We consider real-valued solutions $u=u\left(x\right|s)$, $x\in \mathbb{R}$, of the second Painlevé equation $u_{xx}=xu+2u^3$ which are parameterized in terms of the monodromy data $s\equiv (s_1,s_2,s_3)\subset {\mathbb{C}}^3$ of the associated Flaschka–Newell system of rational differential equations. Our analysis describes the transition, as $x\to -\infty$, between the oscillatory power-like decay asymptotics for $\left|s_1\right|<1$ (Ablowitz–Segur) to the power-like growth behavior for $\left|s_1\right|=1$ (Hastings–McLeod) and from the latter to the singular oscillatory power-like growth for $\left|s_1\right|>1$ (Kapaev). It is shown that the transition asymptotics are of Boutroux type; that is, they are expressed in terms of Jacobi elliptic functions. As applications of our results we obtain asymptotics for the Airy kernel determinant $det(I-\gamma K_{\mathrm{Ai}}){|}_{L^2(x,\infty )}$ in a double scaling limit $x\to -\infty$, $\gamma \uparrow 1$, as well as asymptotics for the spectrum of $K_{\mathrm{Ai}}$.

The simplest case of the Langlands functoriality principle asserts the existence of the symmetric powers ${Sym}^n$ of a cuspidal representation of $GL\left(2\right)$ over the adèles of $F$, where $F$ is a number field. In 1978, Gelbart and Jacquet proved the existence of ${Sym}^2$. After this, progress was slow, eventually leading, through the work of Kim and Shahidi, to the existence of ${Sym}^3$ and ${Sym}^4$. In this series of articles we revisit this problem using recent progress in the deformation theory of modular Galois representations. As a consequence, our methods apply only to classical modular forms on a totally real number field; the present article proves the existence, in this “classical” case, of ${Sym}^6$ and ${Sym}^8$.