We discuss functional and number theoretic extensions of Schanuel's conjecture, with special emphasis on the study of elliptic integrals of the third kind.
Schanuel's conjecture [La] on the layman's exponential function can be viewed as a measure of the defect between an algebraic and a linear dimension. Its functional analogue, be it in Ax's original setting [Ax1], Coleman's [Co], or Zilber's geometric interpretation [Zi], certainly gives ground to this view-point.
The same remark applies to the elliptic version of the conjecture, and to its functional analogue, as studied by Brownawell and Kubota [BK], and by J. Kirby [K1]. Here, the elliptic curve under consideration is constant. In the same spirit, we discuss in the first section of this note Ax's general theorem [Ax2] on the exponential map on a constant semiabelian variety G, where transcendence degrees are controlled by the (linear) dimension of a certain “hull”. We obtain a similar statement for the universal vectorial extension of G, and refer to the recent work of J. Kirby [K2, K3] for further generalizations of Ax's theorem, involving arbitrary differential fields, multiplicative parametrizations, and uniformity questions.
The naïve number-theoretic analogues of these functional results, however, are clearly false. The first counterexample which comes to mind is provided by periods: Riemann-Legendre relations are quadratic, and cannot be tracked back to hulls of the above type.