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

ISBN-10: 3110317885

ISBN-13: 9783110317886

This e-book collects the result of the workshops on functions of Algebraic Curves and functions of Finite Fieldsat the RICAMin 2013. those workshops introduced jointly the main favorite researchers within the region of finite fields and their functions world wide, addressing previous and new difficulties on curves and different facets of finite fields, with emphasis on their various functions to many components of natural and utilized arithmetic.

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

**Example text**

Consider the Artin–Schreier curve ???? with the affine equation ???????? −???? = ????????(????); it has genus (????−1)(???????? +1)/2. 7]. 3 Other nonexistence results To this date, the only nonexistence results about Newton polygons are for Jacobians of curves with automorphisms. Newton polygons of degree ???? covers of ℙ1 have been studied using techniques for exponential sums and Dwork cohomology. For example [26], when ???? = 2, there are no hyperelliptic curves of genus 2???? − 1 which are supersingular. More generally, there are conditions on the first slope of the Newton polygon of a degree ???? cover of the projective line [4].

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. Massimo Giulietti and Gábor Korchmáros, Automorphism groups of algebraic curves with ????rank zero, J. Lond. Math. Soc. 81 (2) (2010), 277–296. Darren Glass, The 2-ranks of hyperelliptic curves with extra automorphisms, Int.

3. 6 (???? = ????2 + ???? + 1, ???? = 2). The Drinfeld modular polynomial of level ????2 + ???? + 1 is given by ????????2 +????+1 (????, ????) = ????5 + ????5 + ????4 ????4 + (????2 + ???? + 1)(????4 ????2 + ????2 ????4 ) + (????2 + ???? + 1)(????4 ???? + ????????4 ) + ????3 (???? + 1)3 (????2 + ???? + 1)(????4 + ????4 ) + ????2 (???? + 1)2 (????2 + ???? + 1)????3 ????3 + (????2 + ????)(????2 + ???? + 1)(????3 + ???? + 1)(????3 + ????2 + 1)(????3 ????2 + ????2 ????3 ) + ????3 (???? + 1)3 (????2 + ???? + 1)(????3 ???? + ????????3 ) + ????6 (???? + 1)6 (????2 + ???? + 1)2 (????3 + ????3 ) + ????5 (???? + 1)5 (????2 + ???? + 1)(????4 + ???? + 1)????2 ????2 + ????6 (???? + 1)6 (????2 + ???? + 1)(????4 + ???? + 1)(????2 ???? + ????????2 ) + ????9 (???? + 1)9 (????2 + ???? + 1)3 (????2 + ????2 ) + ????11 (???? + 1)11 ???????? .

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

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