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## Publications

Publications (21)

For any positive integers $n$ and $m$,
$\mathbb{H}_{n,m}:=\mathbb{H}_n\times\mathbb{C}^{(m,n)}$ is called the
Siegel-Jacobi space, with the Jacobi group acting on it. The Jacobi forms are
defined on this space. In this article we compute the Levi-Civita connection of
the Siegel-Jacobi space and use it to obtain derivations of Jacobi forms. In
degre...

In this paper we present a probabilistic algorithm to compute the
coefficients of modular forms of level one. Focus on the Ramanujan's tau
function, we give out the explicit complexity of the algorithm. From a
practical viewpoint, the algorithm is particularly well suited for
implementations.

We introduce a method in differential geometry to study the derivative
operators of Siegel modular forms. By determining the coefficients of the
invariant Levi-Civita connection on a Siegel upper half plane, and further by
calculating the expressions of the differential forms under this connection, we
get a non-holomorphic derivative operator of th...

Here we show that Rubin’s method of the use of two Euler systems to the proofs of Iwasawa main conjectures of the rational
number field and of an imaginary quadratic field is only proper to these two kinds of number fields. We mainly use the properties
of adéle ring and idéle group in class field theory to get the result.

We call a (q-1)-th Kummer extension of a cyclotomic function field a
quasi-cyclotomic function field if it is Galois, but non-abelian, over the
rational function field with the constant field of q elements. In this paper,
we determine the structure of the Galois groups of a kind of quasi-cyclotomic
function fields over the base field. We also give...

A double covering of a Galois extension K/k in the sense of P. Das (2000) [4] is an extension K˜/K of degree ⩽2 such that K˜/k is Galois. In this paper we determine explicitly all double coverings of any quadratic extensions, the Carlitz cyclotomic extensions of the rational function field over a finite field, and their maximal real subfields. In t...

We call a quadratic extension of a cyclotomic field a quasi-cyclotomic field if it is non-abelian Galois over the rational number field. In this paper, we study the arithmetic of any quasi-cyclotomic field, including to determine the ring of integers of it, the decomposition nature of prime numbers in it, and the structure of the Galois group of it...

Let k be a global function field over the finite field Fq with a fixed place ∞ of degree 1. Let K be a cyclic extension of degree dividing q - 1, in which ∞ is totally ramified. For a certain abelian extension L of k containing K, there are two notions of the group of cyclotomic units arising from sign normalized rank 1 Drinfeld modules on k and on...

We determine all irreducible representations of primary quasi-cyclotomic fields in this paper. The methods can be applied to determine the irreducible representations of any quasi-cyclotomic field. We also compute the Artin L-functions for a class of quasi-cyclotomic fields.

A double covering of a Galois extension K/F in the sense of [3] is an extension
/K of degree ≤2 such that
/F is Galois. In this paper we determine explicitly all double coverings of any cyclotomic extension over the rational number field in the complex number field. We get the results mainly by Galois theory and by using and modifying the results a...

Let k be the field of rational functions over the finite field of q elements. Let k ac be an algebraic closure of k. The maximal abelian extension k ab of k in k ac and its Galois group over k have been described in the obvious manner by Hayes by developing ideas of Carlitz. Let k ab+ε be the composition of all subfields of k ac which are Kummer (q...

We investigate algebraic Γ -monomials of Thakur's positive characteristic Γ -function, by using Anderson and Das' double complex method of computing the sign cohomology of the universal ordinary distribution. We prove that the Γ -monomial associated to an element of the second sign cohomology of the universal ordinary distribution of Fq(T) generate...

Recently Anderson described explicitly the epsilon extension of the maximal abelian ℚab of the rational number field ℚ, which is the compositum of all subfield of ℂ quadratic over ℚab and Galois over ℚ. We have given an analogous description of the epsilon extension of the maximal abelian extension of the
rational function field over a finite fiel...

Let K/k be a finite abelian extension of function fields with Galois group G. Using the Stickelberger elements associated to K/k studied by J. Tate, P. Deligne and D. Hayes, we construct an ideal I in the integral group ring relative to the extension K/k, whose elements annihilate the group of divisor classes of degree zero of K and whose rank is e...

In classical number theory, one has the famous theorem of Kummer-Sinnott giving the index of the cyclotomic units in the total unit group. Using the division values of sgn-normalized rank one Drinfeld modules, we construct the group of extended cyclotomic units for an abelian extension of a global function field and calculate its index in the whole...