Analytic Ax–Schanuel for semi-abelian varieties and Nevanlinna theory

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<p>Let 𝐴 be a semi-abelian variety with an exponential map exp : Lie(𝐴) → 𝐴. The purpose of this paper is to explore Nevanlinna theory of the entire curve \widehat{exp} 𝑓 := (exp 𝑓, 𝑓) : 𝐂 → 𝐴 ×Lie(𝐴) associated with an entire curve 𝑓 : 𝐂 → Lie(𝐴). Firstly we give a Nevanlinna theoretic proof to the analytic Ax–Schanuel Theorem for semi-abelian varieties, which was proved by J. Ax 1972 in the case of formal power series (Ax–Schanuel Theorem). We assume some non-degeneracy condition for 𝑓 such that the elements of the vector-valued function 𝑓(𝑧) − 𝑓(0) ∈ Lie(𝐴) ≅ 𝐂𝑛 are 𝐐-linearly independent in the case of 𝐴 = (𝐂*)𝑛. Our proof is based on the Log Bloch–Ochiai Theorem and a key estimate which we show.</p><p> Our next aim is to establish a Second Main Theorem for \widehat{exp} 𝑓 and its 𝑘-jet lifts with truncated counting functions at level one. We give some applications to a problem of a type raised by S. Lang and the unicity. The results clarify a relationship between the problems of Ax–Schanuel type and Nevanlinna theory.</p>

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