Displaying similar documents to “Kloosterman sums for prime powers in P-adic fields”

On the linear independence of p -adic L -functions modulo p

Bruno Anglès, Gabriele Ranieri (2010)

Annales de l’institut Fourier

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Let p 3 be a prime. Let n such that n 1 , let χ 1 , ... , χ n be characters of conductor d not divided by p and let ω be the Teichmüller character. For all i between 1 and n , for all j between 0 and ( p - 3 ) / 2 , set θ i , j = χ i ω 2 j + 1 if χ i is odd ; χ i ω 2 j if χ i is even . Let K = p ( χ 1 , ... , χ n ) and let π be a prime of the valuation ring 𝒪 K of K . For all i , j let f ( T , θ i , j ) be the Iwasawa series associated to θ i , j and f ( T , θ i , j ) ¯ its reduction modulo ( π ) . Finally let 𝔽 p ¯ be an algebraic closure of 𝔽 p . Our main result is that if the characters χ i are all distinct modulo ( π ) , then 1 and the series...

A generalization of Scholz’s reciprocity law

Mark Budden, Jeremiah Eisenmenger, Jonathan Kish (2007)

Journal de Théorie des Nombres de Bordeaux

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We provide a generalization of Scholz’s reciprocity law using the subfields K 2 t - 1 and K 2 t of ( ζ p ) , of degrees 2 t - 1 and 2 t over , respectively. The proof requires a particular choice of primitive element for K 2 t over K 2 t - 1 and is based upon the splitting of the cyclotomic polynomial Φ p ( x ) over the subfields.

On the trace of the ring of integers of an abelian number field

Kurt Girstmair (1992)

Acta Arithmetica

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Let K, L be algebraic number fields with K ⊆ L, and O K , O L their respective rings of integers. We consider the trace map T = T L / K : L K and the O K -ideal T ( O L ) O K . By I(L/K) we denote the group indexof T ( O L ) in O K (i.e., the norm of T ( O L ) over ℚ). It seems to be difficult to determine I(L/K) in the general case. If K and L are absolutely abelian number fields, however, we obtain a fairly explicit description of the number I(L/K). This is a consequence of our description of the Galois module structure of T ( O L ) (Theorem 1)....

A system of simultaneous congruences arising from trinomial exponential sums

Todd Cochrane, Jeremy Coffelt, Christopher Pinner (2006)

Journal de Théorie des Nombres de Bordeaux

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For a prime p and positive integers < k < h < p with d = ( h , k , , p - 1 ) , we show that M , the number of simultaneous solutions x , y , z , w in p * to x h + y h = z h + w h , x k + y k = z k + w k , x + y = z + w , satisfies M 3 d 2 ( p - 1 ) 2 + 25 h k ( p - 1 ) . When h k = o ( p d 2 ) we obtain a precise asymptotic count on M . This leads to the new twisted exponential sum bound x = 1 p - 1 χ ( x ) e 2 π i f ( x ) / p 3 1 4 d 1 2 p 7 8 + 5 h k 1 4 p 5 8 , for trinomials f = a x h + b x k + c x , and to results on the average size of such sums.

Patterns and periodicity in a family of resultants

Kevin G. Hare, David McKinnon, Christopher D. Sinclair (2009)

Journal de Théorie des Nombres de Bordeaux

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Given a monic degree N polynomial f ( x ) [ x ] and a non-negative integer , we may form a new monic degree N polynomial f ( x ) [ x ] by raising each root of f to the th power. We generalize a lemma of Dobrowolski to show that if m < n and p is prime then p N ( m + 1 ) divides the resultant of f p m and f p n . We then consider the function ( j , k ) Res ( f j , f k ) mod p m . We show that for fixed p and m that this function is periodic in both j and k , and exhibits high levels of symmetry. Some discussion of its structure as a union of lattices is also given. ...

Weber’s class number problem in the cyclotomic 2 -extension of , II

Takashi Fukuda, Keiichi Komatsu (2010)

Journal de Théorie des Nombres de Bordeaux

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Let h n denote the class number of n -th layer of the cyclotomic 2 -extension of . Weber proved that h n ( n 1 ) is odd and Horie proved that h n ( n 1 ) is not divisible by a prime number satisfying 3 , 5 ( mod 8 ) . In a previous paper, the authors showed that h n ( n 1 ) is not divisible by a prime number less than 10 7 . In this paper, by investigating properties of a special unit more precisely, we show that h n ( n 1 ) is not divisible by a prime number less than 1 . 2 · 10 8 . Our argument also leads to the conclusion that h n ( n 1 ) is not divisible by...