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A fundamental problem in the theory of curves over finite fields is to determine the sets M( g):= { N ∈ ℕ / there is a curve over $$ \mathbb{F}_q $$ of genus g with exactly N rational points}. A complete description of M( g) is out of reach. So far, mostly bounds for the numbers N( g):= max M( g) have been studied. In particular, Elkies et al. proved that there is a constant γ > 0 such that for any g ≥ 0 there is some N ∈ M( g) with N ≥ γ g. This implies that lim inf N( g)/ g > 0, and solves a long-standing problem by Serre. We extend the result of Elkies et al. substantially and show that there are constants α, β > 0 such that for all g ≥ 0, the whole interval [0, α g − β] ∩ ℕ is contained in M( g). [ABSTRACT FROM AUTHOR]

The aim of this paper is twofold: Firstly, we generalize well-known formulas for ramification and different exponents in cyclic extensions of function fields over a field K (due to H. Hasse) to extensions E = F( y), where y satisfies an equation of the form f( y) = u · g( y) with polynomials f( y), g( y) ∈ K[ y] and u ∈ F. This result depends essentially on Abhyankar’s Lemma which gives information about ramification in a compositum E = E1 E2 of finite extensions E1, E2 over a function field F. Abhyankar’s Lemma does not hold if both extensions E1/ F and E2/ F are wildly ramified. Our second objective is a generalization of Abhyankar’s Lemma if E1/ F and E2/ F are cyclic extensions of degree p = char( K). This result may be useful for the study of wild towers of function fields over finite fields. [ABSTRACT FROM AUTHOR]