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Conjecture in algebraic geometry
mathematics, specifically arithmetic geometry, the Tate conjecture is a 1963 conjecture of John Tate that would describe the algebraic cycles on a variety
Tate_conjecture
Mathematical conjecture about elliptic curves
In mathematics, the Sato–Tate conjecture is a statistical statement about the family of elliptic curves Ep obtained from an elliptic curve E over the
Sato–Tate_conjecture
Mathematical conjecture
specifically in algebraic K-theory, the Birch–Tate conjecture is a conjecture proposed by Bryan John Birch and John Tate relating the K 2 {\displaystyle K_{2}}
Birch–Tate_conjecture
Group in arithmetic geometry
is trivial if the Tate–Shafarevich conjecture is true. Tate extended the pairing to general abelian varieties, as a variation of Tate duality. A choice
Tate–Shafarevich_group
American mathematician (1925–2019)
Taira Honda and Tate (the Honda–Tate theorem). The Tate conjectures are the equivalent for étale cohomology of the Hodge conjecture. They relate to the
John_Tate_(mathematician)
number conjecture is a major research problem. Tate conjecture The Tate conjecture (John Tate, 1963) provided an analogue to the Hodge conjecture, also
Glossary of arithmetic and diophantine geometry
Glossary_of_arithmetic_and_diophantine_geometry
Aharoni-Korman conjecture also known as the fishbone conjecture Atiyah conjecture (not a conjecture to start with) Borsuk's conjecture Bunkbed conjecture Chinese
List_of_conjectures
Mathematics concept
the Galois image. This conjecture is known only in particular cases. Through generalisations of this conjecture, the Mumford–Tate group has been connected
Mumford–Tate_group
Structure in algebraic geometry
for open problems such as the Hodge conjecture and Tate conjecture. The theory of motives was originally conjectured as an attempt to unify a rapidly multiplying
Motive_(algebraic_geometry)
British-American mathematician (born 1962)
including the Taniyama–Weil conjecture, the local Langlands conjecture for general linear groups, and the Sato–Tate conjecture." He was elected a Fellow
Richard Taylor (mathematician)
Richard_Taylor_(mathematician)
Unsolved problem in geometry
conjecture. Tate conjecture Hodge theory Hodge structure Period mapping Shioda, Tetsuji (July 13–24, 1981). "What is known about the Hodge Conjecture
Hodge_conjecture
Abelian group related to division algebras
of the Brauer group for surfaces in that case is equivalent to the Tate conjecture for divisors on X, one of the main problems in the theory of algebraic
Brauer_group
their Tate modules as Galois modules. It also makes them harder to deal with in terms of the conjectural algebraic geometry (Hodge conjecture and Tate conjecture)
Arithmetic of abelian varieties
Arithmetic_of_abelian_varieties
Curves of genus > 1 over the rationals have only finitely many rational points
been conjectured: The Mordell conjecture that a curve of genus greater than 1 over a number field has only finitely many rational points; Tate's isogeny
Faltings'_theorem
polarized. The method was introduced by Zarhin (1974) in his proof of the Tate conjecture over global fields of positive characteristic. Zarhin, Ju. G. (1974)
Zarhin_trick
тетрадь) lists unsolved problems in algebra and model theory. Birch–Tate conjecture on the relation between the order of the center of the Steinberg group
List of unsolved problems in mathematics
List_of_unsolved_problems_in_mathematics
Mathematical concept
and Madapusi Pera, which together also completed the proof of the Tate conjecture for K3 surfaces in those characteristics. K3 surfaces with Picard number
Supersingular_variety
Mathematics award
and his solution with Fernando Codá Marques of the 50-year-old Willmore Conjecture." Larry Guth – "For ingenious and surprising solutions to long standing
Breakthrough Prize in Mathematics
Breakthrough_Prize_in_Mathematics
Japanese mathematician (1928–2023)
infinite dimension. In number theory, he and John Tate independently posed the Sato–Tate conjecture on L-functions around 1960. Pierre Schapira remarked
Mikio_Sato
Algebraic structure
theorem states that μ is zero. Tate conjecture Tate twist Iwasawa theory Murty 2000, Proposition 13.4 Murty 2000, §13.8 Tate 1966 Faltings 1983 Manin & Panchishkin
Tate_module
Set of conjectures in algebraic geometry
Künneth conjectures and conjecture D for varieties over fields of characteristic zero. The Tate conjecture implies Lefschetz, Künneth, and conjecture D for
Standard conjectures on algebraic cycles
Standard_conjectures_on_algebraic_cycles
British mathematician (born 1931)
1971. In later work, Birch contributed to algebraic K-theory (Birch–Tate conjecture). He then formulated ideas on the role of Heegner points (he was one
Bryan_John_Birch
British mathematician (born 1980)
Breuil–Mézard conjecture for potentially Barsotti–Tate representations, and with Thomas Barnet-Lamb and David Geraghty, he proved the Sato–Tate conjecture for Hilbert
Toby_Gee
Study of algorithms for performing number theoretic computations
Swinnerton-Dyer conjecture, the ABC conjecture, the modularity conjecture, the Sato-Tate conjecture, and explicit aspects of the Langlands program. Magma computer
Computational_number_theory
Unproved conjecture in mathematics
mathematics, the Birch and Swinnerton-Dyer conjecture (often called the Birch–Swinnerton-Dyer conjecture) describes the set of rational solutions to
Birch and Swinnerton-Dyer conjecture
Birch_and_Swinnerton-Dyer_conjecture
Mathematical theory
algebraic de Rham cohomology and p-adic étale cohomology (the Hodge–Tate conjecture, also called CHT). Specifically, let CK be the completion of an algebraic
P-adic_Hodge_theory
Mathematics award
Clozel and Shepherd-Barron, culminating in the solution of the Sato-Tate conjecture for elliptic curves with non-integral j-invariants" 2006 not awarded
Clay_Research_Award
German mathematician (born 1954)
Fields Medal in 1986 for proving the Tate conjecture for Abelian varieties over number fields, the Shafarevich conjecture for Abelian varieties over number
Gerd_Faltings
Russian-American mathematician
conjecture for K-groups of number rings, the Hodge conjecture, the Tate conjecture about algebraic cycles, the Birch and Swinnerton-Dyer conjecture about
Alexander_Beilinson
French mathematician (born 1959)
Retrieved 22 November 2022. Andre, Yves (1996). "On the Shafarevich and Tate conjectures for hyperkähler varieties". Mathematische Annalen. 305 (1). Springer
Yves_André
Monster and modular connection
series for gh. In 1996, Borcherds and Ryba reinterpreted the conjecture as a statement about Tate cohomology of a self-dual integral form of V ♮ {\displaystyle
Monstrous_moonshine
conjecture is a conjecture, named after Fedor Bogomolov, in arithmetic geometry about algebraic curves that generalizes the Manin–Mumford conjecture in
Bogomolov_conjecture
Jiuya (2023). "The Brumer--Stark Conjecture over Z". arXiv:2310.16399v1 [math.NT]. Tate, John (1984). Les conjectures de Stark sur les fonctions L d'Artin
Brumer–Stark_conjecture
American mathematician
algorithms to numerically investigate generalizations of the Sato-Tate conjecture regarding the distribution of point counts for a curve (or abelian
Andrew Sutherland (mathematician)
Andrew_Sutherland_(mathematician)
Prime number with a certain relationship to an elliptic curve
(moonshine theory) Supersingular elliptic curve Lang–Trotter conjecture Sato–Tate conjecture Silverman 1986, pp. 137–144. Deuring 1941. Serre 1998, p. I-25
Supersingular prime (algebraic number theory)
Supersingular_prime_(algebraic_number_theory)
algebraic cycles. The Tate conjecture makes a similar prediction for étale cohomology. Alexander Grothendieck's standard conjectures on algebraic cycles
Algebraic_cycle
Two unsolved conjectures in algebraic number theory
conjecture, Birch–Tate conjecture, all of which are also unsolved. The conjecture, also referred to as Greenberg's invariants conjecture, firstly appeared
Greenberg's_conjectures
Ramanujan–Petersson conjecture Birch and Swinnerton-Dyer conjecture Automorphic form Selberg trace formula Artin conjecture Sato–Tate conjecture Langlands program
List_of_number_theory_topics
Topics referred to by the same term
mathematics, the Shafarevich conjecture, named for Igor Shafarevich, may refer to: The Tate–Shafarevich conjecture that the Tate–Shafarevich group is finite
Shafarevich_conjecture
American mathematician (born 1934)
Peter Swinnerton-Dyer, he provided a resolution of the Shafarevich-Tate conjecture for elliptic K3 surfaces and the pencil of elliptic curves over finite
Michael_Artin
In number theory, the Stark conjectures, introduced by Stark (1971, 1975, 1976, 1980) and later expanded by Tate (1984), give conjectural information about
Stark_conjectures
Estimates the number of points on an elliptic curve over a finite field
of the Weil conjectures, originally proposed by André Weil in 1949 and proved by André Weil in the case of curves. Sato–Tate conjecture Schoof's algorithm
Hasse's theorem on elliptic curves
Hasse's_theorem_on_elliptic_curves
Term used in transcendental number theory
Theorem published by Masser and Wüstholz. A direct consequence is the Tate conjecture for abelian varieties which Gerd Faltings had proved with totally different
Analytic_subgroup_theorem
In mathematics, the Fontaine–Mazur conjectures are some conjectures introduced by Fontaine and Mazur (1995) about when p-adic representations of Galois
Fontaine–Mazur_conjecture
17th-century conjecture proved by Andrew Wiles in 1994
In number theory, Fermat's Last Theorem (sometimes called Fermat's conjecture, especially in older texts) states that there are no positive integers a
Fermat's_Last_Theorem
Algebraic curve in mathematics
) = 1 − a T + q T 2 {\displaystyle L(E(K),T)=1-aT+qT^{2}} The Sato–Tate conjecture is a statement about how the error term 2 q {\displaystyle 2{\sqrt
Elliptic_curve
Mathematical conjecture
In mathematics, the Zilber–Pink conjecture is a far-reaching generalisation of many famous Diophantine conjectures and statements, such as André–Oort,
Zilber–Pink_conjecture
Mathematical surface
surface with Picard number 22 must be supersingular. Conversely, the Tate conjecture would imply that every supersingular K3 surface over an algebraically
Supersingular_K3_surface
Seven mathematical problems with a US$1 million prize for each solution
unsolved mathematical problems, the Birch and Swinnerton-Dyer conjecture, Hodge conjecture, Navier–Stokes existence and smoothness, P versus NP problem
Millennium_Prize_Problems
Mathematical curves that are isomorphic over algebraic closures
and when generalized to hyperelliptic curves, the study of the Sato–Tate conjecture. First assume K {\displaystyle K} is a field of characteristic different
Twists_of_elliptic_curves
in defining the Néron–Tate height, and the height used in the statement of the Birch–Swinnerton-Dyer conjecture is the Néron–Tate height associated to
Néron–Tate_height
Probability distribution
the Wigner distribution is sometimes called the Sato–Tate distribution. See Sato–Tate conjecture. Marchenko–Pastur distribution or Free Poisson distribution
Wigner semicircle distribution
Wigner_semicircle_distribution
Major unsolved problem in transcendental number theory
mathematics, specifically transcendental number theory, Schanuel's conjecture is a conjecture about the transcendence degree of certain field extensions of
Schanuel's_conjecture
Monoidal category
representations and motives through Tannakian categories. Mumford-Tate conjecture proposes that the algebraic groups arising from the Hodge strucuture
Tannakian_formalism
French mathematician (born 1953)
Taylor, Nicholas Shepherd-Barron, and Michael Harris he proved the Sato–Tate conjecture. Arthur, James; Clozel, Laurent (1989). Simple algebras, base change
Laurent_Clozel
Type of zeta function
number of irreducible components of X with maximal dimension. Secondly, Tate conjectured o r d s = n − 1 ζ X ( s ) = r k O X × ( X ) − r k P i c ( X ) {\displaystyle
Arithmetic_zeta_function
Analogs of homology groups for algebraic varieties
finitely generated field (such as a finite field or number field), the Tate conjecture predicts the image (tensored with Ql) of the cycle map from Chow groups
Chow_group
American mathematician
Richard Taylor proved the local Langlands conjecture for GL(n) over a p-adic local field The Sato–Tate conjecture and its generalization to all totally real
Michael Harris (mathematician)
Michael_Harris_(mathematician)
American criminal and cult leader (1934–2017)
was the founder of the Manson Family. He gained notoriety for ordering the Tate–LaBianca murders, where his followers murdered nine people around Los Angeles
Charles_Manson
Type of smooth complex surface of kodaira dimension 0
surfaces by Daniel Burns and Michael Rapoport (1975). Enriques surface Tate conjecture Mathieu moonshine, a mysterious relationship between K3 surfaces and
K3_surface
Jewish Museum (1983–1995). Mikio Sato, 94, Japanese mathematician (Sato–Tate conjecture, Bernstein–Sato polynomial). Tim Schadla-Hall, 75, British archaeologist
Deaths_in_January_2023
British mathematician
Harris and Richard Taylor, he proved the original version of the Sato–Tate conjecture and its generalization to totally real fields, under mild assumptions
Nicholas_Shepherd-Barron
Relates rational elliptic curves to modular forms
statement was known as the Taniyama–Shimura conjecture, Taniyama–Shimura–Weil conjecture, or the modularity conjecture for elliptic curves. The theorem states
Modularity_theorem
(sometimes known as "Serre's Conjecture" or "Serre's problem") Serre's Conjecture concerning Galois representations Serre's "Conjecture II" concerning linear
List of things named after Jean-Pierre Serre
List_of_things_named_after_Jean-Pierre_Serre
the Mordell conjecture (a conjecture dating back to 1922). Other theorems proved in this paper include an instance of the Tate conjecture (relating the
List of publications in mathematics
List_of_publications_in_mathematics
Indian mathematician (born 1960)
doi:10.1016/j.aim.2017.03.017. Prasad, Dipendra (2019). "A mod-p Artin–Tate conjecture, and generalizing the Herbrand–Ribet theorem". Pacific Journal of Mathematics
Dipendra_Prasad
Particular kind of exponential sum
prime, there are no known simple formula for K(a, b; p), and the Sato–Tate conjecture suggests that none exist. The lifting formulas below, however, are
Kloosterman_sum
Artin conductor Artin's conjecture for conjectures by Artin. These include Artin's conjecture on primitive roots Artin conjecture on L-functions Artin group
List of things named after Emil Artin
List_of_things_named_after_Emil_Artin
Conjecture on zeros of the zeta function
problems in mathematics In mathematics, the Riemann hypothesis is the conjecture that the Riemann zeta function has its zeros only at the negative even
Riemann_hypothesis
Theorem in algebraic number theory relating p-adic L-functions and ideal class groups
and Kolyvagin, this gave a conditional proof (on the Tate–Shafarevich conjecture) of the conjecture that E has infinitely many rational points if and only
Main conjecture of Iwasawa theory
Main_conjecture_of_Iwasawa_theory
Korean educator (born 1978)
trace formula in the conditional proofs of generalizations of the Sato–Tate conjecture by Harris (for products of non-isogenous elliptic curves) and
Sug_Woo_Shin
Mathematics professorship at the University of Oxford
has worked on Langlands program and, with others, proved the Sato–Tate conjecture, and collaborated with Andrew Wiles on the solution to Fermat's Last
Savilian Professor of Geometry
Savilian_Professor_of_Geometry
Riemann–Hurwitz formula Riemann–Roch theorem Riemann surface Road curve Sato–Tate conjecture secant Singular solution Sinuosity Slope Space curve Spinode Square
List_of_curves_topics
Japanese mathematician
geometry. His contributions include his solution of the Grothendieck conjecture in anabelian geometry about hyperbolic curves over number fields. Mochizuki
Shinichi_Mochizuki
Conjecture in number theory
In number theory, Szpiro's conjecture relates the conductor of an elliptic curve to its discriminant. In a slightly modified form, it is equivalent to
Szpiro's_conjecture
German mathematician
associated to Tate motives – Matthias Flach and D. Burns, King's College London On the Equivariant Tamagawa Number Conjecture for Tate Motives, Part II
Matthias Flach (mathematician)
Matthias_Flach_(mathematician)
Commune, gang, and cult in California led by Charles Manson
"Tex" Watson, and Patricia Krenwinkel entered the home of actress Sharon Tate and murdered her and four others. Linda Kasabian was also present but did
Manson_Family
Result concerning properties of Galois representations associated with modular forms
Ribet's theorem (earlier called the epsilon conjecture or ε-conjecture) is part of number theory. It concerns properties of Galois representations associated
Ribet's_theorem
Construct in mathematics
the p {\displaystyle p} -component of the Tate–Shafarevich group is finite. It is conjectured that the Tate–Shafarevich group is in fact finite, in which
Selmer_group
Conjecture in number theory
In algebraic geometry and number theory, the torsion conjecture or uniform boundedness conjecture for torsion points for abelian varieties states that
Torsion_conjecture
American mathematician
functions, and in particular for a proof of the first part of the Weil conjectures: the rationality of the zeta function of a variety over a finite field
Bernard_Dwork
French mathematician
Fargues has formulated a general geometric conjecture which refines the classical local Langlands conjecture, and at the same time introduces extra structure
Laurent_Fargues
French mathematician (born 1962)
-adic analog of Dirichlet's analytic class number formula. A conjecture: the Colmez conjecture relating Artin L-functions at s = 0 {\displaystyle s=0} and
Pierre_Colmez
medalist (1976).[citation needed] John Tate, 94, American mathematician (Tate's thesis, Tate conjecture, Tate cohomology group), Abel Prize winner (2010)
Deaths_in_October_2019
Mathematics of varieties with integer coordinates
modern examples include the André–Oort conjecture, the Bogomolov conjecture and also the uniform Mordell conjecture. Serge Lang published a book Diophantine
Diophantine_geometry
American mathematician (born 1972)
and Kolyvagin, this gave a conditional proof (on the Tate–Shafarevich conjecture) of the conjecture that E has infinitely many rational points if and only
Christopher_Skinner
American mathematician (born 1937)
topology. In an elementary fashion, he proved the generalized Schoenflies conjecture (his complete proof required an additional result by Marston Morse), around
Barry_Mazur
Theory in number theory
maps between the curves. A first version of Grothendieck's anabelian conjecture was solved by Hiroaki Nakamura and Akio Tamagawa (for affine curves),
Anabelian_geometry
Special point on a modular curve in mathematics
computations, arXiv:math.NT/0506325v2. Brown, Mark (1994), "On a conjecture of Tate for elliptic surfaces over finite fields", Proc. London Math. Soc
Heegner_point
Chinese-American mathematician (born 1962)
Birch-Swinnerton-Dyer conjecture for modular abelian varieties of GL(2) type over totally real fields through his work relating the Néron–Tate height of Heegner
Shou-Wu_Zhang
Dutch mathematician
known as the André–Oort conjecture (generalizing a conjecture made in 1989 by Yves André). In 2000 Oort proved a conjecture made by Grothendieck in 1970
Frans_Oort
Number of independent rational basis points with infinite order
Katz–Sarnak conjectured that in a suitable asymptotic sense (see below), the rank of elliptic curves should be 1/2 on average. An even stronger conjecture is that
Rank_of_an_elliptic_curve
Mathematical functions that quantify complexity
rational points on algebraic varieties, such as the Manin conjecture and Vojta's conjecture, have far-reaching implications for problems in Diophantine
Height_function
Certain polynomial equations in enough variables over a finite field have solutions
by Chevalley (1935). Chevalley's theorem implied Artin's and Dickson's conjecture that finite fields are quasi-algebraically closed fields (Artin 1982,
Chevalley–Warning_theorem
Soviet American mathematician
(with A. M. Levin) Goncharov, A. B.; Levin, A. M. (1998). "Zagier's conjecture on L(E,2)". Inventiones Mathematicae. 132 (2): 393–432. Bibcode:1998InMat
Alexander_Goncharov
French mathematician
Rueil-Malmaison, France. In 1983, Raynaud published a proof of the Manin–Mumford conjecture. In 1985, he proved Raynaud's isogeny theorem on Faltings heights of isogenous
Michel_Raynaud
American mathematician (born 1947)
Swinnerton-Dyer conjecture includes the proof of an estimate for a partial Euler product associated to an elliptic curve, bounds for the order of the Tate–Shafarevich
Dorian_M._Goldfeld
New Zealand mathematician
entitled "The conjectures of Birch and Swinnerton-Dyer for constant abelian varieties over function fields," he proved the conjecture of Birch and Swinnerton–Dyer
James_Milne_(mathematician)
mathematician. Hodge algebra Hodge–Arakelov theory Hodge bundle Hodge conjecture Hodge cycle Hodge–de Rham spectral sequence Hodge diamond Hodge duality
List of things named after W. V. D. Hodge
List_of_things_named_after_W._V._D._Hodge
American mathematician
Much of Farrell's work lies around the Borel conjecture. He and his co-authors have verified the conjecture for various cases, most notably flat manifolds
F._Thomas_Farrell
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