Cohomology of integer matrices and local-global divisibility on the torus

Marco Illengo

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Displaying similar documents to “Cohomology of integer matrices and local-global divisibility on the torus”

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 $f (x) \in ℤ [x]$ and a non-negative integer $ℓ$ , we may form a new monic degree $N$ polynomial $f_{ℓ} (x) \in ℤ [x]$ by raising each root of $f$ to the $ℓ$ 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 $(j, k) \mapsto Res (f_{j}, f_{k}) mod p^{m}$ . 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. ...

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 $ℚ (ζ_{p})$ , of degrees $2^{t - 1}$ and $2^{t}$ over $ℚ$ , 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.

Kloosterman sums for prime powers in -adic fields

Stanley J. Gurak (2009)

Journal de Théorie des Nombres de Bordeaux

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Let $K$ be a field of degree $n$ over $Q_{p}$ , the field of rational $p$ -adic numbers, say with residue degree $f$ , ramification index $e$ and differential exponent $d$ . Let $O$ be the ring of integers of $K$ and $P$ its unique prime ideal. The trace and norm maps for $K / Q_{p}$ are denoted $T r$ and $N$ , respectively. Fix $q = p^{r}$ , a power of a prime $p$ , and let $η$ be a numerical character defined modulo $q$ and of order $o (η)$ . The character $η$ extends to the ring of $p$ -adic integers $ℤ_{p}$ in the natural way; namely $η (u) = η (\tilde{u})$ , where $\tilde{u}$ denotes the residue...

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 $T = T_{L / K} : L \to K$ and the $O_{K}$ -ideal $T (O_{L}) \subseteq O_{K}$ . 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)....

On integers not of the form n - φ (n)

J. Browkin, A. Schinzel (1995)

Colloquium Mathematicae

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W. Sierpiński asked in 1959 (see [4], pp. 200-201, cf. [2]) whether there exist infinitely many positive integers not of the form n - φ(n), where φ is the Euler function. We answer this question in the affirmative by proving Theorem. None of the numbers $2^{k} \cdot 509203$ (k = 1, 2,...) is of the form n - φ(n).

On the Number of Partitions of an Integer in the $m$ -bonacci Base

Marcia Edson, Luca Q. Zamboni (2006)

Annales de l’institut Fourier

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For each $m \geq 2,$ we consider the $m$ -bonacci numbers defined by $F_{k} = 2^{k}$ for $0 \leq k \leq m - 1$ and $F_{k} = F_{k - 1} + F_{k - 2} + \dots + F_{k - m}$ for $k \geq m .$ When $m = 2,$ these are the usual Fibonacci numbers. Every positive integer $n$ may be expressed as a sum of distinct $m$ -bonacci numbers in one or more different ways. Let $R_{m} (n)$ be the number of partitions of $n$ as a sum of distinct $m$ -bonacci numbers. Using a theorem of Fine and Wilf, we obtain a formula for $R_{m} (n)$ involving sums of binomial coefficients modulo $2 .$ In addition we show that this formula may be used to determine the...

Displaying similar documents to “Cohomology of integer matrices and local-global divisibility on the torus”

Patterns and periodicity in a family of resultants

A generalization of Scholz’s reciprocity law

Kloosterman sums for prime powers in -adic fields

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

On integers not of the form n - φ (n)

On the Number of Partitions of an Integer in the m -bonacci Base

On the Number of Partitions of an Integer in the $m$ -bonacci Base