##
**Representation masses of spinor genera.**
*(English)*
Zbl 1021.11008

Let \(f\) be an integral quadratic form of rank \(n\). The Minkowski-Siegel mass formula provides a means of determining the representation measure (or mass) of an integer \(c\) by the genus of \(f\) by expressing this quantity as a product of local factors. This method has subsequently been extended, via a mass formula involving quadratic characters, to study the corresponding representation measures of \(c\) by the spinor genera within the genus of \(f\). When \(n \geq 4\), these measures are equal for all spinor genera in the genus of \(f\). However, when \(n=3\) it can happen that two different values of this measure occur within a genus, each occurring for exactly half of the spinor genera in the genus. In this case, the integer \(c\) is referred to as a splitting integer for the genus. In [J. Reine Angew. Math. 352, 114-132 (1984; Zbl 0533.10016)], R. Schulze-Pillot studied the difference between the values of the two measures in such a case, expressing this difference as a product of local factors.

The present paper extends much of this previous theory to the general context of representations of a quadratic lattice of arbitrary rank by the spinor genera within a genus of lattices of greater or equal rank, and goes on to investigate the case in which the represented lattice has codimension less than 2. More specifically, let \(F\) be an algebraic number field with ring of integers \(o_F\), let \(L\) be an \(o_F\)-lattice of rank \(n\) on a nondegenerate quadratic space \(V\) over \(F\), and let \(K\) be a sublattice of \(L\) of rank \(k\). The author considers the values taken on by the representation measure \(r(K, \text{spn}^+(M))\), as \(\text{spn}^+(M)\) ranges over the proper spinor genera in the genus gen(\(L\)).

The paper is divided into two parts. The first part deals with the codimension 2 case (i.e., \(n-k=2\)). Here \(r(K, \text{spn}^+(M))\) takes on at most two distinct values as \(M\) ranges over the lattices in gen(\(L\)); \(K\) is said to be a splitting lattice for gen(\(L\)) when two distinct values are realized. The formula given by Schulze-Pillot is generalized here to produce an expression as a product of local factors for the difference between the representation measures of \(K\) by two proper spinor genera in gen(\(L\)). Criteria are then given for the determination of the local factors appearing in the resulting formula. In the special case when \(n=3\), these criteria are made completely explicit, thus providing a means for determining the splitting integers for a ternary genus.

The second part of the paper treats the case when the codimension is less than or equal to one. The quadratic characters on a suitable factor group of the idèle group of \(F\) are used to produce a mass formula for spinor genera in this case. It is no longer true that there are at most two distinct values for \(r(K,\text{spn}^+(M))\) within gen(\(L\)). The author obtains an upper bound for the number of different values that this quantity can take on within a given genus, expressed as a group index involving local spinor norm groups.

The present paper extends much of this previous theory to the general context of representations of a quadratic lattice of arbitrary rank by the spinor genera within a genus of lattices of greater or equal rank, and goes on to investigate the case in which the represented lattice has codimension less than 2. More specifically, let \(F\) be an algebraic number field with ring of integers \(o_F\), let \(L\) be an \(o_F\)-lattice of rank \(n\) on a nondegenerate quadratic space \(V\) over \(F\), and let \(K\) be a sublattice of \(L\) of rank \(k\). The author considers the values taken on by the representation measure \(r(K, \text{spn}^+(M))\), as \(\text{spn}^+(M)\) ranges over the proper spinor genera in the genus gen(\(L\)).

The paper is divided into two parts. The first part deals with the codimension 2 case (i.e., \(n-k=2\)). Here \(r(K, \text{spn}^+(M))\) takes on at most two distinct values as \(M\) ranges over the lattices in gen(\(L\)); \(K\) is said to be a splitting lattice for gen(\(L\)) when two distinct values are realized. The formula given by Schulze-Pillot is generalized here to produce an expression as a product of local factors for the difference between the representation measures of \(K\) by two proper spinor genera in gen(\(L\)). Criteria are then given for the determination of the local factors appearing in the resulting formula. In the special case when \(n=3\), these criteria are made completely explicit, thus providing a means for determining the splitting integers for a ternary genus.

The second part of the paper treats the case when the codimension is less than or equal to one. The quadratic characters on a suitable factor group of the idèle group of \(F\) are used to produce a mass formula for spinor genera in this case. It is no longer true that there are at most two distinct values for \(r(K,\text{spn}^+(M))\) within gen(\(L\)). The author obtains an upper bound for the number of different values that this quantity can take on within a given genus, expressed as a group index involving local spinor norm groups.

Reviewer: Andrew G.Earnest (Carbondale)

### MSC:

11E12 | Quadratic forms over global rings and fields |

11E08 | Quadratic forms over local rings and fields |

11E25 | Sums of squares and representations by other particular quadratic forms |

### Citations:

Zbl 0533.10016
Full Text:
DOI

### References:

[1] | M. Eichler, Die Ähnlichkeitsklassen indefiniter Gitter , Math. Z. 55 (1952), 216–252. · Zbl 0049.31201 |

[2] | J. S. Hsia, Spinor norms of local integral rotations, I , Pacific J. Math. 57 (1975), 199–206. · Zbl 0283.10009 |

[3] | –. –. –. –., Representations by spinor genera , Pacific J. Math. 63 (1976), 147–152. · Zbl 0328.10018 |

[4] | –. –. –. –., “Arithmetic of indefinite quadratic forms” in Integral Quadratic Forms and Lattices (Seoul, 1998) , Contemp. Math. 249 , Amer. Math. Soc., Providence, 1999, 1–15. · Zbl 0992.11032 |

[5] | J. S. Hsia, Y. Kitaoka, and M. Kneser, Representations of positive definite quadratic forms , J. Reine Angew. Math. 301 (1978), 132–141. · Zbl 0374.10013 |

[6] | J. S. Hsia, Y. Y. Shao, and F. Xu, Representations of indefinite quadratic forms , J. Reine Angew. Math. 494 (1998), 129–140. · Zbl 0883.11016 |

[7] | ——–, Spinor norms of relative local rotations , in preparation. |

[8] | Y. Kitaoka, Arithmetic of Quadratic Forms , Cambridge Tracts in Math. 106 , Cambridge Univ. Press, Cambridge, 1993. · Zbl 0785.11021 |

[9] | M. Kneser, Klassenzahlen indefiniter quadratischer Formen in drei oder mehr Veränderlichen , Arch. Math. 7 (1956), 323–332. · Zbl 0071.27205 |

[10] | –. –. –. –., Darstellungsmasse indefiniter quadratischer Formen , Math. Z. 77 (1961), 188–194. · Zbl 0100.03601 |

[11] | ——–, Quadratische Formen , Vorlesungsausarbeitung, Universität Göttingen, 1974. |

[12] | –. –. –. –., “Representations of integral quadratic forms” in Quadratic and Hermitian Forms (Hamilton, Ontario, 1983) , CMS Conf. Proc. 4 , Amer. Math. Soc., Providence, 1984, 159–172. · Zbl 0554.10012 |

[13] | O. T. O’Meara, The integral representations of quadratic forms over local fields , Amer. J. Math. 80 (1958), 843–878. JSTOR: · Zbl 0085.02801 |

[14] | ——–, Introduction to Quadratic Forms , Grundlehren Math. Wiss. 117 , Springer, Berlin, 1973. · Zbl 0259.10018 |

[15] | T. Ono, “On Tamagawa numbers” in Algebraic Groups and Discontinuous Subgroups (Boulder, Colo., 1965) , Proc. Sympos. Pure Math. 9 , Amer. Math. Soc., Providence, 1966, 122–132. |

[16] | F. Satō, Siegel’s main theorem of homogeneous spaces , Comment. Math. Univ. St. Paul. 41 (1992), 141–167. · Zbl 0789.11030 |

[17] | R. Schulze-Pillot, Darstellung durch Spinorgeschlechter ternärer quadratischer Formen , J. Number Theory 12 (1980), 529–540. · Zbl 0443.10017 |

[18] | –. –. –. –., Darstellungsmaße von Spinorgeschlechtern ternärer quadratischer Formen , J. Reine Angew. Math. 352 (1984), 114–132. · Zbl 0533.10016 |

[19] | –. –. –. –., Thetareihen positiv definiter quadratischer Formen , Invent. Math. 75 (1984), 283–299. · Zbl 0533.10021 |

[20] | –. –. –. –., Ternary quadratic forms and Brandt matrices , Nagoya Math. J. 102 (1986), 117–126. · Zbl 0566.10015 |

[21] | C. L. Siegel, Über die analytische Theorie der quadratischen Formen, I, II, III , Ann. of Math. (2) 36 (1935), 527–606.; 37 (1936), 230–263.; 38 (1937), 212–291. JSTOR: · Zbl 0016.01205 |

[22] | A. Weil, “Sur la théorie des formes quadratiques” in Colloque sur la théorie des groupes algébriques (Brussels, 1962) , Librairie universitaire, Louvain, 1962, 9–22. · Zbl 0146.05804 |

[23] | ——–, Basic Number Theory , 3d ed., Grundlehren Math. Wiss. 144 , Springer, New York, 1974. · Zbl 0326.12001 |

[24] | F. Xu, Integral spinor norms in dyadic local fields, I , Pacific J. Math. 157 (1993), 179–200. · Zbl 0806.11019 |

[25] | –. –. –. –., Generation of integral orthogonal groups over dyadic local fields , Pacific J. Math. 167 (1995), 385–398. · Zbl 0829.11021 |

[26] | –. –. –. –., “Arithmetic Springer theorem on quadratic forms under field extensions of odd degree” in Integral Quadratic Forms and Lattices (Seoul, 1998) , Contemp. Math. 249 , Amer. Math. Soc., Providence, 1999, 175–197. · Zbl 0955.11010 |

[27] | –. –. –. –., Representations of indefinite ternary quadratic forms over number fields , Math. Z. 234 (2000), 115–144. · Zbl 0990.11016 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.