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Comment: 12 pages. http://www2.math.kyushu-u.ac.jp/~j.shigezumi/

Comment: Updated references

Comment: 11 pages, some typos are corrected; Introduction and Section 3 are
revised; some remarks are added; main results are unchanged

In this paper we clarify some asymptotic formulas given by Jaulent-Maire,
which relate orders of finite quotients of S-infinitesimal T-classes l-groups
$Cl^S_T(K_n)$ associated to finite layers $K_n$ of a Zl-extension $K_\infty/K$
over a number field to the structural invariants of the Iwasawa module
$X^S_T:=\varprojlim \Cl^S_T(K_n)$. We especially show that the lambda invariant
$\tilde\lambda^S_T$ of those quotients sensibly differs from the structural
invariant $\lambda^S_T$, and we illustrate this fact with explicit examples,
where it can be made as large as desired, positive or negative.

Recently, Bruinier and Ono classified cusp forms $f(z) := \sum_{n=0}^{\infty}
a_f(n)q ^n \in S_{\lambda+1/2}(\Gamma_0(N),\chi)\cap \mathbb{Z}[[q]]$ that does
not satisfy a certain distribution property for modulo odd primes $p$. In this
paper, using Rankin-Cohen Bracket, we extend this result to modular forms of
half integral weight for primes $p \geq 5$. As applications of our main theorem
we derive distribution properties, for modulo primes $p\geq5$, of traces of
singular moduli and Hurwitz class number. We also study an analogue of Newman's
conjecture for overpartitions.

Serre obtained the p-adic limit of the integral Fourier coefficient of
modular forms on $SL_2(\mathbb{Z})$ for $p=2,3,5,7$. In this paper, we extend
the result of Serre to weakly holomorphic modular forms of half integral weight
on $\Gamma_{0}(4N)$ for $N=1,2,4$. A proof is based on linear relations among
Fourier coefficients of modular forms of half integral weight. As applications
we obtain congruences of Borcherds exponents, congruences of quotient of
Eisentein series and congruences of values of $L$-functions at a certain point
are also studied. Furthermore, the congruences of the Fourier coefficients of
Siegel modular forms on Maass Space are obtained using Ikeda lifting.

A recent result of Balandraud shows that for every subset S of an abelian
group G, there exists a non trivial subgroup H such that |TS| <= |T|+|S|-2
holds only if the stabilizer of TS contains H. Notice that Kneser's Theorem
says only that the stabilizer of TS must be a non-zero subgroup.
This strong form of Kneser's theorem follows from some nice properties of a
certain poset investigated by Balandraud. We consider an analogous poset for
nonabelian groups and, by using classical tools from Additive Number Theory,
extend some of the above results. In particular we obtain short proofs of
Balandraud's results in the abelian case.

Let F be a finite extension of Qp and G be GL(2,F). When V is the tensor
product of three admissible, irreducible, finite dimensional representations of
G, the space of G-invariant linear forms has dimension at most one. When a non
zero linear form exists, one wants to find an element of V which is not in its
kernel: this is a test vector. Gross and Prasad found explicit test vectors for
some triple of representations. In this paper, others are found, and they
almost complete the case when the conductor of each representation is at most
1.

Comment: 11 pages, no figures