Glossary of arithmetic and diophantine geometry
Template:Short description 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. Much of the theory is in the form of proposed conjectures, which can be related at various levels of generality.
Diophantine geometry in general is the study of algebraic varieties 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; 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 can be more generally defined as the study of schemes of finite type over the spectrum of the ring of integers.[1] Arithmetic geometry has also been defined as the application of the techniques of algebraic geometry to problems in number theory.[2]
See also the glossary of number theory terms at Glossary of number theory.
A
<dt id="Script error: No such module "delink"." >abc conjecture Template:Defn <dt id="Script error: No such module "delink"." >Arakelov class group Template:Defn <dt id="Script error: No such module "delink"." >Arakelov divisor Template:Defn <dt id="Script error: No such module "delink"." >Arakelov height Template:Defn <dt id="Script error: No such module "delink"." >Arakelov theory Template:Defn <dt id="Script error: No such module "delink"." >Arithmetic of abelian varieties Template:Defn <dt id="Script error: No such module "delink"." >Artin L-functions Template:Defn
B
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-
<dt id="Script error: No such module "delink"." >Bad reduction
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<dt id="Script error: No such module "delink"." >Birch and Swinnerton-Dyer conjecture
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C
<dt id="Script error: No such module "delink"." >Canonical height Template:Defn <dt id="Script error: No such module "delink"." >Chabauty's method Template:Defn <dt id="Script error: No such module "delink"." >Coates–Wiles theorem Template:Defn <dt id="Script error: No such module "delink"." >Crystalline cohomology Template:Defn
D
<dt id="Script error: No such module "delink"." >Diagonal forms Template:Defn <dt id="Script error: No such module "delink"." >Diophantine dimension Template:Defn <dt id="Script error: No such module "delink"." >Discriminant of a point Template:Defn <dt id="Script error: No such module "delink"." >Dwork's method Template:Defn
E
<dt id="Script error: No such module "delink"." >Étale cohomology Template:Defn
F
<dt id="Script error: No such module "delink"." >Faltings height Template:Defn <dt id="Script error: No such module "delink"." >Fermat's Last Theorem Template:Defn <dt id="Script error: No such module "delink"." >Flat cohomology Template:Defn <dt id="Script error: No such module "delink"." >Function field analogy Template:Defn
G
<dt id="Script error: No such module "delink"." >Geometric class field theory Template:Defn <dt id="Script error: No such module "delink"." >Good reduction Template:Defn <dt id="Script error: No such module "delink"." >Grothendieck–Katz conjecture Template:Defn
H
<dt id="Script error: No such module "delink"." >Hasse principle Template:Defn <dt id="Script error: No such module "delink"." >Hasse–Weil L-function Template:Defn <dt id="Script error: No such module "delink"." >Height function Template:Defn <dt id="Script error: No such module "delink"." >Hilbertian fields Template:Defn
I
<dt id="Script error: No such module "delink"." >Igusa zeta-function Template:Defn <dt id="Script error: No such module "delink"." >Infinite descent Template:Defn <dt id="Script error: No such module "delink"." >Iwasawa theory Template:Defn
K
<dt id="Script error: No such module "delink"." >K-theory Template:Defn
L
<dt id="Script error: No such module "delink"." >Lang conjecture Template:Defn <dt id="Script error: No such module "delink"." >Linear torus Template:Defn <dt id="Script error: No such module "delink"." >Local zeta-function Template:Defn
M
<dt id="Script error: No such module "delink"." >Manin–Mumford conjecture Template:Defn <dt id="Script error: No such module "delink"." >Mordell conjecture Template:Defn <dt id="Script error: No such module "delink"." >Mordell–Lang conjecture Template:Defn <dt id="Script error: No such module "delink"." >Mordell–Weil theorem Template:Defn <dt id="Script error: No such module "delink"." >Mordellic variety Template:Defn
N
<dt id="Script error: No such module "delink"." >Naive height Template:Defn <dt id="Script error: No such module "delink"." >Néron symbol Template:Defn <dt id="Script error: No such module "delink"." >Néron–Tate height Template:Defn <dt id="Script error: No such module "delink"." >Nevanlinna invariant Template:Defn
O
<dt id="Script error: No such module "delink"." >Ordinary reduction Template:Defn
Q
<dt id="Script error: No such module "delink"." >Quasi-algebraic closure Template:Defn
R
<dt id="Script error: No such module "delink"." >Reduction modulo a prime number or ideal Template:Defn <dt id="Script error: No such module "delink"." >Replete ideal Template:Defn
S
<dt id="Script error: No such module "delink"." >Sato–Tate conjecture Template:Defn <dt id="Script error: No such module "delink"." >Skolem's method Template:Defn <dt id="Script error: No such module "delink"." >Special set Template:Defn <dt id="Script error: No such module "delink"." >Subspace theorem Template:Defn
T
<dt id="Script error: No such module "delink"." >Tamagawa numbers Template:Defn <dt id="Script error: No such module "delink"." >Tate conjecture Template:Defn <dt id="Script error: No such module "delink"." >Tate curve Template:Defn <dt id="Script error: No such module "delink"." >Tsen rank Template:Defn
U
<dt id="Script error: No such module "delink"." >Uniformity conjecture Template:Defn <dt id="Script error: No such module "delink"." >Unlikely intersection Template:Defn
V
<dt id="Script error: No such module "delink"." >Vojta conjecture Template:Defn
W
<dt id="Script error: No such module "delink"." >Weights Template:Defn <dt id="Script error: No such module "delink"." >Weil cohomology Template:Defn <dt id="Script error: No such module "delink"." >Weil conjectures Template:Defn <dt id="Script error: No such module "delink"." >Weil distributions on algebraic varieties Template:Defn <dt id="Script error: No such module "delink"." >Weil function Template:Defn <dt id="Script error: No such module "delink"." >Weil height machine Template:Defn Template:Compact TOC
See also
References
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- ↑ Template:Nlab
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Further reading
- Dino Lorenzini (1996), An invitation to arithmetic geometry, AMS Bookstore, Template:ISBN