منابع مشابه
Prime Specialization in Genus 0
For a prime polynomial f(T ) ∈ Z[T ], a classical conjecture predicts how often f has prime values. For a finite field κ and a prime polynomial f(T ) ∈ κ[u][T ], the natural analogue of this conjecture (a prediction for how often f takes prime values on κ[u]) is not generally true when f(T ) is a polynomial in T p (p the characteristic of κ). The explanation rests on a new global obstruction wh...
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A classical conjecture predicts how often a polynomial in Z[T ] takes prime values. The natural analogous conjecture for prime values of a polynomial f(T ) ∈ κ[u][T ], where κ is a finite field, is false. The conjecture over κ[u] was modified in earlier work by introducing a correction factor that encodes unexpected periodicity of the Möbius function at the values of f on κ[u] when f ∈ κ[u][T ]...
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We continue the development of the theory of higher-genus Möbius periodicity that was studied in Part I for odd characteristic, now treating asymptotic questions and the case of characteristic 2. The extra difficulties in characteristic 2 are overcome via rigid geometry in characteristic 0. The results on Möbius periodicity in any positive characteristic are used to incorporate a correction fac...
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For counting points of jacobians of genus 2 curves over a large prime field, the best known approach is essentially an extension of Schoof’s genus 1 algorithm. We propose various practical improvements to this method and illustrate them with a large scale computation: we counted hundreds of curves, until one was found that is suitable for cryptographic use, with a state-ofthe-art security level...
متن کاملOn Gunning ’ s Prime Form in Genus 2
Using a classical generalization of Jacobi’s derivative formula, we give an explicit expression for Gunning’s prime form in genus 2 in terms of theta functions and their derivatives. Let X be a compact Riemann surface of genus g > 0. Let X̃ denote the universal cover of X, Π : X̃ → X denote the projection, and Γ be the group of covering transformations of X̃ over X. By a prime form for X wemean a ...
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ژورنال
عنوان ژورنال: Transactions of the American Mathematical Society
سال: 2008
ISSN: 0002-9947
DOI: 10.1090/s0002-9947-08-04283-9