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2 results

  • ON ARTIN'S CONJECTURE, II: PAIRS OF ADDITIVE FORMS

  • J. BRÜDERN, H. GODINHO
  • Journal:
    Proceedings of the London Mathematical Society / Volume 84 / Issue 3 / May 2002
    Published online by Cambridge University Press:
    29 April 2002, pp. 513-538
    Print publication:
    May 2002

  • It is shown that the system of two additive equations a_1 x_1^k + \ldots + a_s x_s^k = b_1 x_1^k + \ldots + b_s x_s^k =0 where $k \ge 2$ and $a_j$, $b_j$ are any given integers, has non-trivial solutions in all $p$-adic fields provided only that $s > 8k^2$. The constant 8 can be reduced when $k$ is not a power of 2. It is expected, in accordance with a classical conjecture of Artin, that the bound $8k^2$ can be replaced by $2k^2$.

    2000 Mathematical Subject Classification:11D72.

  • Isocrystals with additional structure. II

  • ROBERT E. KOTTWITZ
  • Journal:
    Compositio Mathematica / Volume 109 / Issue 3 / December 1997
    Published online by Cambridge University Press:
    04 December 2007, pp. 255-339
    Print publication:
    December 1997
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  • Let $F$ be a $p$-adic field, let $L$ be the completion of a maximal unramified extension of $F$, and let $\sigma$ be the Frobenius automorphism of $L$ over $F$. For any connected reductive group $G$ over $F$ one denotes by $B(G)$ the set of $\sigma$-conjugacy classes in $G(L)$ (elements $x,y$ in $G(L)$ are said to be $\sigma$-conjugate if there exists $g$ in $G(L)$ such that $g^-1 \kappa \sigma(g)=y$. One of the main results of this paper is a concrete description of the set $B(G)$ (previously this was known only in the quasi-split case).