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Let $k$ be a global field of characteristic unequal to two. Let $C$$:$ $y^2=f(x)$ be a nonsingular projective curve over $k$, where $f(x)$ is a quartic polynomial over $k$ with nonzero discriminant, and $K=k(C)$ be the function field of $C$. For each prime spot $\frp$ on $k$, let $\kp$ denote the corresponding completion of $k$ and $\kp(C)$ the function field of $C\times_k \kp$. Consider the map $$h : \, \br(K) \, \ra \, \Prod{\frp} \br\bigl(\kp(C)\bigr),$$ where $\frp$ ranges over all the prime spots of $k$. In this paper, we explicitly describe all the constant classes (coming from $\br(k)$) lying in the kernel of the map $h$, which is an obstruction to the Hasse principle for the Brauer groups of the curve. The kernel of $h$ can be expressed in terms of quaternion algebras with their prime spots. We also provide specific examples over $\qq$, the rationals, for this kernel.
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참고문헌 (9)
참고문헌 신청
D. Haile / 2007 / On an algebra determined by a quartic curve of genus one / J. Algebra 313(2):811~823
D. Haile / 2012 / Curves C that are cyclic twists of Y 3 = X3 + c and the relative Brauer groups Br(k(C)/k) / Trans. Amer. Math. Soc 364(9):4875~4908
I. Han / 2003 / Relative Brauer Groups of Function Fields of Curves of Genus One / Comm. Algebra 31(9):4301~4328
P. Gille / 2006 / Central Simple Algebras and Galois Cohomology / Cambridge University Press
R. Parimala / 1996 / Hasse principle for Witt groups of function fields with special reference to elliptic curves / Duke Math. J 85(3):555~582
D. Haile / 2012 / Curves C that are cyclic twists of Y 3 = X3 + c and the relative Brauer groups Br(k(C)/k) / Trans. Amer. Math. Soc 364(9):4875~4908
I. Han / 2003 / Relative Brauer Groups of Function Fields of Curves of Genus One / Comm. Algebra 31(9):4301~4328
P. Gille / 2006 / Central Simple Algebras and Galois Cohomology / Cambridge University Press
R. Parimala / 1996 / Hasse principle for Witt groups of function fields with special reference to elliptic curves / Duke Math. J 85(3):555~582
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