By Harald Niederreiter, Alina Ostafe, Daniel Panario, Arne Winterhof

This booklet collects the result of the workshops on purposes of Algebraic Curves and functions of Finite Fieldsat the RICAMin 2013. those workshops introduced jointly the main admired researchers within the region of finite fields and their purposes world wide, addressing previous and new difficulties on curves and different features of finite fields, with emphasis on their various purposes to many parts of natural and utilized arithmetic.

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**Example text**

5 ] generated by the above two sets of polynomials, but we can simplify the system of polynomial equations first. Taking for example the last of each set of polynomials, 256 64 ????1 := ℎ64 + ????116 + ????1 and ????2 := ℎ16 1 + ℎ1 + ????1 1 + ℎ1 + ????1 + ????1 , we find that ????3 := ????1 − ????24 = ℎ41 + ℎ1 + ????116 + ????14 + ????1 is an element of the ideal ????. Moreover, since ????2 = ????3 + ????34 and ????1 = ????3 + ????34 + ????316 , we can replace ????1 and ????2 by ????3 when generating the ideal ????. Also we can eliminate the variables ℎ???? altogether, since they can be expressed in terms of ????1 , ????2 , ????3 using the first five generators of ????.

This observation led Elkies to construct a number of recursively defined towers (????0 (???????? ))????≥2 of modular curves in [7, 8]. In [7] several models defined over ℚ of classical modular curves are given, while in [8] the reduction mod ????−1 of the Drinfeld modular tower ????0 (???????? )????≥2 was described. We consider the function field of ????0 (????????). We have ????(????0 (????????)) = ????(????0 (????), ????1 (????), . . , ????????−1 (????), ???????? (????)) . So we can think of ????(????0 (???????? )) as iteratively obtained from ????(????0 (????)) by adjoining the elements ????1 (????), ????2 (????), .

With this approach, finding explicit recursive towers turns out to be an easy task, once the corresponding modular polynomials are known. To illustrate this, we work out the equations for a few cases of Drinfeld modular towers. Good towers of function fields | 25 In the above Drinfeld modular theory was considered over the polynomial ring ???????? [????]. In the last section of the chapter, we study a variation where this ring is replaced by the coordinate ring of an elliptic curve. We illustrate the ideas by going through a specific example in detail.