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The ''Hasse invariant'' is an arithmetic invariant of various objects.
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==== Central simple algebras ====
  
 
The Hasse invariant $h(A)$ of a central simple algebra $A$ over a local
 
The Hasse invariant $h(A)$ of a central simple algebra $A$ over a local
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The Hasse invariant was introduced by H. Hasse
 
The Hasse invariant was introduced by H. Hasse
[[#References|[1]]] and
+
{{Cite|Ha}}, {{Cite|Ha2}}.
[[#References|[2]]].
 
  
====References====
+
==== Quadratic forms ====
<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,
 
The Hasse invariant, the Hasse–Minkowski invariant, Hasse's symbol,
Line 97: Line 87:
 
under the isomorphism $K_\nu\to \R$).
 
under the isomorphism $K_\nu\to \R$).
  
The Hasse invariant was introduced by H. Hasse .
+
Cf. {{Cite|Ha3}}, {{Cite|Ha4}}, {{Cite|Ha5}}, {{Cite|Ha6}}, {{Cite|Ha7}},
 +
{{Cite|OM}}, {{Cite|La}}, {{Cite|We}}
  
====References====
+
 
<table><TR><TD valign="top">[1a]</TD> <TD
+
==== Elliptic curves ====
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 $X$ over a field $K$ of
 
The Hasse invariant of an elliptic curve $X$ over a field $K$ of
Line 131: Line 100:
 
supersingular.
 
supersingular.
  
====References====
+
Cf. {{Cite|Ma}}
<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>
 
 
 
 
 
 
 
====Comments====
 
 
 
  
 
====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>
+
|valign="top"|{{Ref|Ca}}||valign="top"| J.W.S. Cassels, "Rational quadratic forms", Acad. Press (1978)  {{MR|0522835}}  {{ZBL|0395.10029}}     
 +
|-
 +
|valign="top"|{{Ref|CaFr}}||valign="top"| J.W.S. Cassels (ed.)  A. Fröhlich (ed.), ''Algebraic number theory'', Acad. Press (1967)  {{MR|0215665}}  {{ZBL|0153.07403}}     
 +
|-
 +
|valign="top"|{{Ref|Ha}}||valign="top"| H. Hasse, "Ueber p-adische Schiefkörper und ihre Bedeutung für die Arithmetik hyperkomplexer Zahlsysteme" ''Math. Ann.'', '''104''' (1931) pp. 495–534        {{ZBL|0001.19805}}
 +
|-
 +
|valign="top"|{{Ref|Ha2}}||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  {{MR|1512823}}       
 +
|-
 +
|valign="top"|{{Ref|Ha3}}||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  JFM {{ZBL|49.0102.01}}     
 +
|-
 +
|valign="top"|{{Ref|Ha4}}||valign="top"| H. Hasse, "Ueber die Aequivalenz quadratischer Formen im Körper der rationalen Zahlen" ''J. Reine Angew. Math.'', '''152''' (1923) pp. 205–224  JFM {{ZBL|49.0102.02}}     
 +
|-
 +
|valign="top"|{{Ref|Ha5}}||valign="top"| H. Hasse, "Symmetrische Matrizen im Körper der rationalen Zahlen" ''J. Reine Angew. Math.'', '''153''' (1923) pp. 12–43  JFM {{ZBL|49.0104.01}}     
 +
|-
 +
|valign="top"|{{Ref|Ha6}}||valign="top"| H. Hasse, "Darstellbarkeit von Zahlen durch quadratische Formen in einem beliebigen algebraischen Zahlkörper" ''J. Reine Angew. Math.'', '''153''' (1923) pp. 113–130  JFM {{ZBL|49.0114.01}}     
 +
|-
 +
|valign="top"|{{Ref|Ha7}}||valign="top"| H. Hasse, "Aequivalenz quadratischer Formen in einem beliebigen algebraischen Zahlkörper" ''J. Reine Angew. Math.'', '''153''' (1924) pp. 158–162  JFM {{ZBL|50.0104.03}}     
 +
|-
 +
|valign="top"|{{Ref|Ha8}}||valign="top"| R. Hartshorne, "Algebraic geometry", Springer (1977)  {{MR|0463157}}  {{ZBL|0367.14001}}     
 +
|-
 +
|valign="top"|{{Ref|OM}}||valign="top"| O.T. O'Meara, "Introduction to quadratic forms" , Springer (1963)  {{MR|0152507}} {{ZBL|0107.03301}}
 +
|-
 +
|valign="top"|{{Ref|La}}||valign="top"| T.Y. Lam, "The algebraic theory of quadratic forms", Benjamin (1973)  {{MR|0396410}}  {{ZBL|0259.10019}}     
 +
|-
 +
|valign="top"|{{Ref|Ma}}||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)       
 +
|-
 +
|valign="top"|{{Ref|Si}}||valign="top"| J.H. Silverman, "The arithmetic of elliptic curves", Springer (1986)  {{MR|0817210}}  {{ZBL|0585.14026}}     
 +
|-
 +
|valign="top"|{{Ref|We}}||valign="top"| A. Weil, "Basic number theory", Springer (1967) {{MR|0234930}}  {{ZBL|0176.33601}}     
 +
|-
 +
|}

Revision as of 13:22, 5 March 2012

2020 Mathematics Subject Classification: Primary: 11Rxx Secondary: 11Sxx [MSN][ZBL]

The Hasse invariant is an arithmetic invariant of various objects.

Central simple algebras

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 [Ha], [Ha2].

Quadratic forms

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 $K_\nu\to \R$).

Cf. [Ha3], [Ha4], [Ha5], [Ha6], [Ha7], [OM], [La], [We]


Elliptic curves

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.

Cf. [Ma]

References

[Ca] J.W.S. Cassels, "Rational quadratic forms", Acad. Press (1978) MR0522835 Zbl 0395.10029
[CaFr] J.W.S. Cassels (ed.) A. Fröhlich (ed.), Algebraic number theory, Acad. Press (1967) MR0215665 Zbl 0153.07403
[Ha] H. Hasse, "Ueber p-adische Schiefkörper und ihre Bedeutung für die Arithmetik hyperkomplexer Zahlsysteme" Math. Ann., 104 (1931) pp. 495–534 Zbl 0001.19805
[Ha2] 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 MR1512823
[Ha3] H. Hasse, "Ueber die Darstellbarkeit von Zahlen durch quadratische Formen im Körper der rationalen Zahlen" J. Reine Angew. Math., 152 (1923) pp. 129–148 JFM Zbl 49.0102.01
[Ha4] H. Hasse, "Ueber die Aequivalenz quadratischer Formen im Körper der rationalen Zahlen" J. Reine Angew. Math., 152 (1923) pp. 205–224 JFM Zbl 49.0102.02
[Ha5] H. Hasse, "Symmetrische Matrizen im Körper der rationalen Zahlen" J. Reine Angew. Math., 153 (1923) pp. 12–43 JFM Zbl 49.0104.01
[Ha6] H. Hasse, "Darstellbarkeit von Zahlen durch quadratische Formen in einem beliebigen algebraischen Zahlkörper" J. Reine Angew. Math., 153 (1923) pp. 113–130 JFM Zbl 49.0114.01
[Ha7] H. Hasse, "Aequivalenz quadratischer Formen in einem beliebigen algebraischen Zahlkörper" J. Reine Angew. Math., 153 (1924) pp. 158–162 JFM Zbl 50.0104.03
[Ha8] R. Hartshorne, "Algebraic geometry", Springer (1977) MR0463157 Zbl 0367.14001
[OM] O.T. O'Meara, "Introduction to quadratic forms" , Springer (1963) MR0152507 Zbl 0107.03301
[La] T.Y. Lam, "The algebraic theory of quadratic forms", Benjamin (1973) MR0396410 Zbl 0259.10019
[Ma] 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)
[Si] J.H. Silverman, "The arithmetic of elliptic curves", Springer (1986) MR0817210 Zbl 0585.14026
[We] A. Weil, "Basic number theory", Springer (1967) MR0234930 Zbl 0176.33601
How to Cite This Entry:
Hasse invariant. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Hasse_invariant&oldid=21512
This article was adapted from an original article by E.B. Vinberg (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article