Sungmun Cho

The celebrated Smith–Minkowski–Siegel mass formula expresses the mass of a quadratic lattice $(L,Q)$ as a product of local factors, called the local densities of $(L,Q)$ . This mass formula is an essential tool for the classification of integral quadratic lattices. In this paper, we will describe the local density formula explicitly by observing the existence of a smooth affine group scheme $\underline{G}$ over $\mathbb{Z}_{2}$ with generic fiber $\text{Aut}_{\mathbb{Q}_{2}}(L,Q)$ , which satisf

The obstruction to the local-global principle for a hermitian lattice [math] can be quantified by computing the mass of [math] . The mass formula expresses the mass of [math] as a product of local factors, called the local densities of [math] . The local density formula is known except in the case of a ramified hermitian lattice of residue characteristic 2. ¶ Let [math] be a finite unramified field extension of [math] . Ramified quadratic extensions [math] fall into two cases that we call Case 1

Abstract This paper is the complementary work of [S. Cho, Group schemes and local densities of ramified hermitian lattices in residue characteristic 2: Part I, Algebra Number Theory 10 2016, 3, 451–532]. Ramified quadratic extensions <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>E</m:mi> <m:mo>/</m:mo> <m:mi>F</m:mi> </m:mrow> </m:math> {E/F} , where F is a finite unramified field extension of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>ℚ</m:mi> <m

In this paper, we will explain a conceptual reformulation and inductive formula of the Siegel series. Using this, we will explain that both sides of the local intersection multiplicities of [GK93] and the Siegel series have the same inherent structures, beyond matching values. As an application, we will prove a new identity between the intersection number of two modular correspondences over Fp and the sum of the Fourier coefficients of the Siegel-Eisenstein series for Sp_4 of weight 2, which is

T. Ikeda and H. Katsurada have developed the theory of the Gross-Keating invariant of a quadratic form in their recent papers [IK1] and [IK2]. In particular, they prove that the local factor of the Fourier coefficients of the Siegel-Eisenstein series is completely determined by the Gross-Keating invariant with extra datum, called the extended GK datum, in [IK2]. On the other hand, such local factor is a special case of the local densities for a pair of two quadratic forms. Thus we propose a gene

In this paper, we give a formula for the extended Gross-Keating datum of a quadratic form defined over a finite extension of $\mathbb{Z}_p$ (for $p&gt;2$) or a finite unramified extension of $\mathbb{Z}_2$. As an application, we describe an explicit formula for the Siegel series for $\mathbb{Z}_p$. We also present the details of algorithms implemented in a Mathematica package to compute the extended Gross-Keating datum and the Siegel series.

This work is motivated by an investigation into whether, and if so how, certain well known facts about Lie groups manifest in the context of group schemes over rings of integers of local fields. There are the following well-known relations among unitary, orthogonal and symplectic groups: U(n)=O(2n) \cap GL(n, C)=Sp(2n) \cap GL(n, C). Therefore, it is natural to ask whether or not there exist such relations among smooth integral models of unitary, orthogonal and symplectic groups defined over a l

In this paper, we give a formula for the extended Gross-Keating datum of a half-integral symmetric matrix over a finite extension of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper Q Subscript p"> <mml:semantics> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">Q</mml:mi> </mml:mrow> <mml:mi>p</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">\mathbb {Q}_p</mml:annot

A main goal of this paper is to introduce a new description of the stable orbital integral for a regular semisimple element and for the unit element of the Hecke algebra in the case of $\mathfrak{gl}_{n,F}$, $\mathfrak{u}_{n,F}$, and $\mathfrak{sp}_{2n,F}$, by assigning a certain stratification and then smoothening each stratum, where $F$ is a non-Archimedean local field of any characteristic. As applications, we will provide a closed formula for the stable orbital integral for $\mathfrak{gl}_{2

In this paper we provide an alternative but more straightforward inductive formula to compute the Gross-Keating invariant for a quadratic form defined over an unramified finite extension of $\mathbb{Z}_2$.

A Bass order is an order of a number field whose fractional ideals are generated by two elements. The majority of number fields contain infinitely many Bass orders. For example, any order of a number field which contains the maximal order of a subfield with degree 2 or whose discriminant is fourth-power-free in $\mathbb{Z}$, is a Bass order. In this paper, we will propose a closed formula for the number of fractional ideals of a Bass order $R$, up to its invertible ideals, using the conductor of

This is a survey article to explain a main result of the author s recent preprint (joint work with T. Yamauchi) [CY].