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G. Miller in his seminal paper from the mid 1970s has proven that the problem of factoring integers reduces to computing Euler's totient function φ under the Extended Riemann Hypothesis. We show, unconditionally, that such a deterministic polynomial reduction exists for a large class of integers.

Rank of Elliptic Curves and Exponential Sums. Let y2 = x 3 + a(t)x + b(t) be the equation of an elliptic curve over Q, where a(t) and b(t) are polynomials over Z. We give an upper bound on average of the rank of such curves when t varies over Y under the assumption that the L-function attached to these curves satisfies classical assumptions. The proof is based on estimates of exponential sums, over varieties, which are treated by Deligne's theorem.

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