Galois Representations Associated To Drinfeid Modules And A Kind Of Rankin-selberg Functions

This thesis mainly includes two parts.Firstly, we study Galois representations associated to Drinfeld modules. A cele-brated theorem of Serre asserts the image of the adelic representation associated to anelliptic curve over a number field without complex multiplication is open. In the lastdecades, Serre’s open image theorem has received a lot of attention. Recent work ofPink and Ru¨tsche has described the image of Galois representations associated to Drin-feld modules. This result can be viewed as a Drinfeld module analogue of Serre’s openimage theorem on elliptic curves. Jones has studied expectional numbers and minimalexpectional numbers for elliptic curves. In this thesis, we extend Jones’ work to Drinfeldmodules. We define expectional polynomials and minimal expectional polynomials forDrinfeld modules. We study minimal expectional polynomials for Drinfeld modules andthe surjectivity of the Galois representations associated to Drinfeld modules.Secondly, we show a new kind of Rankin–Selberg functions. The Rankin–Selbergconvolution for L-functions associated to automorphic forms on GL(2) was independentlydiscovered by Rankin and Selberg, and it is one of the most important constructions in thetheory of L-functions. The Rankin–Selberg convolution for GL(n)×GL(n′)(1≤n≤n′)has been obtained by Godement, Jacquet and Shalika. In this thesis, we define a newkind of Rankin–Selberg functions for the case GL(r)×GL(r+s)×GL(s), and we provethe functional equation and holomorphic continuation. As an application, we give a newVoronoi formula.