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

ISBN-10: 3110317885

ISBN-13: 9783110317886

This publication collects the result of the workshops on purposes of Algebraic Curves and purposes of Finite Fieldsat the RICAMin 2013. those workshops introduced jointly the main well known researchers within the zone of finite fields and their purposes all over the world, addressing outdated and new difficulties on curves and different elements of finite fields, with emphasis on their assorted functions to many parts of natural and utilized arithmetic.

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**Additional resources for Algebraic Curves and Finite Fields: Cryptography and Other Applications**

**Sample text**

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.

7) As noted before, the parameter ???? plays the same role as ????0 from Section 2. Similarly ???? plays the same role as ????1 and the polynomial ????(????, ????, ????) can be seen as an analogue of 38 | Alp Bassa, Peter Beelen, and Nhut Nguyen a Drinfeld modular polynomial ???????? (????, ????). For completeness, let us note that whereas ???? was a polynomial before, its role is now taken by the ideal ⟨???? + 1, ???? + 1⟩ ⊂ ????, which implicitly played a role in the construction of the isogeny ????. 3 Obtaining a tower Just as for the towers from Section 2, we need a quadratic extension of the constant field in order to obtain many rational places.

Hardy. 5 in [3] for more details). The Rogers–Ramanujan continued fraction can be seen as a modular function for the full modular group ????(5) and defines a uniformizing element of the function field ℚ(????(5)). 2) by extending the first function field of that tower to the function field of ????(5). Also by direct computation one sees that the extension ℚ(????5 )(????, ????)/ℚ(????5 )(????) is a Galois extension (it is in fact the Galois closure of ℚ(????5 )(????, ????)/ℚ(????5 )(????)). For any prime number ???? different from 5 the curves have good reduction, meaning that we may reduce the equations modulo such primes ????.

### Algebraic Curves and Finite Fields: Cryptography and Other Applications by Harald Niederreiter, Alina Ostafe, Daniel Panario, Arne Winterhof

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