Displaying similar documents to “Sieve methods for varieties over finite fields and arithmetic schemes”

The probability that a complete intersection is smooth

Alina Bucur, Kiran S. Kedlaya (2012)

Journal de Théorie des Nombres de Bordeaux

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Given a smooth subscheme of a projective space over a finite field, we compute the probability that its intersection with a fixed number of hypersurface sections of large degree is smooth of the expected dimension. This generalizes the case of a single hypersurface, due to Poonen. We use this result to give a probabilistic model for the number of rational points of such a complete intersection. A somewhat surprising corollary is that the number of rational points on a random smooth intersection...

Counting monic irreducible polynomials P in 𝔽 q [ X ] for which order of X ( mod P ) is odd

Christian Ballot (2007)

Journal de Théorie des Nombres de Bordeaux

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Hasse showed the existence and computed the Dirichlet density of the set of primes p for which the order of 2 ( mod p ) is odd; it is 7 / 24 . Here we mimic successfully Hasse’s method to compute the density δ q of monic irreducibles P in 𝔽 q [ X ] for which the order of X ( mod P ) is odd. But on the way, we are also led to a new and elementary proof of these densities. More observations are made, and averages are considered, in particular, an average of the δ p ’s as p varies through all rational primes.

On the generation of the coefficient field of a newform by a single Hecke eigenvalue

Koopa Tak-Lun Koo, William Stein, Gabor Wiese (2008)

Journal de Théorie des Nombres de Bordeaux

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Let f be a non-CM newform of weight k ≥ 2 . Let L be a subfield of the coefficient field of  f . We completely settle the question of the density of the set of primes p such that the p -th coefficient of  f generates the field  L . This density is determined by the inner twists of  f . As a particular case, we obtain that in the absence of nontrivial inner twists, the density is  1 for L equal to the whole coefficient field. We also present some new data on reducibility of Hecke polynomials, which...

Restriction theory of the Selberg sieve, with applications

Ben Green, Terence Tao (2006)

Journal de Théorie des Nombres de Bordeaux

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The Selberg sieve provides majorants for certain arithmetic sequences, such as the primes and the twin primes. We prove an L 2 – L p restriction theorem for majorants of this type. An immediate application is to the estimation of exponential sums over prime k -tuples. Let a 1 , ⋯ , a k and b 1 , ⋯ , b k be positive integers. Write h ( θ ) : = ∑ n ∈ X e ( n θ ) , where X is the set of all n ≤ N such that the numbers a 1 n + b 1 , ⋯ , a k n + b k are all prime. We obtain upper bounds for ∥ h ∥ L p ( 𝕋 ) , p > 2 , which are (conditionally on the Hardy-Littlewood prime tuple conjecture) of the correct...

Density of rational points on cyclic covers of ℙ n

Ritabrata Munshi (2009)

Journal de Théorie des Nombres de Bordeaux

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We obtain upper bound for the density of rational points on the cyclic covers of ℙ n . As n → ∞ our estimate tends to the conjectural bound of Serre.