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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 co-ordinates 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. __NOTOC__ ==A== ;abc conjecture :The abc conjecture of Masser and Oesterlé attempts to state as much as possible about repeated prime factors in an equation ''a'' + ''b'' = ''c''. For example 3 + 125 = 128 but the prime powers here are exceptional. ;Arakelov class group :The ''Arakelov class group'' is the analogue of the ideal class group or divisor class group for Arakelov divisors. ;Arakelov divisor :An ''Arakelov divisor'' (or ''replete divisor''〔) on a global field is an extension of the concept of divisor or fractional ideal. It is a formal linear combination of places of the field with finite places having integer coefficients and the infinite places having real coefficients.〔〔Lang (1988) pp.74–75〕 ;Arakelov height :The ''Arakelov height'' on a projective space over the field of algebraic numbers is a global height function with local contributions coming from Fubini–Study metrics on the Archimedean fields and the usual metric on the non-Archimedean fields.〔Bombieri & Gubler (2006) pp.66–67〕〔Lang (1988) pp.156–157〕 ;Arakelov theory :Arakelov theory is an approach to arithmetic geometry that explicitly includes the 'infinite primes'. ;Arithmetic of abelian varieties :''See main article arithmetic of abelian varieties'' ;Artin L-functions :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. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Glossary of arithmetic and Diophantine geometry」の詳細全文を読む スポンサード リンク
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