This page uses content from Wikipedia and is licensed under CC BY-SA.

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.

- 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 holomorphic if and only if all subvarieties are of general type.^{[8]}

- ^
^{a}^{b}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. - ^
^{a}^{b}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- ^
^{a}^{b}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- ^
^{a}^{b}^{c}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- ^
^{a}^{b}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.

- Dino Lorenzini (1996), An invitation to arithmetic geometry, AMS Bookstore, ISBN 978-0-8218-0267-0