Introduction

We give an exposition of various ideas and results related to the fundamental results of [Tamal-2], [Mzkl-2] concerning Grothendieck's Conjecture of Anabelian Geometry (which we refer to as the “Grothendieck Conjecture” for short; see [Mzk2], Introduction, for a brief introduction to this conjecture). Many of these ideas existed prior to the publication of [Tamal-2], [Mzkl-2], but were not discussed in these papers because of their rather elementary nature and secondary importance (by comparison to the main results of these papers). Nevertheless, it is the hope of the author that the reader will find this article useful as a supplement to [Tamal-2], [Mzkl-2]. In particular, we hope that the discussion of this article will serve to clarify the meaning and motivation behind the main result of [Mzk2].

Our main results are the following:

• (1) In Section 1, we take the reverse point of view to the usual one (i.e., that the Grothendieck Conjecture should be regarded as a sort of (anabelian) Tate Conjecture) and show that in a certain case, the Tate Conjecture may be regarded as a sort of Grothendieck Conjecture (see Theorem 1.1, Corollary 1.2). In particular, Corollary 1.2 is interesting in that it allows one to express the fundamental phenomenon involved in the Tate and Grothendieck Conjectures using elementary language that can, in principle, be understood even by high school students (see the Introduction to Section 1; the Remarks following Corollary 1.2).

• (2) In Section 2, we show how the main result of [Mzk2] gives rise to a purely algebro-geometric corollary (i.e., one which has nothing to do with Galois groups, arithmetic considerations, etc.) in characteristic 0 (see Corollary 2.1).