In this article we study an abelian analogue of Schanuel's conjecture. This conjecture falls in the realm of the generalised period conjecture of Y. André. As shown by C. Bertolin, the generalised period conjecture includes Schanuel's conjecture as a special case. Extending methods of Bertolin, it can be shown that the abelian analogue of Schanuel's conjecture we consider, also follows from André's conjecture. C. Cheng et al. showed that the classical Schanuel's conjecture implies the algebraic independence of the values of the iterated exponential function and the values of the iterated logarithmic function, answering a question of M. Waldschmidt. We then investigate a similar question in the setup of abelian varieties.

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Let k = 12m(k) + s ≥ 12 for s ∈ {0, 4, 6, 8, 10, 14}, be an even integer and f be a normalised modular form of weight k with real Fourier coefficients, written asUnder suitable conditions on aj (rectifying an earlier result of Getz), we show that all the zeros of f , in the standard fundamental domain for the action of SL(2, Z) on the upper half plane, lies on the arc A := e iθ : π 2 ≤ θ ≤ 2π 3 . Further, extending a result of Nozaki, we show that for certain family {f k } k of normalised modular forms, the zeros of f k and f k+12 interlace on A • := e iθ : π 2 < θ < 2π 3 .6 1728 .

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