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SUMMARY:Akinari Hoshi (Niigata University)
DTSTART:20241106T160000Z
DTEND:20241106T170000Z
DTSTAMP:20260418T065849Z
UID:LNTS/142
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/LNTS/142/">N
 orm one tori and Hasse norm principle</a>\nby Akinari Hoshi (Niigata Unive
 rsity) as part of London number theory seminar\n\nLecture held in Huxley 1
 40\, Imperial College.\n\nAbstract\nLet $k$ be a field and $T$ be an algeb
 raic $k$-torus. In 1969\, over a global field $k$\, Voskresenskii proved t
 hat there exists an exact sequence $0\\to A(T)\\to H^1(k\,{\\rm Pic}\\\,\\
 overline{X})^\\vee\\to Sha(T)\\to 0$ where $A(T)$ is the kernel of the wea
 k approximation of $T$\, $Sha(T)$ is the Shafarevich-Tate group of $T$\, $
 X$ is a smooth $k$-compactification of $T$\, ${\\rm Pic}\\\,\\overline{X}$
  is the Picard group of $\\overline{X}=X\\times_k\\overline{k}$ and $\\vee
 $ stands for the Pontryagin dual. On the other hand\, in 1963\, Ono proved
  that for the norm one torus $T=R^{(1)}_{K/k}(G_m)$ of $K/k$\, $Sha(T)=0$ 
 if and only if the Hasse norm principle holds for $K/k$. First\, we determ
 ine $H^1(k\,{\\rm Pic}\\\, \\overline{X})$ for algebraic $k$-tori $T$ up t
 o dimension $5$. Second\, we determine $H^1(k\,{\\rm Pic}\\\, \\overline{X
 })$ for norm one tori $T=R^{(1)}_{K/k}(G_m)$ with $[K:k]\\leq 17$. Third\,
  we give a necessary and sufficient condition for the Hasse norm principle
  for $K/k$ with $[K:k]\\leq 15$. We also show that $H^1(k\,{\\rm Pic}\\\, 
 \\overline{X})=0$ or $Z/2Z$ for $T=R^{(1)}_{K/k}(G_m)$ when the Galois gro
 up of the Galois closure of $K/k$ is the Mathieu group $M_{11}$ or the Jan
 ko group $J_1$. As applications of the results\, we get the group $T(k)/R$
  of $R$-equivalence classes over a local field $k$ via Colliot-Th\\'{e}l\\
 `{e}ne and Sansuc's formula and the Tamagawa number $\\tau(T)$ over a numb
 er field $k$ via Ono's formula $\\tau(T)=|H^1(k\,\\widehat{T})|/|Sha(T)|$.
  This is joint work with Kazuki Kanai and Aiichi Yamasaki.\n
LOCATION:https://researchseminars.org/talk/LNTS/142/
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