Generalizing the conjecture of the algebraic independence of $e$ and $\pi$, Schanuel's conjecture (1966) claims that the transcendence degree of $\alpha_1,\dots,\alpha_n,e^{\alpha_1},\dots,e^{\alpha_n}$ is at least $n$ for $\mathbf{Q}$-linearly independent (non-degeneracy condition) complex numbers $\alpha_j \in \mathbf{C}$ ($1 \leq j \leq n$). J. Ax (1971/72) proved a formal function analogue of the conjecture for $(\alpha_j) = ((f_j(t),(e^{f_j(t)}))$ and also dealt with the case of semi-abelian varieties.

In this talk we discuss the problem for an analytic $f:R \to \text{Lie}(A)$ with exponential map $\exp_A: \text{Lie}(A) \to A$ of a semi-abelian variety $A$, where $R$ may be $\mathbf{C}$, an affine algebraic curve, a punctured disk, a parabolic Riemann surface, etc, to say, $R=\mathbf{C}$ here for simplicity. We study the value distribution of $\widehat{\exp}_Af:= (f,\exp_A f):\mathbf{C} \to \text{Lie}(A) \times A$ from the Nevalinna theoretic viewpoint, giving another proof of Ax-Schanuel theorem in the analytic case, such that $\text{tr.deg}_\mathbf{C}\widehat{\exp}_Af \geq \dim(A) +1$.