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]
A
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.[9]
C
D
E
F
G
H
I
K
L
M
N
O
Q
R
S
T
U
V
W
See also
References
- ↑ Arithmetic geometry at the nLab
- ↑ Sutherland, Andrew V. (September 5, 2013). "Introduction to Arithmetic Geometry" (PDF). Retrieved 22 March 2019.
- 1 2 Schoof, René (2008). "Computing Arakelov class groups". In Buhler, J.P.; P., Stevenhagen (eds.). Algorithmic Number Theory: Lattices, Number Fields, Curves and Cryptography. MSRI Publications. Vol. 44. Cambridge University Press. pp. 447–495. ISBN 978-0-521-20833-8. MR 2467554. Zbl 1188.11076.
- 1 2 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. S2CID 12089289. Zbl 1030.11063.
- ↑ Bombieri & Gubler (2006) pp.66–67
- ↑ Lang (1988) pp.156–157
- ↑ Lang (1997) pp.91–96
- ↑ 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. S2CID 189832636. Zbl 0359.14009.
- ↑ Neukirch, Jürgen; Schmidt, Alexander; Wingberg, Kay (2008). Cohomology of Number Fields. Grundlehren der Mathematischen Wissenschaften. Vol. 323 (2nd ed.). Springer-Verlag. p. 361. ISBN 978-3-540-37888-4.
- ↑ Lang (1997) p.146
- 1 2 3 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. S2CID 121049418.
- ↑ 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)
- ↑ 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. S2CID 117772856. Zbl 0287.43007.
- 1 2 Hindry & Silverman (2000) p.479
- ↑ Bombieri & Gubler (2006) pp.82–93
- ↑ Raynaud, Michel (1983). "Sous-variétés d'une variété abélienne et points de torsion". In Artin, Michael; Tate, John (eds.). Arithmetic and geometry. Papers dedicated to I. R. Shafarevich on the occasion of his sixtieth birthday. Vol. I: Arithmetic. Progress in Mathematics (in French). Vol. 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 (eds.). Number fields and function fields — two parallel worlds. Progress in Mathematics. Vol. 239. Birkhäuser. pp. 311–318. ISBN 0-8176-4397-4. Zbl 1098.14030.
- ↑ McQuillan, Michael (1995). "Division points on semi-abelian varieties". Invent. Math. 120 (1): 143–159. Bibcode:1995InMat.120..143M. doi:10.1007/BF01241125. S2CID 120053132.
- ↑ 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. Vol. 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
- 1 2 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. S2CID 119945673. 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:10.1090/S0894-0347-97-00195-1. JSTOR 2152901. Zbl 0872.14017.
- ↑ Zannier, Umberto (2012). Some Problems of Unlikely Intersections in Arithmetic and Geometry. Annals of Mathematics Studies. Vol. 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. Vol. 4. Cambridge University Press. ISBN 978-0-521-71229-3. Zbl 1130.11034.
- Hindry, Marc; Silverman, Joseph H. (2000). Diophantine Geometry: An Introduction. Graduate Texts in Mathematics. Vol. 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. Vol. 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