Suppose F = Q(√-p1 pt) is an imaginary quadratic number field with distinct primes p1,..., pt,where pi≡ 1(mod 4)(i = 1,..., t- 1) and pt ≡ 3(mod 4). We express the possible values of the 8-rank r8 of the class group of F in terms of a quadratic form Q over F2 which is defined by quartic symbols. In particular,we show that r8 is bounded by the isotropy index of Q.
The issue of local and global conjugacy is closely related to the multiplicity one property in representation theory and the Langlands program. In this article we give first families of connected instances for SO2N where the multiplicity one fails in both aspects of representation theory and automorphic forms with certain assumptions on the Langlands functoriality.
Let f(n)be a multiplicative function satisfying |f(n)|≤1,q(≤N^2)be a positive integer and a be an integer with(a,q)= 1.In this paper,we shall prove that ∑n≤N(n,q)=1f(n)e(an/q)■(1/2)(τ(q)/q)N loglog(6N)+ q^(1/4+ε/2)N^(2/1)(log(6N))^(1/2)+N/(1/2)(loglog(6N)),where n is the multiplicative inverse of n such that nn ≡ 1(mod q),e(x)= exp(2πix),and τ(·)is the divisor function.
The Hilbert genus field of the real biquadratic field K=Q（√δ,√d）is described by Yue（2010）and Bae and Yue（2011）explicitly in the case＆=2 or p with p=1 mod 4 a prime and d a squarefree positive integer.In this article,we describe explicitly the Hilbert genus field of the imaginary biquadratic field K=Q（√δ,√d）,whereδ=-1,-2 or-p with p=3 mod 4 a prime and d any squarefree integer.This completes the explicit construction of the Hilbert genus field of any biquadratic field which contains an imaginary quadratic subfield of odd class number.
Let K_(0)=Q(√δ)beaquadraticfield.Forthose K_(0) withoddclassnumber,much work has been done on the explicit construction of the Hilbert genus field of a biquadratic extension K=Q(√δ,√d)over Q.Whenδ=2 or p with p≡1 mod 4 a prime and K is real,it was described in Yue(Ramanujan J 21:17–25,2010)and Bae and Yue(Ramanujan J 24:161–181,2011).In this paper,we describe the Hilbert genus field of K explicitly when K_(0) is real and K is imaginary.In fact,we give the explicit construction of the Hilbert genus field of any imaginary biquadratic field which contains a real quadratic subfield of odd class number.
In this paper, we use the 2-descent method to find a series of odd non-congruent numbers = 1 （nmd 8） whose prime factors are = 1 （rood 4） such that the congruent elliptic curves have second lowest Selmer groups, which include Li and Tian＇s result as special cases.
