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

This booklet collects the result of the workshops on functions of Algebraic Curves and purposes of Finite Fieldsat the RICAMin 2013. those workshops introduced jointly the main famous researchers within the quarter of finite fields and their purposes all over the world, addressing previous and new difficulties on curves and different points of finite fields, with emphasis on their various functions to many components of natural and utilized arithmetic.

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Math. 24 (1974), 95–119. Frans Oort, Hyperelliptic supersingular curves, Arithmetic algebraic geometry (Texel, 1989), Progr. Math. 89, Birkhäuser Boston, Boston, MA, 1991, pp. 247–284. Frans Oort, Problems from the Workshop on Automorphisms of Curves, Rend. Sem. Mat. Univ. Padova 113 (2005), 129–177. Rachel Pries and Hui June Zhu, The ????-rank stratification of Artin-Schreier curves, Ann. Inst. Fourier (Grenoble) 62 (2012), 707–726. Jasper Scholten and Hui June Zhu, Hyperelliptic curves in characteristic 2, Int.

Math. (2) 176 (2012), 589–635. Richard M. Crew, Etale ????-covers in characteristic ????, Compositio Math. 52 (1984), 31–45. A. J. de Jong and F. Oort, Purity of the stratification by Newton polygons, J. Amer. Math. Soc. 13 (2000), 209–241. P. Deligne and D. Mumford, The irreducibility of the space of curves of given genus, Inst. Hautes Études Sci. Publ. Math. 36 (1969), 75–109. Carel Faber and Gerard van der Geer, Complete subvarieties of moduli spaces and the Prym map, J. Reine Angew. Math. 573 (2004), 117–137.

This modular curve is known to have genus zero and to be irreducible, but not absolutely irreducible, see [14]. There it is also shown that the number of components is equal to the class number ℎ???? , over which extension field these components are defined and how the Galois group of this extension acts on the components. In our case we obtain that there are five components defined over ????32 and that the Frobenius map of ????32 /????2 acts transitively on these five components. One such component is determined by the following relation between ???? and ????2 : ????213 + (????5 ???? + ????14 )????212 + (????4 ????2 + ????19 ???? + ????7 )????211 + (????9 ????3 + ????18 ????2 + ????9 ???? + ????21 )????210 + (????10 ????4 + ????21 ????3 + ????16 ????2 + ????18 ???? + ????8 )????29 + (????15 ????5 + ????29 ????4 + ????10 ????3 + ????27 ????2 + ????25 ???? + ????8 )????28 + (????6 + ????28 ????5 + ????6 ????4 + ????11 ????3 + ????6 ????2 + ????28 ???? + ????9 )????27 + (????5 ????7 + ????23 ????6 + ????2 ????5 + ????15 ????4 + ????12 ????3 + ????4 ????2 + ????6 ???? + ????25 )????26 + (????4 ????8 + ????30 ????7 + ????18 ????6 + ????3 ????5 + ????15 ????4 + ????12 ????3 + ????23 ????2 + ????29 ???? + ????10 )????25 + (????9 ????9 + ????25 ????8 + ????8 ????7 + ????????6 + ????7 ????5 + ????25 ????4 + ????23 ????3 + ????15 ????2 + ???????? + ????26 )????24 + (????4 ????10 + ????27 ????9 + ????15 ????8 + ????11 ????7 + ????5 ????6 + ????26 ????5 + ????18 ????4 + ????9 ????3 + ????11 ????2 + ????30 ????)????23 + (????9 ????11 + ????30 ????10 + ????10 ????9 + ????15 ????8 + ????12 ????7 + ????6 ????6 + ????2 ????5 + ????26 ????4 + ????15 ????3 + ????6 ????2 + ????13 ???? + ????30 )????22 36 | Alp Bassa, Peter Beelen, and Nhut Nguyen + (????10 ????12 + ????16 ????11 + ????4 ????10 + ????12 ????9 + ????18 ????8 + ????28 ????7 + ????2 ????6 + ????9 ????5 + ????3 ????4 + ????8 ????3 + ????10 ????2 + ????17 ????)????2 + ????15 ????13 + ????5 ????12 + ????24 ????11 + ????4 ????10 + ????11 ????9 + ????8 ????8 + ????12 ????7 + ????27 ????6 + ????5 + ????23 ????4 + ????19 ????3 + ????8 ????2 + ????24 ???? + 1 , with ????5 + ????2 + 1 = 0.

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