit_theorem
Function studied by Ramanujan
Séminaire Delange-Pisot-Poitou, 14 Swinnerton-Dyer, H. P. F. (1973). "On ℓ {\displaystyle \ell } -adic representations and congruences for coefficients of
Ramanujan_tau_function
German mathematician (born 1958)
general L-functions are also defined by Euler products, involving, at each finite place, the determinant of the Frobenius endomorphism acting on l-adic cohomology
Christopher_Deninger
unit root zeta function, named after Bernard Dwork, is the L-function attached to the p-adic Galois representation arising from the p-adic etale cohomology
Dwork_conjecture
Mathematical terminology
group is zero). If X is a smooth proper scheme over a field K then the ℓ-adic cohomology groups of its geometric fibre are Galois modules for the absolute
Galois_representation
Israeli mathematician and professor
construction of p-adic L-functions for modular forms on GL(2) over any number field. He gave a formula for the explicit sums of arithmetic functions expressing
Shai_Haran
French mathematician (born 1962)
He works on special values of L-functions and p {\displaystyle p} -adic representations of p {\displaystyle p} -adic groups at the meeting point of Fontaine's
Pierre_Colmez
The Ihara zeta function was first defined by Yasutaka Ihara in the 1960s in the context of discrete subgroups of the two-by-two p-adic special linear
Ihara_zeta_function
American mathematician (born 1945)
the p-adic Bessel function. The arithmetic information that Sperber's work produced included determining the degree of the associated L-function, proving
Steven_Sperber
Sheaf cohomology on the étale site
1960 using p-adic methods), and the remaining conjecture, the analogue of the Riemann hypothesis was proved by Pierre Deligne (1974) using ℓ-adic cohomology
Étale_cohomology
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
Gross–Stark conjecture, a p-adic analogue of the Stark conjectures relating derivatives of Deligne–Ribet p-adic L-functions (for totally even characters
Stark_conjectures
French mathematician and historian of mathematics
completed a doctorate of the third cycle in 1981, with a dissertation on p-adic L-functions and Iwasawa theory supervised by John H. Coates. She worked at the
Catherine_Goldstein
-\mathbb {N} } . The Ihara zeta function is considered a p-adic (and a graph-theoretic) analogue of the Selberg zeta function. For the case where the surface
Selberg_zeta_function
Indian mathematician
a thesis titled Iwasawa Theory of Lubin-Tate Division Towers and p-Adic L-Functions under the supervision of John Coates. Since his PhD, Saikia has held
Anupam_Saikia
Japanese mathematician
Lectures on p-adic L-functions / by Kenkichi Iwasawa (1972) Local class field theory / Kenkichi Iwasawa (1986) ISBN 0-19-504030-9 Algebraic functions / Kenkichi
Kenkichi_Iwasawa
Australian mathematician (1945–2022)
research at the University of Cambridge, his doctoral dissertation being on p-adic analogues of Baker's method. In 1969, Coates was appointed assistant professor
John_H._Coates
Unsolved problem in mathematics
for the L-function obtained by the substitution u = p − s {\displaystyle u=p^{-s}} : 1 + τ ( p ) u − p 11 u 2 = 0. {\displaystyle 1+\tau (p)u-p^{11}u^{2}=0
Ramanujan–Petersson conjecture
Ramanujan–Petersson_conjecture
American mathematician
Coleman, Robert F.; de Shalit, Ehud (1988), "p-adic regulators on curves and special values of p-adic L-functions", Invent. Math., 93 (2): 239–266, Bibcode:1988InMat
Robert_F._Coleman
Belgian mathematician
important results on the l-adic representations attached to modular forms, and the conjectural functional equations of L-functions. Deligne also focused
Pierre_Deligne
Weil cohomology theory for schemes X over a base field k
the work on p-adic L-functions. Crystalline cohomology, from the point of view of number theory, fills a gap in ℓ {\displaystyle \ell } -adic cohomology
Crystalline_cohomology
Natural number
Adrien-Marie Legendre to express the asymptotic behavior of the prime-counting function. The Weil's conjecture on Tamagawa numbers states that the Tamagawa number
1
Conjectures connecting number theory and geometry
see p-adic numbers.) Langlands attached automorphic L-functions to these automorphic representations, and conjectured that every Artin L-function arising
Langlands_program
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
On generating functions from counting points on algebraic varieties over finite fields
I − TF on the ℓ-adic cohomology group Hi. The rationality of the zeta function follows immediately. The functional equation for the zeta function follows from
Weil_conjectures
{\displaystyle \mathbb {Q} _{p}} and Z p {\displaystyle \mathbb {Z} _{p}} are the field of p-adic numbers and ring of p-adic integers respectively. The
Schwartz–Bruhat_function
Japanese mathematician (born 1952)
Fellowship. Hida received in 1992 for his research on p-adic L-functions of algebraic groups and p-adic Hecke rings the Spring Prize of the Mathematical Society
Haruzo_Hida
Euler system p-adic L-function Arithmetic geometry Complex multiplication Abelian variety of CM-type Chowla–Selberg formula Hasse–Weil zeta function
List of algebraic number theory topics
List_of_algebraic_number_theory_topics
Japanese mathematician (1930–2020)
contributions include works on p-adic L functions and real-analytic automorphic forms. His work on p-adic L-functions, later recognised as an aspect of
Tomio_Kubota
Vietnamese mathematician (born 1946)
p-adic interpolation, in Mat. Zametki, 26 (1979), no.1 (in Russian), AMS translation in Mathematical Notes, 26 (1980), 541-549. On p-adic L-functions
Hà_Huy_Khoái
Field of mathematics
properties of integer, rational, p-adic, or algebraic points under repeated application of a polynomial or rational function. A fundamental goal is to describe
Arithmetic_dynamics
South Korean mathematician (born 1963)
inaugural fellow of the UK Academy for the Mathematical Sciences . "p-adic L-functions and Selmer varieties associated to elliptic curves with complex multiplication"
Minhyong_Kim
Function named after Harish Chandra
similar 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
Harish-Chandra's_c-function
q} elements, and Frobq is the geometric Frobenius acting on ℓ {\displaystyle \ell } -adic étale cohomology with compact supports of X ¯ {\displaystyle
Local_zeta_function
Type of character in number theory
to construct a class of L-functions larger than Dirichlet L-functions, and a natural setting for the Dedekind zeta-functions and certain others which
Hecke_character
Extension of the factorial function
gamma function Multivariate gamma function p-adic gamma function Pochhammer k-symbol Polygamma function q-gamma function Ramanujan's master theorem Spouge's
Gamma_function
Topological space in mathematics
continuous image of an interval. L ∗ {\displaystyle L^{*}} is not a manifold and is not first countable. There exists a p-adic analog of the long line, which
Long_line_(topology)
Finite extension of the rationals
the same way, now giving functions mapping to Q p {\displaystyle \mathbb {Q} _{p}} . By using this p {\displaystyle p} -adic norm map N f i {\displaystyle
Algebraic_number_field
Fraction with denominator a power of two
a subsystem of the 2-adic numbers as well as of the reals, and can represent the fractional parts of 2-adic numbers. Functions from natural numbers to
Dyadic_rational
Some remarkable congruences for the partition function
the following P {\displaystyle P} function in the l-adic topology: P ℓ ( b ; z ) := ∑ n = 0 ∞ p ( ℓ b n + 1 24 ) q n / 24 . {\displaystyle P_{\ell }(b;z):=\sum
Ramanujan's_congruences
Special numbers in mathematics
that can be used in the construction of two-variable p-adic L-functions. They are related to critical L-values of Hecke characters. When A is the area of
Eisenstein–Kronecker_number
American mathematician at Duke University
theory, in particular the theory of special values of classical and p-adic L-functions". Darmon, Henri; Dasgupta, Samit (2006). "Elliptic units for real
Samit_Dasgupta
Romanian-Canadian mathematician
Romanian-Canadian mathematician, specializing in arithmetic algebraic geometry and p-adic cohomology theories. Born in Timișoara, Romania, Iovita received in 1978
Adrian_Ioviță
Number-theoretic concept
{\displaystyle p} runs over all prime numbers, and Z p {\displaystyle \mathbb {Z} _{p}} is the ring of p-adic integers. This group is important because of its
Profinite_integer
Type of zeta function
function is a zeta function associated with a scheme of finite type over integers. The arithmetic zeta function generalizes the Riemann zeta function
Arithmetic_zeta_function
modification of the usual notion of differentiability of functions that is particularly suited to p-adic analysis. In short, the definition is made more restrictive
Strict_differentiability
Product of numbers from 1 to n
the non-positive integers. In the p-adic numbers, it is not possible to continuously interpolate the factorial function directly, because the factorials
Factorial
American mathematician (born 1944)
Mazur–Tate–Teitelbaum conjecture as well as a formula for the derivative of a p-adic Dirichlet L-function at s = 0 {\displaystyle s=0} (joint with Bruce Ferrero). Greenberg
Ralph_Greenberg
function. The most common examples are the complex numbers and all its subfields, and the valued fields, which include the p {\displaystyle p} -adic fields
Topological_ring
Number divisible only by 1 and itself
rational numbers can be measured by their p {\displaystyle p} -adic distance, the p {\displaystyle p} -adic absolute value of their difference. For
Prime_number
Open problem on 3x+1 and x/2 functions
{2}}\right)2^{k}.} The function Q is a 2-adic isometry. Consequently, every infinite parity sequence occurs for exactly one 2-adic integer, so that almost
Collatz_conjecture
Graduate-level textbooks in mathematics
Edmund F., "The Leroy P Steele Prize of the AMS", MacTutor History of Mathematics Archive, University of St Andrews "Leroy P. Steele Prize for Mathematical
Annals_of_Mathematics_Studies
mathematics, motivic L-functions are a generalization of Hasse–Weil L-functions to general motives over global fields. The local L-factor at a finite place
Motivic_L-function
Special functions of several complex variables
define the Theta functions over other fields where the exponential function might not be everywhere defined, such as fields of p-adic numbers. The Jacobi
Theta_function
operators on the symmetric space G/K. For semisimple p-adic Lie groups, the theory of zonal spherical functions and Hecke algebras was first developed by Satake
Zonal_spherical_function
Russian mathematician (1937–2008)
a Russian mathematician working in the field of analytic number theory, p-adic numbers and Dirichlet series. For most of his student and professional life
Anatoly_Karatsuba
American mathematician
Washington wrote a standard work on cyclotomic fields. He also worked on p-adic L-functions. He wrote a treatise with Allan Adler on their discovery of a connection
Lawrence_C._Washington
Mathematical function, inverse of an exponential function
(multi-valued) inverse function of the matrix exponential. Another example is the p-adic logarithm, the inverse function of the p-adic exponential. Both are
Logarithm
Conjecture in the representation theory of Lie groups
conjecture is a restriction problem in the representation theory of real or p-adic Lie groups posed by Gan Wee Teck, Benedict Gross, and Dipendra Prasad. The
Gan–Gross–Prasad_conjecture
Israeli mathematician (born 1955)
Ehud (1988). " p {\displaystyle p} -adic regulators on curves and special values of p {\displaystyle p} -adic L {\displaystyle L} -functions". Inventiones
Ehud_de_Shalit
German mathematician (1927–2011)
investigated p-adic L-functions (now named after them). These functions are a component of Iwasawa theory and are a p-adic version of the Dirichlet L-functions. With
Heinrich-Wolfgang_Leopoldt
German algebraic number theorist
number theorist at ETH Zurich. Her research interests include L-functions, modular forms, p-adic Hodge theory, and Iwasawa theory, and her work has led to
Sarah_Zerbes
Unproved conjecture in mathematics
modulo each prime p {\displaystyle p} . This L {\displaystyle L} -function is analogous to the Riemann zeta function and the Dirichlet L-series that is defined
Birch and Swinnerton-Dyer conjecture
Birch_and_Swinnerton-Dyer_conjecture
Mathematical formal group law
(1987), Iwasawa theory of elliptic curves with complex multiplication. p-adic L functions, Perspectives in Mathematics, vol. 3, Academic Press, ISBN 0-12-210255-X
Lubin–Tate_formal_group_law
Mathematics prize
original on July 23, 2024. Retrieved September 13, 2017. Case & Leggett 2005, p. 97. "Educational Awards: Ruth Satter". Association for Women in Science.
Ruth Lyttle Satter Prize in Mathematics
Ruth_Lyttle_Satter_Prize_in_Mathematics
Concept in number theory
include the real numbers and the fields of p {\displaystyle p} -adic numbers for all prime numbers p {\displaystyle p} . More generally, if K {\displaystyle
Adele_ring
Complementary series for p-adic groups, Annals of Mathematics 132 (1990), 273–330. F. Shahidi, Eisenstein Series and Automorphic L-functions, Colloquium Publications
Langlands–Shahidi_method
expression appearing in the functional equation of an Artin L-function. Suppose that L {\displaystyle L} is a finite Galois extension of the local field K {\displaystyle
Artin_conductor
Branch of number theory
value function |·| : Q → R, there are p-adic absolute value functions |·|p : Q → R, defined for each prime number p, which measure divisibility by p. Ostrowski's
Algebraic_number_theory
In mathematics, the Moy–Prasad filtration is a family of filtrations of p-adic reductive groups and their Lie algebras, named after Allen Moy and Gopal
Moy–Prasad_filtration
Mathematical function, denoted exp(x) or e^x
Mittag-Leffler function, a generalization of the exponential function p-adic exponential function Padé table for exponential function – Padé approximation
Exponential_function
Prize awarded by the American Mathematical Society
JSTOR 1971100. MR 0498828. Zbl 0393.20011. Fintzen, Jessica (2021). "Types for tame p-adic groups". Annals of Mathematics. 193 (1): 303–346. doi:10.4007/annals.2021
Cole_Prize
Problem about mathematical number fields
fields and describe leading coefficients of Artin L-functions. In 2021, Dasgupta and Kakde announced a p-adic solution to finding the maximal abelian extension
Hilbert's_twelfth_problem
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
Numeral system in which every non-negative integer can be represented in exactly one way
power of k. Smullyan (1961) calls this notation k-adic, but it should not be confused with the p-adic numbers: bijective numerals are a system for representing
Bijective_numeration
Algebraic structure with addition, multiplication, and division
fields, such as fields of rational functions, algebraic function fields, algebraic number fields, finite fields, and p-adic fields are commonly used and studied
Field_(mathematics)
Number theory expression
Legendre's formula that the p-adic exponential function has radius of convergence p − 1 / ( p − 1 ) {\displaystyle p^{-1/(p-1)}} . Legendre, A. M. (1830)
Legendre's_formula
Semiring with minimum and addition replacing addition and multiplication
non-Archimedean local field, such as the p-adic numbers Q p {\displaystyle \mathbb {Q} _{p}} with the p-adic valuation extending the one on Q {\displaystyle
Tropical_semiring
main conjecture of Iwasawa theory is a deep relationship between p-adic L-functions and ideal class groups of cyclotomic fields, proved by Iwasawa for
List of Japanese inventions and discoveries
List_of_Japanese_inventions_and_discoveries
Number
zero function (or zero map) on a domain D. This is the constant function with 0 as its only possible output value, that is, it is the function f defined
0
Identity obeyed by many special functions related to the gamma function
functions, for which the identity follows from a p-adic relation over a finite field. For example, the multiplication theorem for the gamma function follows
Multiplication_theorem
French mathematician
gave on "Fonctions L p-adiques" ("p-adic L-functions"). Perrin-Riou's research is in number theory, concentrating on p-adic L-functions and Iwasawa theory
Bernadette_Perrin-Riou
P ADIC-L-FUNCTION
P ADIC-L-FUNCTION
Male
French
French form of Greek Ioel (Hebrew Yowel), JOËL means "Jehovah is God" or "to whom Jehovah is God."
Male
Scottish
Scottish form of Latin Paulus, PÀL means "small."
Male
French
Masculine form of French Gaëlle, GAËL means "holy and generous."
Male
Hungarian
Hungarian form of Greek Paulos, PÃL means "small."
Male
English
Variant spelling of English Eric, ARIC means "ever-ruler."
Boy/Male
Hebrew
Attractive; handsome; pleasure given. Adin was a biblical exile who returned to Israel from Babylon.
Male
Irish
Irish Gaelic form of Greek MichaÄ“l, MÃCHEÃL means "who is like God?"
Female
English
(עֲדִי) Hebrew unisex name ADI means "my ornament" or "my witness."
Male
Swedish
Swedish form of Greek Paulos, PÃ…L means "small."
Male
French
French name derived from Latin natalis dies, NOËL means "day of birth."
Boy/Male
Indian
A companion of the prophet, Also the name of the son of Hatim tiay known for his generosity, Also the son of Thabit had this name
Male
Norwegian
Norwegian variant form of Scandinavian Njal, NJÃ…L means "champion."
Male
Hungarian
Hungarian form of Roman Latin Cornelius, KORNÉL means "of a horn."
Male
French
French form of Hebrew Rephael, RAPHAËL means "healed of God" or "whom God has healed."
Male
Irish
Irish form of Greek Paulos, PÓL means "small."
Male
English
Anglicized form of Hebrew Adiyn, ADIN means "dainty, delicate." In the bible, this is the name of an ancestor of a family of exiles who returned with Zerubbabel.
Male
Dutch
, God's judge.
Boy/Male
Irish
Rooster.
Male
Hungarian
Hungarian form of English Philip, FÜLÖP means "lover of horses."
Male
English
Short form of English Alexander, ALIC means "defender of mankind."
P ADIC-L-FUNCTION
P ADIC-L-FUNCTION
Girl/Female
Indian
Boy/Male
American, British, English
From the Deer's Stream
Boy/Male
African, American, Christian, French, Hebrew, Hindu, Indian
One Vigour (Nothing is True); Man; Human Being
Surname or Lastname
English
English : regional name from the county of Wiltshire in southwest central England, which gets its name from Wilton (once the county’s principal town) + Old English scīr ‘district’, ‘administrative division’.
Girl/Female
Hindu, Indian, Marathi, Sanskrit
Night
Boy/Male
Tamil
South indian dynasty
Boy/Male
Hindu, Indian, Sanskrit, Thai
Root; A Knot; The Place Where the Three Main Nadis Join; Crying
Male
English
Anglicized form of Hebrew Achazyah, AHAZIAH means "God holds" or "whom God holds." In the bible, this is the name of a son of Ahab and a son of Jehoram.
Boy/Male
Hindi
Baby with a full head of hair.
Girl/Female
Indian, Telugu
Literature
P ADIC-L-FUNCTION
P ADIC-L-FUNCTION
P ADIC-L-FUNCTION
P ADIC-L-FUNCTION
P ADIC-L-FUNCTION
a.
Related to, or derived, ammonia; -- used chiefly as a suffix; as, amic acid; phosphamic acid.
a.
Pertaining to, or derived from, the cod (Gadus); -- applied to an acid obtained from cod-liver oil, viz., gadic acid.
n.
See L.
n.
An extension at right angles to the length of a main building, giving to the ground plan a form resembling the letter L; sometimes less properly applied to a narrower, or lower, extension in the direction of the length of the main building; a wing.
n.
A short right-angled pipe fitting, used in connecting two pipes at right angles.