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Expresses the number of points of a variety over a finite field
algebraic geometry, the Grothendieck trace formula expresses the number of points of a variety over a finite field in terms of the trace of the Frobenius endomorphism
Grothendieck_trace_formula
Topics referred to by the same term
Jacquet's relative trace formula, simple trace formula, stable trace formula Grothendieck trace formula, an analogue in algebraic geometry of the Lefschetz
Trace_formula
Extension of Lidskii's theorem
Alexander Grothendieck. Lidskii's theorem does not hold in general for Banach spaces. The theorem should not be confused with the Grothendieck trace formula from
Grothendieck_trace_theorem
sequence Grothendieck–Springer resolution Grothendieck–Teichmüller group Grothendieck–Teichmüller theory Grothendieck-Witt ring Grothendieck trace formula Grothendieck
List of things named after Alexander Grothendieck
List_of_things_named_after_Alexander_Grothendieck
In algebraic geometry, Behrend's trace formula is a generalization of the Grothendieck–Lefschetz trace formula to a smooth algebraic stack over a finite
Behrend's_trace_formula
Mapping theorem in topology
also algebraic geometry counterpart of this theorem called Lefschetz trace formula that allows to express number of points of variety over finite field
Lefschetz_fixed-point_theorem
Belgian mathematician
supervision of Alexander Grothendieck, with a thesis titled Théorie de Hodge. Starting in 1965, Deligne worked with Grothendieck at the Institut des Hautes
Pierre_Deligne
Norm on a vector space of matrices
which is itself equivalent to another norm, called the Grothendieck norm. To define the Grothendieck norm, first note that a linear operator K1 → K1 is just
Matrix_norm
Clausen (2009). Altman, Allen; Kleiman, Steven (1970), Introduction to Grothendieck duality theory, Lecture Notes in Mathematics, vol. 146, Springer, doi:10
Sum_of_residues_formula
class of X. Behrend's trace formula Behrend's trace formula generalizes Grothendieck's trace formula; both formulas compute the trace of the Frobenius on
Glossary of algebraic geometry
Glossary_of_algebraic_geometry
Type of zeta function
{1}{1-q^{-s}}}.} For a variety X over a finite field, it is known by Grothendieck's trace formula that ζ X ( s ) = Z ( X , q − s ) {\displaystyle \zeta _{X}(s)=Z(X
Arithmetic_zeta_function
and so the trace is not uniquely defined. However, if the order q ≤ 2/3, then there is a unique trace, as given by a theorem of Grothendieck. If L : B
Fredholm_kernel
Moduli spaces of ramified covers
questions (approached with combinatorial means), and to use the Grothendieck trace formula and Deligne's estimations of eigenvalues of Frobenius (as explained
Hurwitz_space
On generating functions from counting points on algebraic varieties over finite fields
conjecture) (Grothendieck 1965). The general theorems about étale cohomology allowed Grothendieck to prove an analogue of the Lefschetz fixed-point formula for
Weil_conjectures
Subject area in mathematics
1950s by Alexander Grothendieck in his study of intersection theory on algebraic varieties. In the modern language, Grothendieck defined only K0, the
Algebraic_K-theory
Theorem in abstract algebra
conjectures using the Arthur–Selberg trace formula, but in order for this approach to work, the geometric sides of the trace formula for different groups must be
Fundamental lemma (Langlands program)
Fundamental_lemma_(Langlands_program)
German mathematician
function). He is also known for Behrend's formula, the generalization of the Grothendieck–Lefschetz trace formula to algebraic stacks. He is the recipient
Kai_Behrend
Diagram that shows all possible logical relations between a collection of sets
which are considered the precursors of Venn diagrams, can also be clearly traced back to the 16th century. Pioneers in this tradition of Euler diagrams included
Venn_diagram
Sheaf cohomology on the étale site
groups with finite coefficients of a topological space, introduced by Grothendieck in order to prove the Weil conjectures. Étale cohomology theory can be
Étale_cohomology
inequality Gagliardo–Nirenberg interpolation inequality Gårding's inequality Grothendieck inequality Grunsky's inequalities Hanner's inequalities Hardy's inequality
List_of_inequalities
Conjecture in algebraic geometry
and planned to be completed in a second volume using the Grothendieck-Lefschetz trace formula and the Ran space. Ono (1965) used the Weil conjecture to
Weil's conjecture on Tamagawa numbers
Weil's_conjecture_on_Tamagawa_numbers
Algebraic geometry analog of a principal bundle in algebraic topology
Groupes algébriques, Tome I. Let T {\displaystyle {\mathcal {T}}} be a Grothendieck topology and X {\displaystyle X} a scheme. Moreover let G {\displaystyle
Torsor_(algebraic_geometry)
Form of mathematical proof
Nevertheless, his argument in al-Fakhri is the earliest extant proof of the sum formula for integral cubes. In India, early implicit proofs by mathematical induction
Mathematical_induction
Symbol representing a mathematical object
real. The set of points (x, y) in the 2D plane satisfying this equation trace out the graph of a parabola. Here, a, b and c are regarded as constants
Variable_(mathematics)
formulae of the general theory.) It is a consequence of the Lefschetz trace formula for the Frobenius morphism that Z ( X , t ) = ∏ i = 0 2 dim X det
Local_zeta_function
Algebraic manipulation of "true" and "false"
variables of a given Boolean (propositional) formula can be assigned in such a way as to make the formula evaluate to true is called the Boolean satisfiability
Boolean_algebra
French mathematician
Drinfeld, formule des traces d'Arthur-Selberg et correspondance de Langlands. [Drinfelʹd shtukas, Arthur-Selberg trace formula and Langlands correspondence]
Laurent_Lafforgue
Mathematical result in differential geometry
by the Chow ring of a smooth variety, and the Grothendieck group on the left is given by the Grothendieck group of algebraic vector bundles. Due to (Teleman
Atiyah–Singer_index_theorem
Theorem in algebraic geometry
applies to vector bundles on a smooth projective variety, but Alexander Grothendieck found wide generalizations, for example to singular varieties. On an
Serre_duality
Concept in algebraic geometry
over an algebraically closed field, but that restriction was removed by Grothendieck). The analogs of Cartan's theorems hold in great generality: if F {\displaystyle
Coherent_sheaf_cohomology
^{G}(X)).} In particular, K 0 G ( C ) {\displaystyle K_{0}^{G}(C)} is the Grothendieck group of Coh G ( X ) {\displaystyle \operatorname {Coh} ^{G}(X)} .
Equivariant algebraic K-theory
Equivariant_algebraic_K-theory
Embedding of categories into functor categories
Vakil, Ravi (2026). The rising sea: Foundations of algebraic geometry. Grothendieck, Alexander; Dieudonné, Jean (1961). "Éléments de géométrie algébrique
Yoneda_lemma
Function computable with bounded loops
formally in mathematics before, but the construction of primitive recursion is traced back to Richard Dedekind's theorem 126 of his Was sind und was sollen die
Primitive_recursive_function
Basic notion of sameness in mathematics
Jean-Jacques (eds.). The Mathematical and Philosophical Legacy of Alexander Grothendieck. Cham: Springer Nature Switzerland. pp. 337–354. doi:10.1007/978-3-031-68934-5_13
Equality_(mathematics)
a total of 890 pages. 1983 – Selberg trace formula. Hejhal's proof of a general form of the Selberg trace formula consisted of 2 volumes with a total length
List of long mathematical proofs
List_of_long_mathematical_proofs
Finite extension of the rationals
Kato, Kazuya (1990), "L-functions and Tamagawa numbers of motives", The Grothendieck Festschrift, Vol. I, Progr. Math., vol. 86, Boston, MA: Birkhäuser Boston
Algebraic_number_field
Theory of truth in the philosophy of language
satisfaction can be defined recursively on the construction of formulas. Atomic Formula: R ( t 1 , … , t n ) {\displaystyle R(t_{1},\ldots ,t_{n})} , M
Semantic_theory_of_truth
Relationship between programs and proofs
calculus correspond to relevant logic. The local truth (∇) modality in Grothendieck topology or the equivalent "lax" modality (◯) of Benton, Bierman, and
Curry–Howard_correspondence
Type of logic diagram
the four basic categorical propositions. The origin of the square can be traced back to Aristotle's tractate On Interpretation and its distinction between
Square_of_opposition
Maximal proper filter
U\subseteq {\mathcal {P}}(X)} is ultra and Y {\displaystyle Y} is a set. The trace U | Y := { B ∩ Y : B ∈ U } {\displaystyle U\vert _{Y}:=\{B\cap Y:B\in U\}}
Ultrafilter_on_a_set
History of maths
space. 1957 Alexander Grothendieck Grothendieck's relative point of view, S-schemes. 1957 Alexander Grothendieck Grothendieck–Hirzebruch–Riemann–Roch
Timeline of category theory and related mathematics
Timeline_of_category_theory_and_related_mathematics
Computation model defining an abstract machine
computation—the current state of the total system. What Turing called "the state formula" includes both the current instruction and all the symbols on the tape:
Turing_machine
American mathematician
Flicker is the author of a number of books including: Arthur's Invariant Trace Formula and Comparison of Inner Forms (2016) Drinfeld Moduli Schemes and Automorphic
Yuval_Flicker
Relationship between two functors abstracting many common constructions
abelianization which assigns to every group G the quotient group Gab=G/[G,G]. The Grothendieck group. In K-theory, the point of departure is to observe that the category
Adjoint_functors
Branch of mathematics
instead the lattice of open sets as the basic notion of the theory, while Grothendieck topologies are structures defined on arbitrary categories that allow
Topology
bilinéaires sur les espaces de Banach, d'après Grothendieck (Grothendieck's inequality) Alexander Grothendieck, Techniques de construction et théorèmes d'existence
Séminaire Nicolas Bourbaki (1960–1969)
Séminaire_Nicolas_Bourbaki_(1960–1969)
Mathematical category whose hom sets form Abelian groups
that composition of morphisms distributes over the group operation. In formulas: f ∘ ( g + h ) = ( f ∘ g ) + ( f ∘ h ) {\displaystyle f\circ (g+h)=(f\circ
Preadditive_category
Central object of study in category theory
commutes. Set η G ( a ) = a − 1 {\displaystyle \eta _{G}(a)=a^{-1}} . The formulas ( a ∗ b ) − 1 = b − 1 ∗ a − 1 = a − 1 ∗ op b − 1 {\displaystyle
Natural_transformation
Axiomatic set theories based on the principles of mathematical constructivism
principle that implies P E M {\displaystyle {\mathrm {PEM} }} for the formulas permitted in one's adopted Separation schema, by Diaconescu's theorem.
Constructive_set_theory
Type of category in category theory
by P in C/Y can be expressed in terms of the dependent product by the formula Q P ≅ Π p ( p ∗ ( Q ) ) {\displaystyle Q^{P}\cong \Pi _{p}(p^{*}(Q))}
Cartesian_closed_category
Branch of mathematics
collaborators, use localization theory, categories of quasicoherent sheaves and Grothendieck-topological ideas to formulate noncommutative schemes. A major goal of
Noncommutative_geometry
Category theory constructs
object b of B. Dually, right Kan extensions can be computed by the end formula ( Ran F X ) b = ∫ a X a B ( b , F a ) . {\displaystyle (\operatorname
Kan_extension
Algebraic variety that is a moduli space for principally polarized abelian varieties
James; Ellwood, David; Kottwitz, Robert (eds.). Harmonic Analysis, the Trace Formula, and Shimura Varieties. Clay Mathematics Proceedings. Vol. 4. American
Siegel_modular_variety
Surjective homomorphism
1007/978-1-4020-1962-3. ISBN 0-521-83414-7. MR 2056583. Définition 2.2. in Alexander Grothendieck, Technique de descente et théorèmes d’existence en géométrie algébrique
Epimorphism
Size of a set in mathematics
V κ {\displaystyle V_{\kappa }} can serve as a model of ZFC (cf. Grothendieck universe). Stronger and stronger large cardinal axioms assert the existence
Cardinality
Mathematical space
Example 1.24. Milnor & Stasheff (1974), pp. 57–59. Milnor & Stasheff 1974. Grothendieck, Alexander (1971). Éléments de géométrie algébrique. Vol. 1 (2nd ed.)
Grassmannian
Representation of data of various types in lambda calculus
{\displaystyle \operatorname {pred} _{2}} definition here. Indeed, if we trace its execution, we arrive at the new, even more streamlined, yet fully equivalent
Church_encoding
Differential algebra
"etale morphism of schemes in nLab". ncatlab.org. Retrieved 2024-09-29. Grothendieck, Alexander (1964). "Éléments de géométrie algébrique : IV. Étude locale
Weyl_algebra
Mathematical term; concerning axioms used to derive theorems
ISBN 978-3-7643-7524-9. Fantechi, Barbara, ed. (2005). Fundamental Algebraic Geometry: Grothendieck's FGA Explained. American Mathematical Soc. p. 248. ISBN 978-0-8218-4245-4
Axiomatic_system
Category admitting tensor products
the categorical structure by the identity morphism and the composition formula in C, respectively. If c ≤ c ′ {\displaystyle c\leq c'} and c ′ ≤ c {\displaystyle
Monoidal_category
Historical development of geometry
fields as demonstrated by the works of among others André Weil, Alexander Grothendieck, and Jean-Pierre Serre as well as over the real or complex numbers. Finite
History_of_geometry
abstract structure was itself abstracted and led to category theory. Grothendieck and Serre recast algebraic geometry using sheaf theory. Large advances
History_of_mathematics
variety over a field k of characteristic zero can be computed in terms of Grothendieck's algebraic de Rham complex. In particular, if the variety V=Spec A is
Cyclic_homology
Branch of mathematics that studies abstract algebraic structures
modular forms. Important results in the theory include the Selberg trace formula and the realization by Robert Langlands that the Riemann–Roch theorem
Representation_theory
Branch of mathematics
undergone major foundational development, with the introduction by Alexander Grothendieck of scheme theory, which allows using topological methods, including cohomology
Geometry
Equalities for combinations of sets
denotes the universe set, which means that all sets that are used in the formula are subsets of X . {\displaystyle X.} In particular, the complement of
List of set identities and relations
List_of_set_identities_and_relations
Sheaf consisting of modules on a ringed space; generalizing vector bundles
Proposition 6.9. Hartshorne, Robin. Algebraic Geometry. pp. 233–235. Grothendieck, Alexandre; Dieudonné, Jean (1960). "Éléments de géométrie algébrique:
Sheaf_of_modules
Mathematical object studied in the field of algebraic geometry
Compactifications and cohomology of modular varieties. In Harmonic analysis, the trace formula, and Shimura varieties, volume 4 of Clay Math. Proc., pages 551–582
Algebraic_variety
Locally convex topological vector space
this class forms a category with properties similar to those of Ste. Grothendieck space A generalization which has some of the properties of reflexive
Reflexive_space
First article on transfinite set theory
in the article he submitted — he added it during proofreading. They have traced this and other facts about the article to the influence of Karl Weierstrass
Cantor's first set theory article
Cantor's_first_set_theory_article
{\displaystyle A} is a commutative C*-algebra. Grothendieck 1. Grothendieck's inequality. 2. Grothendieck's factorization theorem. Hahn–Banach The Hahn–Banach
Glossary of functional analysis
Glossary_of_functional_analysis
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GROTHENDIECK TRACE-FORMULA
GROTHENDIECK TRACE-FORMULA
GROTHENDIECK TRACE-FORMULA
GROTHENDIECK TRACE-FORMULA
GROTHENDIECK TRACE-FORMULA
GROTHENDIECK TRACE-FORMULA
GROTHENDIECK TRACE-FORMULA
GROTHENDIECK TRACE-FORMULA
GROTHENDIECK TRACE-FORMULA
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