[過去ﾛｸﾞ] 現代数学の系譜 工学物理雑談 古典ガロア理論も読む78 (1002ﾚｽ)

現代数学の系譜 工学物理雑談 古典ガロア理論も読む78 http://rio2016.5ch.net/test/read.cgi/math/1571400076/

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468: 現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE [] 2019/11/04(月) 18:04:22.55 ID:Qu1TcOyQ >>467 つづき 1.3. A systematic approach. In §4, we offer a systematic approach, already started in [Leg13b], to give more examples of non H-parametric extensions over k of group G containing H. Given a k-regular Galois extension E1/k(T) of group H and a k-regular Galois extension E2/k(T) of group G, we provide two sufficient conditions which each guarantees that there exist some specializations of E1/k(T) of group H which cannot be specializations of E2/k(T) (and so E2/k(T) is not H-parametric over k). The first one (Branch Point Hypothesis) involves the branch point arithmetic while the second one (Inertia Hypothesis) is a more geometric condition on the inertia of the two extensions E1/k(T) and E2/k(T). Theorem 4.2 is our precise result. We work over base fields k which are quotient fields of any Dedekind domain of characteristic zero with infinitely many distinct primes2 , additionaly assumed to be hilbertian. Number fields or finite extensions of rational function fields k(X), with k an arbitrary field of characteristic zero (and X an indeterminate), are typical examples. つづく http://rio2016.5ch.net/test/read.cgi/math/1571400076/468

469: 現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE [] 2019/11/04(月) 18:04:38.17 ID:Qu1TcOyQ >>468 つづき 7.3. Some other cases H is a non abelian simple group. We now give some examples involving some k-regular Galois extensions of k(T) provided by the rigidity method. We use below standard Atlas [C+85] notation for conjugacy classes of finite groups. 7.3.1. Examples with PSL2(Fp). Let p ? 5 be a prime such that ( 2/p) = ?1 (resp. ( 3/p) = ?1) and E1/k(T) (resp. E2/k(T)) a k-regular Galois extension of group PSL2(Fp) and inertia canonical invariant (2A, pA, pB) (resp. (3A, pA, pB)) [Ser92, propositions 7.4.3-4 and theorem 8.2.2]. Corollary 7.4. Assume that k is hilbertian and (?1)(p?1)/2 p is a square in k. Then the extensions E1/k(T) (if (2/p) = ?1) and E2/k(T) (if (3/p) = ?1) satisfy the (geometric non PSL2(Fp)-parametricity) condition. (引用終り) http://rio2016.5ch.net/test/read.cgi/math/1571400076/469

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