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The Hasse invariant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046660/h0466601.png" /> of a central simple algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046660/h0466602.png" /> over a local field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046660/h0466603.png" /> (or over the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046660/h0466604.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046660/h0466605.png" />) is the image of the class of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046660/h0466606.png" /> under the canonical isomorphism of the [[Brauer group|Brauer group]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046660/h0466607.png" /> onto the group of all complex roots of unity (or onto the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046660/h0466608.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046660/h0466609.png" />). For a cyclic algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046660/h04666010.png" /> with generators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046660/h04666011.png" /> and defining relations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046660/h04666012.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046660/h04666013.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046660/h04666014.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046660/h04666015.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046660/h04666016.png" /> is a primitive <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046660/h04666017.png" />-th root of unity, the Hasse invariant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046660/h04666018.png" /> is the same as the [[Norm-residue symbol|norm-residue symbol]] (Hilbert symbol) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046660/h04666019.png" />. In particular, the Hasse invariant of the quaternion algebra is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046660/h04666020.png" />.
+
The Hasse invariant $h(A)$ of a central simple algebra $A$ over a local
 +
field $K$ (or over the field $K=\R$ or $\C$) is the image of the class of
 +
$A$ under the canonical isomorphism of the
 +
[[Brauer group|Brauer group]] of $K$ onto the group of all complex
 +
roots of unity (or onto the group $\{\pm1\}$ or $\{1\}$). For a cyclic algebra
 +
$A$ with generators $a,b$ and defining relations $a^n=x$, $b^n=y$, $ba=\epsilon ab$, where
 +
$x,y\in K^*$ and $\epsilon\in K$ is a primitive $n$-th root of unity, the Hasse invariant
 +
$h(A)$ is the same as the
 +
[[Norm-residue symbol|norm-residue symbol]] (Hilbert symbol) $(x,y)_n$. In
 +
particular, the Hasse invariant of the quaternion algebra is $-1$.
  
For a [[Central algebra|central algebra]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046660/h04666021.png" /> over a global field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046660/h04666022.png" /> and any [[Valuation|valuation]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046660/h04666023.png" /> of this field the local Hasse invariant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046660/h04666024.png" /> is defined as the Hasse invariant of the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046660/h04666025.png" /> over the completion <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046660/h04666026.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046660/h04666027.png" /> in the topology determined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046660/h04666028.png" />. The local Hasse invariants determine the class of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046660/h04666029.png" /> uniquely. They are related by the following conditions: 1) there are only finitely-many valuations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046660/h04666030.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046660/h04666031.png" />; and 2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046660/h04666032.png" /> (the reciprocity law). Apart from these conditions they can assume arbitrary values.
+
For a
 +
[[Central algebra|central algebra]] $A$ over a global field $K$ and
 +
any
 +
[[Valuation|valuation]] $\nu$ of this field the local Hasse invariant
 +
$h_\nu(A)$ is defined as the Hasse invariant of the algebra $A\otimes K_\nu$ over the
 +
completion $K_\nu$ of $K$ in the topology determined by $\nu$. The local
 +
Hasse invariants determine the class of $A$ uniquely. They are related
 +
by the following conditions: 1) there are only finitely-many
 +
valuations $\nu$ for which $h_\nu(A)\ne 1$; and 2) $\prod_\nu h_\nu(A) = 1$ (the reciprocity law). Apart
 +
from these conditions they can assume arbitrary values.
  
The Hasse invariant was introduced by H. Hasse [[#References|[1]]] and [[#References|[2]]].
+
The Hasse invariant was introduced by H. Hasse
 +
[[#References|[1]]] and
 +
[[#References|[2]]].
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> H. Hasse,   "Ueber <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046660/h04666033.png" />-adische Schiefkörper und ihre Bedeutung für die Arithmetik hyperkomplexen Zahlsysteme" ''Math. Ann.'' , '''104''' (1931) pp. 495–534</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> H. Hasse,   "Die Struktur der R. Brauerschen Algebrenklassengruppe über einem algebraischen Zahlkörper. Inbesondere Begründung der Theorie des Normenrestsymbols und Herleitung des Reziprozitätsgesetzes mit nichtkommutativen Hilfsmitteln" ''Math. Ann.'' , '''107''' (1933) pp. 731–760</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> J.W.S. Cassels (ed.)  A. Fröhlich (ed.) , ''Algebraic number theory'' , Acad. Press (1986)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> A. Weil,   "Basic number theory" , Springer (1967)</TD></TR></table>
+
<table><TR><TD valign="top">[1]</TD> <TD
 +
valign="top"> H. Hasse, "Ueber $p$-adische Schiefkörper und ihre
 +
Bedeutung für die Arithmetik hyperkomplexen Zahlsysteme"
 +
''Math. Ann.'' , '''104''' (1931) pp. 495–534</TD></TR><TR><TD
 +
valign="top">[2]</TD> <TD valign="top"> H. Hasse, "Die Struktur der
 +
R. Brauerschen Algebrenklassengruppe über einem algebraischen
 +
Zahlkörper. Inbesondere Begründung der Theorie des Normenrestsymbols
 +
und Herleitung des Reziprozitätsgesetzes mit nichtkommutativen
 +
Hilfsmitteln" ''Math. Ann.'' , '''107''' (1933)
 +
pp. 731–760</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">
 +
J.W.S. Cassels (ed.)  A. Fröhlich (ed.) , ''Algebraic number theory''
 +
, Acad. Press (1986)</TD></TR><TR><TD valign="top">[4]</TD> <TD
 +
valign="top"> A. Weil, "Basic number theory" , Springer
 +
(1967)</TD></TR></table>
  
The Hasse invariant, the Hasse–Minkowski invariant, Hasse's symbol, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046660/h04666034.png" />, of a non-degenerate [[Quadratic form|quadratic form]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046660/h04666035.png" /> over a local field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046660/h04666036.png" /> of characteristic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046660/h04666037.png" /> (or over the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046660/h04666038.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046660/h04666039.png" />) is the product
+
The Hasse invariant, the Hasse–Minkowski invariant, Hasse's symbol,
 +
$\epsilon(f)$, of a non-degenerate
 +
[[Quadratic form|quadratic form]] $f\sim a_1 x_1^2 + \cdots + a_n
 +
x_n^2$
 +
over a local field $K$ of
 +
characteristic $\ne 2$ (or over the field $K=\R$ or $\C$) is the product $$\prod_{i<j} (a_i,a_j) = \pm 1$$
 +
where $(\ ,\ )$ is the quadratic Hilbert symbol, that is, $(a,b) = 1$ if the
 +
quadratic form $ax^2+by^2$ represents 1 in the field $K$ and $(a,b) = -1$
 +
otherwise. The Hasse invariant depends only on the equivalence class
 +
of the form $f$, and not on the choice of a diagonal form in this
 +
class. Sometimes the Hasse invariant is defined as the product $\prod_{i\le j}(a_i,a_j)$,
 +
which differs from the definition above by the factor $(d(f),d(f))$, where $d(f)$
 +
is the
 +
[[Discriminant|discriminant]] of the form $f$.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046660/h04666040.png" /></td> </tr></table>
+
In the case of a local field $K$ the number $n$ of variables, the
 +
discriminant and the Hasse invariant determine the class of the form
 +
$f$. For $n\ge 3$, the invariants $d(f)$ and $\epsilon(f)$ can take arbitrary values
 +
independently of each other; for $n=2$ the case $d(f)=-1$, $\epsilon(f) = -1$ is excluded;
 +
for $n=1$ one always has $\epsilon(f) = 1$.
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046660/h04666041.png" /> is the quadratic Hilbert symbol, that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046660/h04666042.png" /> if the quadratic form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046660/h04666043.png" /> represents 1 in the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046660/h04666044.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046660/h04666045.png" /> otherwise. The Hasse invariant depends only on the equivalence class of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046660/h04666046.png" />, and not on the choice of a diagonal form in this class. Sometimes the Hasse invariant is defined as the product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046660/h04666047.png" />, which differs from the definition above by the factor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046660/h04666048.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046660/h04666049.png" /> is the [[Discriminant|discriminant]] of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046660/h04666050.png" />.
+
When $K=\R$, the Hasse invariant can be expressed in terms of the
 +
[[Signature|signature]], namely, $$\def\e{\epsilon} \e(f) = (-1)^{s(s-1)/2}$$ where $s$ is the negative index
 +
of inertia of the form $f$. When $K=\C$, one has $\e(f)=1$.
  
In the case of a local field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046660/h04666051.png" /> the number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046660/h04666052.png" /> of variables, the discriminant and the Hasse invariant determine the class of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046660/h04666053.png" />. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046660/h04666054.png" />, the invariants <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046660/h04666055.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046660/h04666056.png" /> can take arbitrary values independently of each other; for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046660/h04666057.png" /> the case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046660/h04666058.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046660/h04666059.png" /> is excluded; for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046660/h04666060.png" /> one always has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046660/h04666061.png" />.
+
For a non-degenerate quadratic form $f$ over a global field $K$ of
 +
characteristic $\ne 2$ and any valuation $\nu$ of $K$ the local Hasse
 +
invariant $\e_\nu(f)$ is defined as the Hasse invariant of the quadratic form
 +
$f$ regarded over the completion $K_\nu$ of $K$ in the topology determined
 +
by $\nu$. The number of variables, the discriminant, the local Hasse
 +
invariants, and the signatures over the real completions of $K$
 +
determine the class of $f$.
  
When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046660/h04666062.png" />, the Hasse invariant can be expressed in terms of the [[Signature|signature]], namely,
+
Necessary and sufficient conditions for the existence of a
 +
non-degenerate quadratic form in $n$ variables over a global field $K$
 +
of characteristic $\ne 2$ having a given discriminant $d\ne 0$, given the local
 +
Hasse invariants $\e_\nu$, and, for real valuations $\nu$, given the negative
 +
indices of inertia $s_\nu$, are as follows:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046660/h04666063.png" /></td> </tr></table>
+
a) $\e_\nu \ne 1$ for only finitely-many valuations $\nu$;
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046660/h04666064.png" /> is the negative index of inertia of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046660/h04666065.png" />. When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046660/h04666066.png" />, one has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046660/h04666067.png" />.
+
b) $\prod_\nu \e_\nu = 1$ (a consequence of the
 +
[[Quadratic reciprocity law|quadratic reciprocity law]]);
  
For a non-degenerate quadratic form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046660/h04666068.png" /> over a global field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046660/h04666069.png" /> of characteristic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046660/h04666070.png" /> and any valuation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046660/h04666071.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046660/h04666072.png" /> the local Hasse invariant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046660/h04666073.png" /> is defined as the Hasse invariant of the quadratic form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046660/h04666074.png" /> regarded over the completion <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046660/h04666075.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046660/h04666076.png" /> in the topology determined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046660/h04666077.png" />. The number of variables, the discriminant, the local Hasse invariants, and the signatures over the real completions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046660/h04666078.png" /> determine the class of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046660/h04666079.png" />.
+
c) $\e_\nu = 1$ if $n=1$ or if $n=2$ and $d\in (-1)(K_\nu^*)^2$;
  
Necessary and sufficient conditions for the existence of a non-degenerate quadratic form in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046660/h04666080.png" /> variables over a global field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046660/h04666081.png" /> of characteristic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046660/h04666082.png" /> having a given discriminant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046660/h04666083.png" />, given the local Hasse invariants <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046660/h04666084.png" />, and, for real valuations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046660/h04666085.png" />, given the negative indices of inertia <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046660/h04666086.png" />, are as follows:
+
d) $\e_\nu = (-1)^{s_\nu(s_\nu-1)/2}$ for every real valuation $\nu$;
  
a) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046660/h04666087.png" /> for only finitely-many valuations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046660/h04666088.png" />;
+
e) $\e_\nu=1$ for every complex valuation $\nu$;
  
b) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046660/h04666089.png" /> (a consequence of the [[Quadratic reciprocity law|quadratic reciprocity law]]);
+
f) ${\rm sign}\; d_\nu = (-1)^{s_\nu}$ for every real valuation $\nu$ (here $d_\nu$ is the image of $d$
 
+
under the isomorphism $K_\nu\to \R$).
c) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046660/h04666090.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046660/h04666091.png" /> or if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046660/h04666092.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046660/h04666093.png" />;
 
 
 
d) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046660/h04666094.png" /> for every real valuation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046660/h04666095.png" />;
 
 
 
e) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046660/h04666096.png" /> for every complex valuation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046660/h04666097.png" />;
 
 
 
f) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046660/h04666098.png" /> for every real valuation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046660/h04666099.png" /> (here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046660/h046660100.png" /> is the image of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046660/h046660101.png" /> under the isomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046660/h046660102.png" />).
 
  
 
The Hasse invariant was introduced by H. Hasse .
 
The Hasse invariant was introduced by H. Hasse .
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1a]</TD> <TD valign="top"> H. Hasse,   "Ueber die Darstellbarkeit von Zahlen durch quadratische Formen im Körper der rationalen Zahlen" ''J. Reine Angew. Math.'' , '''152''' (1923) pp. 129–148</TD></TR><TR><TD valign="top">[1b]</TD> <TD valign="top"> H. Hasse,   "Ueber die Aequivalenz quadratischer Formen im Körper der rationalen Zahlen" ''J. Reine Angew. Math.'' , '''152''' (1923) pp. 205–224</TD></TR><TR><TD valign="top">[1c]</TD> <TD valign="top"> H. Hasse,   "Symmetrische Matrizen im Körper der rationalen Zahlen" ''J. Reine Angew. Math.'' , '''153''' (1924) pp. 12–43</TD></TR><TR><TD valign="top">[1d]</TD> <TD valign="top"> H. Hasse,   "Darstellbarkeit von Zahlen durch quadratische Formen in einem beliebigen algebraischen Zahlkörper" ''J. Reine Angew. Math.'' , '''153''' (1924) pp. 113–130</TD></TR><TR><TD valign="top">[1e]</TD> <TD valign="top"> H. Hasse,   "Aequivalenz quadratischer Formen in einem beliebigen algebraischen Zahlkörper" ''J. Reine Angew. Math.'' , '''153''' (1924) pp. 158–162</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> O.T. O'Meara,   "Introduction to quadratic forms" , Springer (1963)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> T.Y. Lam,   "The algebraic theory of quadratic forms" , Benjamin (1973)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> J.W.S. Cassels,   "Rational quadratic forms" , Acad. Press (1978)</TD></TR></table>
+
<table><TR><TD valign="top">[1a]</TD> <TD
 +
valign="top"> H. Hasse, "Ueber die Darstellbarkeit von Zahlen durch
 +
quadratische Formen im Körper der rationalen Zahlen" ''J. Reine
 +
Angew. Math.'' , '''152''' (1923) pp. 129–148</TD></TR><TR><TD
 +
valign="top">[1b]</TD> <TD valign="top"> H. Hasse, "Ueber die
 +
Aequivalenz quadratischer Formen im Körper der rationalen Zahlen"
 +
''J. Reine Angew. Math.'' , '''152''' (1923)
 +
pp. 205–224</TD></TR><TR><TD valign="top">[1c]</TD> <TD valign="top">
 +
H. Hasse, "Symmetrische Matrizen im Körper der rationalen Zahlen"
 +
''J. Reine Angew. Math.'' , '''153''' (1924)
 +
pp. 12–43</TD></TR><TR><TD valign="top">[1d]</TD> <TD valign="top">
 +
H. Hasse, "Darstellbarkeit von Zahlen durch quadratische Formen in
 +
einem beliebigen algebraischen Zahlkörper" ''J. Reine Angew. Math.'' ,
 +
'''153''' (1924) pp. 113–130</TD></TR><TR><TD valign="top">[1e]</TD>
 +
<TD valign="top"> H. Hasse, "Aequivalenz quadratischer Formen in einem
 +
beliebigen algebraischen Zahlkörper" ''J. Reine Angew. Math.'' ,
 +
'''153''' (1924) pp. 158–162</TD></TR><TR><TD valign="top">[2]</TD>
 +
<TD valign="top"> O.T. O'Meara, "Introduction to quadratic forms" ,
 +
Springer (1963)</TD></TR><TR><TD valign="top">[3]</TD> <TD
 +
valign="top"> T.Y. Lam, "The algebraic theory of quadratic forms" ,
 +
Benjamin (1973)</TD></TR><TR><TD valign="top">[4]</TD> <TD
 +
valign="top"> J.W.S. Cassels, "Rational quadratic forms" , Acad. Press
 +
(1978)</TD></TR></table>
  
The Hasse invariant of an elliptic curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046660/h046660103.png" /> over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046660/h046660104.png" /> of characteristic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046660/h046660105.png" /> is the number 0 or 1 depending on whether the endomorphism of the cohomology group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046660/h046660106.png" /> induced by the [[Frobenius endomorphism|Frobenius endomorphism]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046660/h046660107.png" /> is null or bijective. Curves for which the Hasse invariant is zero are called supersingular.
+
The Hasse invariant of an elliptic curve $X$ over a field $K$ of
 +
characteristic $p>0$ is the number 0 or 1 depending on whether the
 +
endomorphism of the cohomology group $H^1(X,\mathcal{O}_X)$ induced by the
 +
[[Frobenius endomorphism|Frobenius endomorphism]] of $X$ is null or
 +
bijective. Curves for which the Hasse invariant is zero are called
 +
supersingular.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> R. Hartshorne,   "Algebraic geometry" , Springer (1977)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> Yu.I. Manin,   "On the Hasse–Witt matrix of an algebraic curve" ''Izv. Akad. Nauk. SSSR Ser. Mat.'' , '''25''' : 1 (1961) pp. 153–172 (In Russian)</TD></TR></table>
+
<table><TR><TD valign="top">[1]</TD> <TD
 +
valign="top"> R. Hartshorne, "Algebraic geometry" , Springer
 +
(1977)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">
 +
Yu.I. Manin, "On the Hasse–Witt matrix of an algebraic curve"
 +
''Izv. Akad. Nauk. SSSR Ser. Mat.'' , '''25''' : 1 (1961) pp. 153–172
 +
(In Russian)</TD></TR></table>
  
  
Line 54: Line 142:
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> J.H. Silverman,   "The arithmetic of elliptic curves" , Springer (1986)</TD></TR></table>
+
<table><TR><TD valign="top">[a1]</TD> <TD
 +
valign="top"> J.H. Silverman, "The arithmetic of elliptic curves" ,
 +
Springer (1986)</TD></TR></table>

Revision as of 20:42, 14 September 2011

The Hasse invariant $h(A)$ of a central simple algebra $A$ over a local field $K$ (or over the field $K=\R$ or $\C$) is the image of the class of $A$ under the canonical isomorphism of the Brauer group of $K$ onto the group of all complex roots of unity (or onto the group $\{\pm1\}$ or $\{1\}$). For a cyclic algebra $A$ with generators $a,b$ and defining relations $a^n=x$, $b^n=y$, $ba=\epsilon ab$, where $x,y\in K^*$ and $\epsilon\in K$ is a primitive $n$-th root of unity, the Hasse invariant $h(A)$ is the same as the norm-residue symbol (Hilbert symbol) $(x,y)_n$. In particular, the Hasse invariant of the quaternion algebra is $-1$.

For a central algebra $A$ over a global field $K$ and any valuation $\nu$ of this field the local Hasse invariant $h_\nu(A)$ is defined as the Hasse invariant of the algebra $A\otimes K_\nu$ over the completion $K_\nu$ of $K$ in the topology determined by $\nu$. The local Hasse invariants determine the class of $A$ uniquely. They are related by the following conditions: 1) there are only finitely-many valuations $\nu$ for which $h_\nu(A)\ne 1$; and 2) $\prod_\nu h_\nu(A) = 1$ (the reciprocity law). Apart from these conditions they can assume arbitrary values.

The Hasse invariant was introduced by H. Hasse [1] and [2].

References

[1] H. Hasse, "Ueber $p$-adische Schiefkörper und ihre

Bedeutung für die Arithmetik hyperkomplexen Zahlsysteme"

Math. Ann. , 104 (1931) pp. 495–534
[2] H. Hasse, "Die Struktur der

R. Brauerschen Algebrenklassengruppe über einem algebraischen Zahlkörper. Inbesondere Begründung der Theorie des Normenrestsymbols und Herleitung des Reziprozitätsgesetzes mit nichtkommutativen Hilfsmitteln" Math. Ann. , 107 (1933)

pp. 731–760
[3]

J.W.S. Cassels (ed.) A. Fröhlich (ed.) , Algebraic number theory

, Acad. Press (1986)
[4] A. Weil, "Basic number theory" , Springer (1967)

The Hasse invariant, the Hasse–Minkowski invariant, Hasse's symbol, $\epsilon(f)$, of a non-degenerate quadratic form $f\sim a_1 x_1^2 + \cdots + a_n x_n^2$ over a local field $K$ of characteristic $\ne 2$ (or over the field $K=\R$ or $\C$) is the product $$\prod_{i<j} (a_i,a_j) = \pm 1$$ where $(\ ,\ )$ is the quadratic Hilbert symbol, that is, $(a,b) = 1$ if the quadratic form $ax^2+by^2$ represents 1 in the field $K$ and $(a,b) = -1$ otherwise. The Hasse invariant depends only on the equivalence class of the form $f$, and not on the choice of a diagonal form in this class. Sometimes the Hasse invariant is defined as the product $\prod_{i\le j}(a_i,a_j)$, which differs from the definition above by the factor $(d(f),d(f))$, where $d(f)$ is the discriminant of the form $f$.

In the case of a local field $K$ the number $n$ of variables, the discriminant and the Hasse invariant determine the class of the form $f$. For $n\ge 3$, the invariants $d(f)$ and $\epsilon(f)$ can take arbitrary values independently of each other; for $n=2$ the case $d(f)=-1$, $\epsilon(f) = -1$ is excluded; for $n=1$ one always has $\epsilon(f) = 1$.

When $K=\R$, the Hasse invariant can be expressed in terms of the signature, namely, $$\def\e{\epsilon} \e(f) = (-1)^{s(s-1)/2}$$ where $s$ is the negative index of inertia of the form $f$. When $K=\C$, one has $\e(f)=1$.

For a non-degenerate quadratic form $f$ over a global field $K$ of characteristic $\ne 2$ and any valuation $\nu$ of $K$ the local Hasse invariant $\e_\nu(f)$ is defined as the Hasse invariant of the quadratic form $f$ regarded over the completion $K_\nu$ of $K$ in the topology determined by $\nu$. The number of variables, the discriminant, the local Hasse invariants, and the signatures over the real completions of $K$ determine the class of $f$.

Necessary and sufficient conditions for the existence of a non-degenerate quadratic form in $n$ variables over a global field $K$ of characteristic $\ne 2$ having a given discriminant $d\ne 0$, given the local Hasse invariants $\e_\nu$, and, for real valuations $\nu$, given the negative indices of inertia $s_\nu$, are as follows:

a) $\e_\nu \ne 1$ for only finitely-many valuations $\nu$;

b) $\prod_\nu \e_\nu = 1$ (a consequence of the quadratic reciprocity law);

c) $\e_\nu = 1$ if $n=1$ or if $n=2$ and $d\in (-1)(K_\nu^*)^2$;

d) $\e_\nu = (-1)^{s_\nu(s_\nu-1)/2}$ for every real valuation $\nu$;

e) $\e_\nu=1$ for every complex valuation $\nu$;

f) ${\rm sign}\; d_\nu = (-1)^{s_\nu}$ for every real valuation $\nu$ (here $d_\nu$ is the image of $d$ under the isomorphism $K_\nu\to \R$).

The Hasse invariant was introduced by H. Hasse .

References

[1a] H. Hasse, "Ueber die Darstellbarkeit von Zahlen durch

quadratische Formen im Körper der rationalen Zahlen" J. Reine

Angew. Math. , 152 (1923) pp. 129–148
[1b] H. Hasse, "Ueber die

Aequivalenz quadratischer Formen im Körper der rationalen Zahlen" J. Reine Angew. Math. , 152 (1923)

pp. 205–224
[1c]

H. Hasse, "Symmetrische Matrizen im Körper der rationalen Zahlen" J. Reine Angew. Math. , 153 (1924)

pp. 12–43
[1d]

H. Hasse, "Darstellbarkeit von Zahlen durch quadratische Formen in einem beliebigen algebraischen Zahlkörper" J. Reine Angew. Math. ,

153 (1924) pp. 113–130
[1e] H. Hasse, "Aequivalenz quadratischer Formen in einem

beliebigen algebraischen Zahlkörper" J. Reine Angew. Math. ,

153 (1924) pp. 158–162
[2] O.T. O'Meara, "Introduction to quadratic forms" , Springer (1963)
[3] T.Y. Lam, "The algebraic theory of quadratic forms" , Benjamin (1973)
[4] J.W.S. Cassels, "Rational quadratic forms" , Acad. Press (1978)

The Hasse invariant of an elliptic curve $X$ over a field $K$ of characteristic $p>0$ is the number 0 or 1 depending on whether the endomorphism of the cohomology group $H^1(X,\mathcal{O}_X)$ induced by the Frobenius endomorphism of $X$ is null or bijective. Curves for which the Hasse invariant is zero are called supersingular.

References

[1] R. Hartshorne, "Algebraic geometry" , Springer (1977)
[2]

Yu.I. Manin, "On the Hasse–Witt matrix of an algebraic curve" Izv. Akad. Nauk. SSSR Ser. Mat. , 25 : 1 (1961) pp. 153–172

(In Russian)


Comments

References

[a1] J.H. Silverman, "The arithmetic of elliptic curves" , Springer (1986)
How to Cite This Entry:
Hasse invariant. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Hasse_invariant&oldid=19597
This article was adapted from an original article by E.B. Vinberg (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article