On the values of Artin L-series at s=1 and annihilation of class groups

Hugo Castillo; Andrew Jones

Displaying similar documents to “On the values of Artin L-series at s=1 and annihilation of class groups”

Galois co-descent for étale wild kernels and capitulation

Manfred Kolster, Abbas Movahhedi (2000)

Annales de l'institut Fourier

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Let $F$ be a number field with ring of integers $o_{F}$ . For a fixed prime number $p$ and $i \geq 2$ the étale wild kernels $W K_{2 i - 2}^{\overset{´}{e} t} (F)$ are defined as kernels of certain localization maps on the $i$ -fold twist of the $p$ -adic étale cohomology groups of $spec o_{F} [\frac{1}{p}]$ . These groups are finite and coincide for $i = 2$ with the $p$ -part of the classical wild kernel $W K_{2} (F)$ . They play a role similar to the $p$ -part of the $p$ -class group of $F$ . For class groups, Galois co-descent in a cyclic extension $L / F$ is described by the ambiguous class formula given...

Quaternion extensions with restricted ramification

Peter Schmid (2014)

Acta Arithmetica

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In any normal number field having Q₈, the quaternion group of order 8, as Galois group over the rationals, at least two finite primes must ramify. The classical example by Dedekind of such a field is extraordinary in that it is totally real and only the primes 2 and 3 are ramified. In this note we describe in detail all Q₈-fields over the rationals where only two (finite) primes are ramified. We also show that, for any integer n>3 and any prime $p \equiv 1 (m o d 2^{n - 1})$ , there exist unique real and complex...

On the ring of $p$ -integers of a cyclic $p$ -extension over a number field

Humio Ichimura (2005)

Journal de Théorie des Nombres de Bordeaux

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Let $p$ be a prime number. A finite Galois extension $N / F$ of a number field $F$ with group $G$ has a normal $p$ -integral basis ( $p$ -NIB for short) when $𝒪_{N}^{'}$ is free of rank one over the group ring $𝒪_{F}^{'} [G]$ . Here, $𝒪_{F}^{'} = 𝒪_{F} [1 / p]$ is the ring of $p$ -integers of $F$ . Let $m = p^{e}$ be a power of $p$ and $N / F$ a cyclic extension of degree $m$ . When $ζ_{m} \in F^{\times}$ , we give a necessary and sufficient condition for $N / F$ to have a $p$ -NIB (Theorem 3). When $ζ_{m} \notin F^{\times}$ and $p ∤ [F (ζ_{m}) : F]$ , we show that $N / F$ has a $p$ -NIB if and only if $N (ζ_{m}) / F (ζ_{m})$ has a $p$ -NIB (Theorem 1). When $p$ divides $[F (ζ_{m}) : F]$ , we show that this...

On a noncommutative Iwasawa main conjecture for varieties over finite fields

Malte Witte (2014)

Journal of the European Mathematical Society

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We formulate and prove an analogue of the noncommutative Iwasawa main conjecture for $ℓ$ -adic Lie extensions of a separated scheme $X$ of finite type over a finite field of characteristic prime to $ℓ$ .

A note on p-adic valuations of Schenker sums

Piotr Miska (2015)

Colloquium Mathematicae

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A prime number p is called a Schenker prime if there exists n ∈ ℕ₊ such that p∤n and p|aₙ, where $a ₙ = \sum_{j = 0}^{n} (n! / j!) n^{j}$ is a so-called Schenker sum. T. Amdeberhan, D. Callan and V. Moll formulated two conjectures concerning p-adic valuations of aₙ when p is a Schenker prime. In particular, they conjectured that for each k ∈ ℕ₊ there exists a unique positive integer $n_{k} < 5^{k}$ such that $v ₅ (a_{m \cdot 5^{k} + n_{k}}) \geq k$ for each nonnegative integer m. We prove that for every k ∈ ℕ₊ the inequality v₅(aₙ) ≥ k has exactly one solution modulo $5^{k}$ . This...

Making sense of capitulation: reciprocal primes

David Folk (2016)

Acta Arithmetica

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Let ℓ be a rational prime, K be a number field that contains a primitive ℓth root of unity, L an abelian extension of K whose degree over K, [L:K], is divisible by ℓ, a prime ideal of K whose ideal class has order ℓ in the ideal class group of K, and $a_{}$ any generator of the principal ideal $^{ℓ}$ . We will call a prime ideal of K ’reciprocal to ’ if its Frobenius element generates $G a l (K (\sqrt[ℓ]{a_{}}) / K)$ for every choice of $a_{}$ . We then show that becomes principal in L if and only if every reciprocal prime is not...

On the ideal class groups of the maximal real subfields of number fields with all roots of unity

Masato Kurihara (1999)

Journal of the European Mathematical Society

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In this paper, for a totally real number field $k$ we show the ideal class group of $k {(\cup_{n > 0} μ_{n})}^{+}$ is trivial. We also study the $p$ -component of the ideal class group of the cyclotomic $𝐙_{p}$ -extension.

Truncatable primes and unavoidable sets of divisors

Artūras Dubickas (2006)

Acta Mathematica Universitatis Ostraviensis

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We are interested whether there is a nonnegative integer $u_{0}$ and an infinite sequence of digits $u_{1}, u_{2}, u_{3}, \dots$ in base $b$ such that the numbers $u_{0} b^{n} + u_{1} b^{n - 1} + \dots + u_{n - 1} b + u_{n},$ where $n = 0, 1, 2, \dots,$ are all prime or at least do not have prime divisors in a finite set of prime numbers $S .$ If any such sequence contains infinitely many elements divisible by at least one prime number $p \in S,$ then we call the set $S$ unavoidable with respect to $b$ . It was proved earlier that unavoidable sets in base $b$ exist if $b \in {2, 3, 4, 6},$ and that no unavoidable set exists in base $b = 5 .$ Now,...

Some remarks on Hilbert-Speiser and Leopoldt fields of given type

James E. Carter (2007)

Colloquium Mathematicae

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Let p be a rational prime, G a group of order p, and K a number field containing a primitive pth root of unity. We show that every tamely ramified Galois extension of K with Galois group isomorphic to G has a normal integral basis if and only if for every Galois extension L/K with Galois group isomorphic to G, the ring of integers $O_{L}$ in L is free as a module over the associated order $_{L / K}$ . We also give examples, some of which show that this result can still hold without the assumption that...

A note on free pro- $p$ -extensions of algebraic number fields

Masakazu Yamagishi (1993)

Journal de théorie des nombres de Bordeaux

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For an algebraic number field $k$ and a prime $p$ , define the number $ρ$ to be the maximal number $d$ such that there exists a Galois extension of $k$ whose Galois group is a free pro- $p$ -group of rank $d$ . The Leopoldt conjecture implies $1 \leq ρ \leq r_{2} + 1$ , ( $r_{2}$ denotes the number of complex places of $k$ ). Some examples of $k$ and $p$ with $ρ = r_{2} + 1$ have been known so far. In this note, the invariant $ρ$ is studied, and among other things some examples with $ρ < r_{2} + 1$ are given.

Explicit Selmer groups for cyclic covers of ℙ¹

Michael Stoll, Ronald van Luijk (2013)

Acta Arithmetica

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For any abelian variety J over a global field k and an isogeny ϕ: J → J, the Selmer group $S e l^{ϕ} (J, k)$ is a subgroup of the Galois cohomology group $H ¹ (G a l (k^{s} / k), J [ϕ])$ , defined in terms of local data. When J is the Jacobian of a cyclic cover of ℙ¹ of prime degree p, the Selmer group has a quotient by a subgroup of order at most p that is isomorphic to the ‘fake Selmer group’, whose definition is more amenable to explicit computations. In this paper we define in the same setting the ‘explicit Selmer group’, which...

On sets which contain a qth power residue for almost all prime modules

Mariusz Ska/lba (2005)

Colloquium Mathematicae

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A classical theorem of M. Fried [2] asserts that if non-zero integers $β ₁, . . ., β_{l}$ have the property that for each prime number p there exists a quadratic residue $β_{j}$ mod p then a certain product of an odd number of them is a square. We provide generalizations for power residues of degree n in two cases: 1) n is a prime, 2) n is a power of an odd prime. The proofs involve some combinatorial properties of finite Abelian groups and arithmetic results of [3].