2011/11/29 by Byungheup Jun, Jungyun Lee, Jun, Byungheup +1
Mathematics · #Analytic Number Theory Research #Algebraic Geometry and Number Theory #Advanced Algebra and Geometry
paper · pdf · doi:10.48550/arxiv.1111.6716
For a family of real quadratic fields \Kn=\FQ(√(f(n)))\n∈ \FN, a Dirichlet character χ modulo q and prescribed ideals \\fbn⊂ Kn\, we investigate the linear behaviour of the special value of partial Hecke's L-function LKn(s,χn:=χ∘ NKn,\fbn) at s=0. We show that for n=qk+r, LKn(0,χn,\fbn) can be written as (1)/(12q2)(Aχ(r)+kBχ(r)), where Aχ(r),Bχ(r)∈ \FZ[χ(1),χ(2),..., χ(q)] if a certain condition on \fbn in terms of its continued fraction is satisfied. Furthermore, we write precisely Aχ(r) and Bχ(r) using values of the Bernoulli polynomials. We describe how the linearity is used in solving class number one problem for some families and recover the proofs in some cases. Finally, we list some families of real quadratic fields with the linearity.