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IUTを読むための用語集資料集スレ http://rio2016.5ch.net/test/read.cgi/math/1592654877/

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366: 現代数学の系譜 雑談 ◆yH25M02vWFhP [] 2020/08/06(木) 16:32:53.72 ID:Jwpd0UuY Conductor 導手 （参考） https://en.wikipedia.org/wiki/Conductor_of_an_abelian_variety Conductor of an abelian variety (抜粋) In mathematics, in Diophantine geometry, the conductor of an abelian variety defined over a local or global field F is a measure of how "bad" the bad reduction at some prime is. It is connected to the ramification in the field generated by the torsion points. Definition For an abelian variety A defined over a field F as above, with ring of integers R, consider the Neron model of A, which is a 'best possible' model of A defined over R. This model may be represented as a scheme over Spec(R) (cf. spectrum of a ring) for which the generic fibre constructed by means of the morphism Spec(F) → Spec(R) gives back A. http://rio2016.5ch.net/test/read.cgi/math/1592654877/366

369: 現代数学の系譜 雑談 ◆yH25M02vWFhP [] 2020/08/06(木) 17:13:34.17 ID:Jwpd0UuY >>366 ”bad reduction” https://en.wikipedia.org/wiki/Glossary_of_arithmetic_and_diophantine_geometry Glossary of arithmetic and diophantine geometry (抜粋) B Bad reduction See good reduction. G Good reduction Fundamental to local analysis in arithmetic problems is to reduce modulo all prime numbers p or, more generally, prime ideals. In the typical situation this presents little difficulty for almost all p; for example denominators of fractions are tricky, in that reduction modulo a prime in the denominator looks like division by zero, but that rules out only finitely many p per fraction. With a little extra sophistication, homogeneous coordinates allow clearing of denominators by multiplying by a common scalar. For a given, single point one can do this and not leave a common factor p. However singularity theory enters: a non-singular point may become a singular point on reduction modulo p, because the Zariski tangent space can become larger when linear terms reduce to 0 (the geometric formulation shows it is not the fault of a single set of coordinates). Good reduction refers to the reduced variety having the same properties as the original, for example, an algebraic curve having the same genus, or a smooth variety remaining smooth. In general there will be a finite set S of primes for a given variety V, assumed smooth, such that there is otherwise a smooth reduced Vp over Z/pZ. For abelian varieties, good reduction is connected with ramification in the field of division points by the Neron?Ogg?Shafarevich criterion. The theory is subtle, in the sense that the freedom to change variables to try to improve matters is rather unobvious: see Neron model, potential good reduction, Tate curve, semistable abelian variety, semistable elliptic curve, Serre?Tate theorem.[16] http://rio2016.5ch.net/test/read.cgi/math/1592654877/369

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