In this talk, we discuss conformal field theories in two dimensions (2d CFTs) and aspects of the theory of modular forms. Physical considerations lead us to study two extensions to the theory of modular forms: modular forms for GL2(Z) that are defined on the double half-plane (in distinction to SL2(Z) modular forms defined on the upper half-plane), and L-functions for modular forms with poles *within* the fundamental domain. We introduce both concepts, and discuss their consistency, both with each other and with the physical considerations which led to them. Finally, we note that very similar physical considerations may apply to finite-temperature path integrals for generic QFTs in higher dimensions, and comment on possible consequences of this.

Temperature manifests itself within quantum field theories (QFTs) and conformal field theories (CFTs) via an identification of points in the Euclidean-time direction, which differ by an integer multiple of 1/T. Today, I will talk about finite-temperature path integrals for general QFTs and for two-dimensional CFTs (2d CFTs) on the compact two-torus. By definition, the latter path integrals are modular invariant. I will discuss why, propose an extension of the modular group from SL_2(\Z) to GL_2(\Z), introduce the notion of modular forms with poles, and discuss general properties of modular forms with and without poles that are defined on the extended group GL_2(\Z). Finally, I will discuss how this extension to GL_2(\Z) may introduce a new source of anomalies/consistency conditions in 2d CFTs (and beyond).