Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

A METHOD OF COMPUTING THE CONSTANT FIELD OBSTRUCTION TO THE HASSE PRINCIPLE FOR THE BRAUER GROUPS OF GENUS ONE CURVES

  • Han, Ilseop (Department of Mathematics California State University)
  • Received : 2015.09.20
  • Published : 2016.11.01
Download PDF
⟨ Previous Next ⟩

Abstract

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 p on k, let ${\hat{k}}_p$ denote the corresponding completion of k and ${\hat{k}}_p(C)$ the function field of $C{\times}_k{\hat{k}}_p$. Consider the map $$h:Br(K){\rightarrow}{\prod\limits_{\mathfrak{p}}}Br({\hat{k}}_p(C))$$, where p 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 ${\mathbb{Q}}$, the rationals, for this kernel.

Keywords

References

  1. D. Haile, and I. Han, On an algebra determined by a quartic curve of genus one, J. Algebra 313 (2007), no. 2, 811-823 https://doi.org/10.1016/j.jalgebra.2006.10.024
  2. D. Haile, I. Han, and A. R. Wadsworth, Curves C that are cyclic twists of $Y^3$ = $X^3$ + c and the relative Brauer groups Br(k(C)/k), Trans. Amer. Math. Soc. 364 (2012), no. 9, 4875-4908. https://doi.org/10.1090/S0002-9947-2012-05526-7
  3. I. Han, Relative Brauer Groups of Function Fields of Curves of Genus One, Comm. Algebra 31 (2003), no. 9, 4301-4328. https://doi.org/10.1081/AGB-120022794
  4. P. Gille and T. Szamuely, Central Simple Algebras and Galois Cohomology, Cambridge University Press, Cambridge, 2006.
  5. R. Parimala and R. Sujatha, Hasse principle for Witt groups of function fields with special reference to elliptic curves, Duke Math. J. 85 (1996), no. 3, 555-582. https://doi.org/10.1215/S0012-7094-96-08521-X
  6. P. Roquette, Splitting of algebras by function fields of one variable, Nagoya Math. J. 27 (1966), 625-642. https://doi.org/10.1017/S0027763000026441
  7. SAGE Mathematical Software, Version 2.6, http://www.sagemath.org.
  8. J. Silverman, The Arithmetic of Elliptic Curves, Springer, New York, 1986.
  9. J. Silverman and J. Tate, Rational Points on Elliptic Curves, Springer, New York, 1992.