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The theory of polarized supersingular abelian varieties (A, λ) is often essentially related to the theory of quaternion hermitian lattices. In this paper, we add one more such relation by giving an adelic description of supersingular loci of low dimensions. Katsura, Li and Oort have shown that the supersingular locus in the moduli of principally polarized abelian varieties is not irreducible and that the number of its irreducible components is equal to the class number of certain maximal quaternion hermitian lattices. Now the locus has certain natural algebraic subsets consisting of A with fixed a-numbers that are defined as the dimensions of embeddings from α_p to A. For low dimensional cases when dim A ≤ 3, we describe configuration of these subsets in the locus by intersection properties of some adelic subgroups of quaternion hermitian groups corresponding to parahoric subgroups locally at characteristic p. In particular, we describe which components each superspecial point lies on. This is proved by using certain liftability property of isogenies of abelian varieties, where the isogenies are interpreted to cosets of GL_n and of parahoric subgroups of the quaternion hermitian groups acting on quaternion hermitian matrices.

This note gives a survey on relations between the theory of quaternion hermitian lattices and that of supersingular abelian varieties, including relations between polarizations, moduli loci, automorphism groups, curves with many rational points, and class numbers, type numbers, lattice automorphisms, algebraic modular forms. For readers' convenience, we give some explicit formulas for related numbers and give a slightly big list of related references. The last section is an announcement of new results on supersingular loci of low dimensions.

We solve two related Gauss problems. In the arithmetic setting, we consider genera of maximal $O$-lattices, where $O$ is the maximal order in a definite quaternion $\mathbb{Q}$-algebra, and we list all cases where they have class number $1$. We also prove a unique orthogonal decomposition result for more general $O$-lattices. In the geometric setting, we study the Siegel modular variety $\mathcal{A}_g \otimes \overline{\mathbb{F}}_p$ of genus $g$, and we list all $x$ in $\mathcal{A}_g \otimes \overline{\mathbb{F}}_p$ for which the corresponding central leaf $\mathcal{C}(x)$ consists of one point, that is, such that $x$ is the unique point in the locus consisting of the points whose associated polarised $p$-divisible groups are isomorphic to that of $x$. The solution to the second Gauss problem involves mass formulae, computations of automorphism groups, and a careful analysis of Ekedahl-Oort strata in genus $g = 4$.

The word \textit{type number} of an algebra means classically the number of isomorphism classes of maximal orders in the algebra, but here we consider quaternion hermitian lattices in a fixed genus and their right orders. Instead of inner isomorphism classes of right orders, we consider isomorphism classes realized by similitudes of the quaternion hermitian forms.The number $T$ of such isomorphism classes are called \textit{type number} or \textit{$G$-type number}, where $G$ is the group of quaternion hermitian similitudes. We express $T$ in terms of traces of some special Hecke operators. This is a generalization of the result announced in [5] (I) from the principal genus to general lattices. We also apply our result to the number of isomorphism classes of any polarized superspecial abelian varieties which have a model over ${\Bbb F}_p$ such that the polarizations are in a "fixed genus of lattices". This is a generalization of [8] and has an application to the number of components in the supersingular locus which are defined over ${\Bbb F}_p$.

In our previous papers, we defined the $G$-type number of any genera of quaternion hermitian lattices as a generalization of the type number of a quaternion algebra. Now we prove in this paper that the $G$-type number of any genus of positive definite binary quaternion hermitian maximal lattices in $B^2$ for a definite quaternion algebra $B$ over $\mathbb Q$ is equal to the class number of some explicitly defined genus of positive definite quinary quadratic lattices. This is a generalization of a part of the results in 1982, where only the principal genus was treated. Explicit formulas for this type number can be obtained by using Asai's class number formula. In particular, in case when the discriminant of $B$ is a prime, we will write down an explicit formula for $T$, $H$ and $2T-H$ for the non-principal genus, where $T$ and $H$ are the type number and the class number. This number was known for the principal genus before. In another paper, our new result is applied to polarized superspecial varieties and irreducible components of supersingular locus in the moduli of principally polarized abelian varieties having a model over a finite prime field, where $2T-H$ plays an important role.

We enumerate all orders in definite quaternion algebras over number fields with the Hermite property; this includes all orders with the cancellation property for locally free modules., Comment: Some typos corrected; final version; 24 pages plus tables

We give a refinement of the Kronecker-Hurwitz class number relation, based on a tesselation of the Euclidean plane into semi-infinite triangles labeled by $PSL_2(\mathbb{Z})$ that may be of independent interest., 6 pages, 3 figures

J. S. Hsia has conjectured an arithmetical version of Springer Theorem, which states that no two spinor genera in the same genus of integral quadratic forms become identified over an odd degree extension. In this paper we prove by examples that the corresponding result for quaternionic skew-hermitian forms does not hold in full generality. We prove that it does hold for unimodular skew-hermitian lattices under all extensions and for lattices whose discriminant is relatively prime to 2 under Galois extensions.

In his masterworkDisquisitiones Arithmeticae, Gauss stated an approximate formula for the average of the class number for negative discriminants. In this paper we improve the known estimates for the error term in Gauss approximate formula. Namely, our result can be written asfor every ∊ > 0, whereH(−n) is, in modern notation,h(−4n). We also consider the average ofh(−n) itself obtaining the same type of result.Proving this formula we transform firstly the problem in a lattice point problem (as probably Gauss did) and we use a functional equation due to Shintani and Dirichlet class number formula to express the error term as a sum of character and exponential sums that can be estimated with techniques introduced in a previous work on the sphere problem.

For an odd prime $p$ let $\rho_p$ be the least odd prime ($\ne p$) which is a~quadratic residue mod $p$. Using the theorems of Heegner--Baker--Stark and Siegel--Tatuzawa on the class number $h=h(-p)$ of the imaginary quadratic number field $\mathbb{Q}(\sqrt{-p})$ it is shown that $\rho_p100$ und $5\le u

The obstruction to the local-global principle for a hermitian lattice (L, H) can be quantified by computing the mass of (L, H). The mass formula expresses the mass of (L, H) as a product of local factors, called the local densities of (L, H). The local density formula is known except in the case of a ramified hermitian lattice of residue characteristic 2. Let F be a finite unramified field extension of Q_2. Ramified quadratic extensions E/F fall into two cases that we call Case 1 and Case 2. In this paper, we obtain the local density formula for a ramified hermitian lattice in Case 1, by constructing a smooth integral group scheme model for an appropriate unitary group. Consequently, this paper, combined with the paper of W. T. Gan and J.-K. Yu, allows the computation of the mass formula for a hermitian lattice (L, H) in Case 1., Comment: 70 pages. The previous version is divided into two papers (Case 1 and Case 2). The current version is for Case 1 and its final version (for Case 1) is to appear in Algebra & Number Theory

We give a new proof of a recent theorem of Kaplansky and use it to revive an old, seemingly forgotten result of Glaisher.

In this paper, we prove that a binary definite quadratic form over $\mathbf{F}_q [t]$, where $q$ is odd, is completely determined up to equivalence by the polynomials it represents up to degree $3m-2$, where $m$ is the degree of its discriminant. We also characterize, when $q>13$, all the definite binary forms over $\mathbf{F}_q [t]$ that have class number one.

Let Λ be an n-dimensional integral lattice of determinant 1. We show that, for all sufficiently large n, the minimal nonzero squared length in Λ does not exceed [ ( n + 6 ) /10 ]. This bound is a consequence of some new conditions on the theta series of these lattices; these conditions also enable us to find the greatest possible minimal squared length in all dimensions n ≤ 33. In particular we settle the ‘‘no-roots’’ problem: there is a determinant 1 lattice containing no vectors of squared length 1 or 2 precisely when n ♢ 23, n © 25. There are also analogues of all these results for codes.

In this paper we consider the quadratic form $\phi [x] = \sum _{i=1}^{n} x_{i}^{2}$ over the vector space $\mathbb{Q}_{n}^{1}$. We take a $\mathbb{Z}$-maximal lattice $L$ in $\mathbb{Q}_{n}^{1}$ with respect to $\phi$. Let $\{L^{(i)}\}_{i=1}^{k(n)}$ be a complete set of representatives for the classes belonging to the genus of $L$. Applying Shimura's mass formula, we determine these representatives $L^{(i)}$ explicitly for $n = 11, 13$, and $14$. Consequently we obtain class numbers $k(11) = 3$, $k(13) = 4$, and $k(14) = 4$.

In a series of recent papers, G. Shimura obtained an exact formula for the mass of a maximal lattice in a quadratic or hermitian space over a totally real number field. Using Bruhat-Tits theory, we obtain a quick and more conceptual proof of his formula when the form is totally definite.

The subject matter of this paper is an old one with a rich history, beginning with the work of Gauss and Eisenstein, maturing at the hands of Smith and Minkowski, and culminating in the fundamental results of Siegel. More precisely, if L is a lattice over Z (for simplicity), equipped with an integral quadratic form Q, the celebrated Smith-Minkowski-Siegel mass formula expresses the total mass of (L,Q), which is a weighted class number of the genus of (L,Q), as a product of local factors. These local factors are known as the local densities of (L,Q). Subsequent work of Kneser, Tamagawa and Weil resulted in an elegant formulation of the subject in terms of Tamagawa measures. In particular, the local density at a non-archimedean place p can be expressed as the integral of a certain volume form ωld over AutZp(L,Q), which is an open compact subgroup of AutQp(L,Q). The question that remains is whether one can find an explicit formula for the local density. Through the work of Pall (for p 6= 2) and Watson (for p = 2), such an explicit formula for the local density is in fact known for an arbitrary lattice over Zp (see [P] and [Wa]). The formula is obviously structured, though [CS] seems to be the first to comment on this. Unfortunately, the known proof (as given in [P] and [K]) does not explain this structure and involves complicated recursions. On the other hand, Conway and Sloane [CS, §13] have given a heuristic explanation of the formula. In this paper, we will give a simple and conceptual proof of the local density formula, for p 6= 2. The view point taken here is similar to that of our earlier work [GHY], and the proof is based on the observation that there exists a smooth affine group scheme G over Zp with generic fiber AutQp(L,Q), which satisfies G(Zp) = AutZp(L,Q). This follows from general results of smoothening [BLR], as we explain in Section 3. For the purpose of obtaining an explicit formula, it is necessary to have an explicit construction of G. The main contribution of this paper is to give such an explicit construction of G (in Section 5), and to determine its special fiber (in Section 6). Finally, by comparing ωld and the canonical volume form ωcan of G, we obtain the explicit formula for the local density in Section 7. The smooth group schemes constructed in this paper should also be of independent interest.

1. Classification of positive norm vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 525 2. Vectors in the lattice II1,25 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 528 3. Lattices with no roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 534 The primitive norm 0 vectors of II1,25 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 537 The norm 2 vectors of II1,25 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 539 The norm 4 vectors of II1,25 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 544

For an integer d > 0 (resp. d < 0) let hd denote the number of Sl2(Z)-equivalence classes of primitive (resp. primitive positive-definite) integral binary quadratic forms of discriminant d. For where t and u are the smallest positive integral solutions of the equation t2 − du2 = 4 if d is a non-square and εd = 1 if d is a square.

Let K be a totally real algebraic number field. In a positive definite quadratic space over K a lattice En is called a unit lattice of rank n if En has an orthonormal basis {e1 …, en}. The class number one problem is to find n and K for which the class number of En is one. Dzewas ([1]), Nebelung ([3]), Pfeuffer ([6], [7]) and Peters ([5]) have settled this problem.