In this paper, we will give the subconvexity bounds for self-dual GL(3) L-functions in the t aspect as well as subconvexity bounds for self-dual GL(3) × GL(2) L-functions in the GL(2) spectral aspect.

Let $L$ be a completely regular frame and $mathcal{R}L$ be the ring of real-valued continuous functions on $L$. We consider the set $$mathcal{R}_{infty}L = {varphi in mathcal{R} L : uparrow varphi( dfrac{-1}{n}, dfrac{1}{n}) mbox{ is a compact frame for any $n in mathbb{N}$}}.$$ Suppose that $C_{infty} (X)$ is the family of all functions $f in C(X)$ for which the set ${x in X: |f(x)|geq dfrac{1...

After a too-brief introduction to adeles and ideles, we sketch proof of analytic continuation and functional equation of Riemann’s zeta, in the modern form due independently to Iwasawa and Tate about 1950. The sketch is repeated for Dedekind zeta functions of number fields, noting some additional complications. The sketch is repeated again for Hecke’s (größencharakter) L-functions, noting furth...

coming from the cokernels of the rst column. These are cyclic (pro-cyclic in the limit) and are generated by the image of q t. So, lim A(K t)=NA(K n;t) = ?=< (q t ; L t =K t) > where (q t ; L t =K t) is the norm residue symbol of q t for the extension L t =K t. We write Q t for the image of (q t ; L t =K t) under the isomorphism ? ! Z p induced by sending to 1. Note that although Q t is not can...

We study inclusive semi-leptonic ¯ B → X c ℓ¯ ν ℓ decay using the power counting m c ∼ Λ QCD m b. Assuming this scaling for the charm-quark mass, the decay kinematics can be chosen to access the shape-function region even in b → c transitions. To apply effective field theory methods in this region we extend SCET to describe massive collinear quarks. Working at tree level and including O(Λ QCD /...