Let $a$, $b$, $c$ be fixed coprime positive integers. In this paper we prove that if $b \equiv 3 \pmod{4}$, $a \equiv -1 \pmod{b^{2l}}$, $a^2 + b^{2l-1} = c$ and $c$ is odd, where $l$ is a positive integer, then the equation $a^x + b^y = c^z$ has only the positive integer solution $(x,y,z) = (2,2l-1,1)$.

We construct an elliptic curve of the form $y^2 = x^3 + kx$ with rank at least 6 over $Q(x_1, x_2, x_3)$.

We construct an elliptic curve of rank at least 4 over $Q(t)$ with a rational point of order 4. We also show an infinite family of elliptic curves of rank at least 5 over $Q$ with a rational point of order4, which is parametrized by the rational points of an elliptic curve of rank at least 1.

The aim of the present note is to develop a study on the feasibility of a unified theory of mean values of automorphic $L$-functions, a desideratum in the field. This is an outcome of the investigation commenced with the part XII ([14]), where a framework was laid on the basis of the theory of automorphic representations, and a general approach to the mean values was envisaged. Specifically, it is shown here that the inner-product method, which was initiated by A. Good [7] and greatly enhanced by M. Jutila [9], ought to be brought to perfection so that the mean square of the $L$-function attached to any cusp form on the upper half-plane could be reached within the notion of automorphy. The Kirillov map is our key implement. Because of its geometric nature, our method appears to extend to bigger linear Lie groups. This note is essentially self-contained.