5 F eb 2 00 6 Circular sets of prime numbers and p - extensions of the rationals
نویسنده
چکیده
Let p be an odd prime number and let S be a finite set of prime numbers congruent to 1 modulo p. We prove that the group GS(Q)(p) has cohomological dimension 2 if the linking diagram attached to S and p satisfies a certain technical condition, and we show that GS(Q)(p) is a duality group in these cases. Furthermore, we investigate the decomposition behaviour of primes in the extension QS(p)/Q and we relate the cohomology of GS(Q)(p) to the étale cohomology of the scheme Spec(Z) − S. Finally, we calculate the dualizing module.
منابع مشابه
Circular sets of prime numbers and p-extensions of the rationals
Let p be an odd prime number and let S be a finite set of prime numbers congruent to 1 modulo p. We prove that the group GS(Q)(p) has cohomological dimension 2 if the linking diagram attached to S and p satisfies a certain technical condition, and we show that GS(Q)(p) is a duality group in these cases. Furthermore, we investigate the decomposition behaviour of primes in the extension QS(p)/Q a...
متن کاملN ov 2 00 4 THE NUMBER OF S 4 FIELDS WITH GIVEN DISCRIMINANT
We prove that the number of S 4-extensions of the rationals of given discriminant d is O(d 1/2+ǫ) for all ǫ > 0. For a prime number C we derive that the number of octahedral modular forms of weight 1 and conductor C is bounded above by O(C 1/2 log(C) 2).
متن کاملσ-sporadic prime ideals and superficial elements
Let $A$ be a Noetherian ring, $I$ be an ideal of $A$ and $sigma$ be a semi-prime operation, different from the identity map on the set of all ideals of $A$. Results of Essan proved that the sets of associated prime ideals of $sigma(I^n)$, which denoted by $Ass(A/sigma(I^n))$, stabilize to $A_{sigma}(I)$. We give some properties of the sets $S^{sigma}_{n}(I)=Ass(A/sigma(I^n))setminus A_{sigma}(I...
متن کاملan 2 00 3 A brief introduction to p - adic numbers Stephen
In this short survey we look at a few basic features of p-adic numbers, somewhat with the point of view of a classical analyst. In particular, with p-adic numbers one has arithmetic operations and a norm, just as for real or complex numbers. Let Z denote the integers, Q denote the rational numbers, R denote the real numbers, and C denote the complex numbers. Also let | · | denote the usual abso...
متن کاملan 2 00 3 A brief introduction to p - adic numbers Stephen Semmes
In this short survey we look at a few basic features of p-adic numbers, somewhat with the point of view of a classical analyst. In particular, with p-adic numbers one has arithmetic operations and a norm, just as for real or complex numbers. Let Z denote the integers, Q denote the rational numbers, R denote the real numbers, and C denote the complex numbers. Also let | · | denote the usual abso...
متن کامل