Generalizing Kedlaya's order counting based on Miura Theory
نویسنده
چکیده
K Kedlaya proposed an method to count the number of Fq rational points in a hyper elliptic curve using the Leschetz xed points formula in Monsky Washinitzer Cohomology The method has been extended to super elliptic curves Gaudry and G urel immediately to characteristic two hyper elliptic curves and to Cab curves J Denef F Vercauteren Based on Miura theory which is associated with how a curve is expressed as an a ne variety this paper applies Kedlaya s method to so called strongly telescopic curves which are no longer plane curves and contain super elliptic curves as special cases Monsky Washinitzer Cohomology Let k Fqi for some i with q pm and p prime R W k the Witt ring of k and K the quotient eld of R Let A the coordinate ring of a smooth a ne variety over k A a smooth R algebra with A R k A and A the p adic completion of A Let vp denote the p adic valuation on R Fix x xn A whose reductions x xn generate A over k De nition Monsky Washinitzer The week completion Ay of A is the substring of A consisting of elements z X l ln al lnx l xn n such that l d z min l ln l vp al ln l c z for some d z Z and c z Let be the Ay module of di erent forms over K generated by symbols dx x Ay R K and subject to the relations d x y dx dy for x y Ay R K d xy xdy ydx for x y Ay R K and dx for x K We de ne the exterior derivative d r r by X l lrdxl dxlr d X d l lr dxl dxlr where l lr Ay the sum runs over l lr n and r denotes the r th exterior power of De nition Monsky Washinitzer In the sequence of homomorphisms
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ورودعنوان ژورنال:
- IACR Cryptology ePrint Archive
دوره 2004 شماره
صفحات -
تاریخ انتشار 2004