Namespaces
Variants
Actions

Hasse invariant

From Encyclopedia of Mathematics
Jump to: navigation, search
The printable version is no longer supported and may have rendering errors. Please update your browser bookmarks and please use the default browser print function instead.

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=21571
This article was adapted from an original article by E.B. Vinberg (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article