Hasse invariant

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
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[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