Genus character LL-functions of quadratic orders in an adelic way and maximal orders of matrix algebras

For a quadratic extension $K$ of ${\mathbb Q}$, we consider orders $O$ in $K$ that are not necessarily maximal and the ideal class group $Cl^+(O)$ in the narrow sense of proper ideals of $O$. Characters of $Cl^+(O)$ of order at most two are traditionally called genus characters. Explicit description of such characters is known classically, but explicit $L$-functions associated to those characters are only recently obtained partially by Chinta and Offen and completely by Kaneko and Mizuno. As remarked in the latter paper, the present author also obtained the formula of such $L$-functions independently. Indeed, here we will give a simple and transparent alternative proof of the formula by rewriting explicit genus characters and their values in an adelic way starting from scratch. We also add an explicit formula for the genus number in the wide sense, which is maybe known but rarely treated. As an appendix we give an ideal-theoretic characterization of isomorphism classes of maximal orders of the matrix algebras $M_n(F)$ over a number field $F$ up to $GL_n(F)$ and $GL_n^+(F)$ conjugation respectively, and apply genus numbers to count them when $n=2$ and $F$ is quadratic. To avoid any misconception, we include some easy known details.Comment: 25 page

Supersingular abelian varieties and quaternion hermitian lattices (Theory and Applications of Supersingular Curves and Supersingular Abelian Varieties)

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

Dimensions of paramodular forms and compact twist modular forms with involutions

We give an explicit dimension formula for paramodular forms of degree two of prime level with plus or minus sign of the Atkin--Lehner involution of weight $\det^k\operatorname{Sym}(j)$ with $k\geq 3$, as well as a dimension formula for algebraic modular forms of any weight associated with the binary quaternion hermitian maximal lattices in non-principal genus of prime discriminant with fixed sign of the involution. These two formulas are essentially equivalent by a recent result of N. Dummigan, A. Pacetti. G. Rama and G. Tornar\'ia on correspondence between algebraic modular forms and paramodular forms with signs. So we give the formula by calculating the latter. When $p$ is odd, our formula for the latter is based on a class number formula of some quinary lattices by T. Asai and its interpretation to the type number of quaternion hermitian forms given in our previous works. On paramodular forms, we also give a dimensional bias between plus and minus eigenspaces, some list of palindromic Hilbert series, numerical examples for small $p$ and $k$, and the complete list of primes $p$ such that there is no paramodular cusp form of level $p$ of weight 3 with plus sign. This last result has geometric meaning on moduli of Kummer surface with $(1,p)$ polarization.Comment: 57 pages: Added dimensional bias of plus and minus eigenspaces, examples of palindromic Hilbert series, and supplied more details of the proof of Theorem 2.

SUPERSINGULAR LOCI OF LOW DIMENSIONS AND PARAHORIC SUBGROUPS

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