Glossary of arithmetic and diophantine geometry
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.
Arithmetical or arithmetic (algebraic) geometry is a field with a less elementary definition. After the advent of scheme theory it could reasonably be defined as the study of Alexander Grothendieck's schemes of finite type over the spectrum of the ring of integers Z. This point of view has been very influential for around half a century; it has very widely been regarded as fulfilling Leopold Kronecker's ambition to have number theory operate only with rings that are quotients of polynomial rings over the integers (to use the current language of commutative algebra). In fact scheme theory uses all sorts of auxiliary constructions that do not appear at all 'finitistic', so that there is little connection with 'constructivist' ideas as such. That scheme theory may not be the last word appears from continuing interest in the 'infinite primes' (the real and complex local fields), which do not come from prime ideals as the p-adic numbers do.
A
Arakelov theory is an approach to arithmetic geometry that explicitly includes the 'infinite primes'.
Artin L-functions are defined for quite general Galois representations. The introduction of étale cohomology in the 1960s meant that Hasse–Weil L-functions could be regarded as Artin L-functions for the Galois representations on l-adic cohomology groups.
B
- Bad reduction
- See good reduction.
- Birch and Swinnerton-Dyer conjecture
- The Birch and Swinnerton-Dyer conjecture on elliptic curves postulates a connection between the rank of an elliptic curve and the order of pole of its Hasse–Weil L-function. It has been an important landmark in Diophantine geometry since the mid-1960s, with results such as the Coates–Wiles theorem, Gross–Zagier theorem and Kolyvagin's theorem.[7]
- Bombieri–Lang conjecture
Enrico Bombieri, Serge Lang and Paul Vojta and Piotr Blass have conjectured that algebraic varieties of general type do not have Zariski dense subsets of K-rational points, for K a finitely-generated field. This circle of ideas includes the understanding of analytic hyperbolicity and the Lang conjectures on that, and the Vojta conjectures. An analytically hyperbolic algebraic variety V over the complex numbers is one such that no holomorphic mapping from the whole complex plane to it exists, that is not constant. Examples include compact Riemann surfaces of genus g > 1. Lang conjectured that V is analytically hyperbolic if and only if all subvarieties are of general type.[8]
C
Chabauty's method, based on p-adic analytic functions, is a special application but capable of proving cases of the Mordell conjecture for curves whose Jacobian's rank is less than its dimension. It developed ideas from Thoralf Skolem's method for an algebraic torus. (Other older methods for Diophantine problems include Runge's method.)
Crystalline cohomology is a p-adic cohomology theory in characteristic p, introduced by Alexander Grothendieck to fill the gap left by étale cohomology which is deficient in using mod p coefficients in this case. It is one of a number of theories deriving in some way from Dwork's method, and has applications outside purely arithmetical questions.
D
Diagonal forms are some of the simplest projective varieties to study from an arithmetic point of view (including the Fermat varieties). Their local zeta-functions are computed in terms of Jacobi sums. Waring's problem is the most classical case.
Bernard Dwork used distinctive methods of p-adic analysis, p-adic algebraic differential equations, Koszul complexes and other techniques that have not all been absorbed into general theories such as crystalline cohomology. He first proved the rationality of local zeta-functions, the initial advance in the direction of the Weil conjectures.
E
F
Fermat's last theorem, the most celebrated conjecture of Diophantine geometry, was proved by Andrew Wiles and Richard Taylor.
Flat cohomology is, for the school of Grothendieck, one terminal point of development. It has the disadvantage of being quite hard to compute with. The reason that the flat topology has been considered the 'right' foundational topos for scheme theory goes back to the fact of faithfully-flat descent, the discovery of Grothendieck that the representable functors are sheaves for it (i.e. a very general gluing axiom holds).
G
H
I
Infinite descent was Pierre de Fermat's classical method for Diophantine equations. It became one half of the standard proof of the Mordell–Weil theorem, with the other being an argument with height functions (q.v.). Descent is something like division by two in a group of principal homogeneous spaces (often called 'descents', when written out by equations); in more modern terms in a Galois cohomology group which is to be proved finite. See Selmer group.
Iwasawa theory builds up from the analytic number theory and Stickelberger's theorem as a theory of ideal class groups as Galois modules and p-adic L-functions (with roots in Kummer congruence on Bernoulli numbers). In its early days in the late 1960s it was called Iwasawa's analogue of the Jacobian. The analogy was with the Jacobian variety J of a curve C over a finite field F (qua Picard variety), where the finite field has roots of unity added to make finite field extensions F′ The local zeta-function (q.v.) of C can be recovered from the points J(F′) as Galois module. In the same way, Iwasawa added pn-power roots of unity for fixed p and with n → ∞, for his analogue, to a number field K, and considered the inverse limit of class groups, finding a p-adic L-function earlier introduced by Kubota and Leopoldt.
K
Algebraic K-theory is on one hand a quite general theory with an abstract algebra flavour, and, on the other hand, implicated in some formulations of arithmetic conjectures. See for example Birch–Tate conjecture, Lichtenbaum conjecture.
L
M
N
O
Q
R
S
T
U
V
W
See also
- Arithmetic topology
- Arithmetic dynamics
References
^ ab Schoof, René (2008). "Computing Arakelov class groups". In Buhler, J.P.; P., Stevenhagen. Algorithmic Number Theory: Lattices, Number Fields, Curves and Cryptography. MSRI Publications. 44. Cambridge University Press. pp. 447–495. ISBN 978-0-521-20833-8. MR 2467554. Zbl 1188.11076..mw-parser-output cite.citationfont-style:inherit.mw-parser-output qquotes:"""""""'""'".mw-parser-output code.cs1-codecolor:inherit;background:inherit;border:inherit;padding:inherit.mw-parser-output .cs1-lock-free abackground:url("//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/9px-Lock-green.svg.png")no-repeat;background-position:right .1em center.mw-parser-output .cs1-lock-limited a,.mw-parser-output .cs1-lock-registration abackground:url("//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Lock-gray-alt-2.svg/9px-Lock-gray-alt-2.svg.png")no-repeat;background-position:right .1em center.mw-parser-output .cs1-lock-subscription abackground:url("//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Lock-red-alt-2.svg/9px-Lock-red-alt-2.svg.png")no-repeat;background-position:right .1em center.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registrationcolor:#555.mw-parser-output .cs1-subscription span,.mw-parser-output .cs1-registration spanborder-bottom:1px dotted;cursor:help.mw-parser-output .cs1-hidden-errordisplay:none;font-size:100%.mw-parser-output .cs1-visible-errorfont-size:100%.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration,.mw-parser-output .cs1-formatfont-size:95%.mw-parser-output .cs1-kern-left,.mw-parser-output .cs1-kern-wl-leftpadding-left:0.2em.mw-parser-output .cs1-kern-right,.mw-parser-output .cs1-kern-wl-rightpadding-right:0.2em
^ ab Neukirch (1999) p.189
^ Lang (1988) pp.74–75
^ van der Geer, G.; Schoof, R. (2000). "Effectivity of Arakelov divisors and the theta divisor of a number field". Selecta Mathematica, New Series. 6 (4): 377–398. arXiv:math/9802121. doi:10.1007/PL00001393. Zbl 1030.11063.
^ Bombieri & Gubler (2006) pp.66–67
^ Lang (1988) pp.156–157
^ Lang (1997) pp.91–96
^ ab Hindry & Silverman (2000) p.479
^ Coates, J.; Wiles, A. (1977). "On the conjecture of Birch and Swinnerton-Dyer". Inventiones Mathematicae. 39 (3): 223–251. Bibcode:1977InMat..39..223C. doi:10.1007/BF01402975. Zbl 0359.14009.
^ Neukirch, Jürgen; Schmidt, Alexander; Wingberg, Kay (2008). Cohomology of Number Fields. Grundlehren der Mathematischen Wissenschaften. 323 (2nd ed.). Springer-Verlag. p. 361. ISBN 3-540-37888-X.
^ Lang (1997) p.146
^ abc Lang (1997) p.171
^ Faltings, Gerd (1983). "Endlichkeitssätze für abelsche Varietäten über Zahlkörpern". Inventiones Mathematicae. 73 (3): 349–366. Bibcode:1983InMat..73..349F. doi:10.1007/BF01388432.
^ Cornell, Gary; Silverman, Joseph H. (1986). Arithmetic geometry. New York: Springer. ISBN 0-387-96311-1. → Contains an English translation of Faltings (1983)
^ Serre, Jean-Pierre; Tate, John (November 1968). "Good reduction of abelian varieties". The Annals of Mathematics. Second. 88 (3): 492–517. doi:10.2307/1970722. JSTOR 1970722. Zbl 0172.46101.
^ Lang (1997) pp.43–67
^ Bombieri & Gubler (2006) pp.15–21
^ Igusa, Jun-Ichi (1974). "Complex powers and asymptotic expansions. I. Functions of certain types". Journal für die reine und angewandte Mathematik. 1974 (268–269): 110–130. doi:10.1515/crll.1974.268-269.110. Zbl 0287.43007.
^ Bombieri & Gubler (2006) pp.82–93
^ Raynaud, Michel (1983). "Sous-variétés d'une variété abélienne et points de torsion". In Artin, Michael; Tate, John. Arithmetic and geometry. Papers dedicated to I. R. Shafarevich on the occasion of his sixtieth birthday. Vol. I: Arithmetic. Progress in Mathematics (in French). 35. Birkhauser-Boston. pp. 327–352. Zbl 0581.14031.
^ Roessler, Damian (2005). "A note on the Manin–Mumford conjecture". In van der Geer, Gerard; Moonen, Ben; Schoof, René. Number fields and function fields — two parallel worlds. Progress in Mathematics. 239. Birkhäuser. pp. 311–318. ISBN 0-8176-4397-4. Zbl 1098.14030.
^ Marcja, Annalisa; Toffalori, Carlo (2003). A Guide to Classical and Modern Model Theory. Trends in Logic. 19. Springer-Verlag. pp. 305–306. ISBN 1402013302.
^ 2 page exposition of the Mordell–Lang conjecture by B. Mazur, 3 Nov. 2005
^ Lang (1997) p.15
^ Baker, Alan; Wüstholz, Gisbert (2007). Logarithmic Forms and Diophantine Geometry. New Mathematical Monographs. 9. Cambridge University Press. p. 3. ISBN 978-0-521-88268-2. Zbl 1145.11004.
^ Bombieri & Gubler (2006) pp.301–314
^ Lang (1988) pp.66–69
^ Lang (1997) p.212
^ ab Lang (1988) p.77
^ Hindry & Silverman (2000) p.488
^ Batyrev, V.V.; Manin, Yu.I. (1990). "On the number of rational points of bounded height on algebraic varieties". Math. Ann. 286: 27–43. doi:10.1007/bf01453564. Zbl 0679.14008.
^ Lang (1997) pp.161–162
^ Neukirch (1999) p.185
^ It is mentioned in J. Tate, Algebraic cycles and poles of zeta functions in the volume (O. F. G. Schilling, editor), Arithmetical Algebraic Geometry, pages 93–110 (1965).
^ Lang (1997) pp.17–23
^ Hindry & Silverman (2000) p.480
^ Lang (1997) p.179
^ Bombieri & Gubler (2006) pp.176–230
^ Tsen, C. (1936). "Zur Stufentheorie der Quasi-algebraisch-Abgeschlossenheit kommutativer Körper". J. Chinese Math. Soc. 171: 81–92. Zbl 0015.38803.
^ Lorenz, Falko (2008). Algebra. Volume II: Fields with Structure, Algebras and Advanced Topics. Springer. pp. 109–126. ISBN 978-0-387-72487-4.
^ Caporaso, Lucia; Harris, Joe; Mazur, Barry (1997). "Uniformity of rational points". Journal of the American Mathematical Society. 10 (1): 1–35. doi:10.2307/2152901. Zbl 0872.14017.
^ Zannier, Umberto (2012). Some Problems of Unlikely Intersections in Arithmetic and Geometry. Annals of Mathematics Studies. 181. Princeton University Press. ISBN 978-0-691-15371-1.
^ Pierre Deligne, Poids dans la cohomologie des variétés algébriques, Actes ICM, Vancouver, 1974, 79–85.
^ Lang (1988) pp.1–9
^ Lang (1997) pp.164,212
^ Hindry & Silverman (2000) 184–185
Bombieri, Enrico; Gubler, Walter (2006). Heights in Diophantine Geometry. New Mathematical Monographs. 4. Cambridge University Press. doi:10.2277/0521846153. ISBN 978-0-521-71229-3. Zbl 1130.11034.
Hindry, Marc; Silverman, Joseph H. (2000). Diophantine Geometry: An Introduction. Graduate Texts in Mathematics. 201. ISBN 0-387-98981-1. Zbl 0948.11023.
Lang, Serge (1988). Introduction to Arakelov theory. New York: Springer-Verlag. ISBN 0-387-96793-1. MR 0969124. Zbl 0667.14001.
Lang, Serge (1997). Survey of Diophantine Geometry. Springer-Verlag. ISBN 3-540-61223-8. Zbl 0869.11051.
Neukirch, Jürgen (1999). Algebraic Number Theory. Grundlehren der Mathematischen Wissenschaften. 322. Springer-Verlag. ISBN 978-3-540-65399-8. Zbl 0956.11021.
Further reading
- Dino Lorenzini (1996), An invitation to arithmetic geometry, AMS Bookstore,
ISBN 978-0-8218-0267-0