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 ${}^{\ell }$. We will call a prime ideal of K ’reciprocal to ’ if its Frobenius element generates $Gal(K\left(\sqrt[\ell ]{a_{}}\right)/K)$ for every choice of $a_{}$. We then show that becomes principal in L if and only if every reciprocal prime is not a norm inside...

Let $X$ be the wonderful compactification of a connected adjoint semisimple group $G$ defined over a number field $K$. We prove Manin’s conjecture on the asymptotic (as $T\to \infty$) of the number of $K$-rational points of $X$ of height less than $T$, and give an explicit construction of a measure on $X\left(𝔸\right)$, generalizing Peyre’s measure, which describes the asymptotic distribution of the rational points $𝐆\left(K\right)$ on $X\left(𝔸\right)$. Our approach is based on the mixing property of $L^2(𝐆\left(K\right)\setminus 𝐆\left(𝔸\right))$ which we obtain with a rate of convergence.

The Manin conjecture is established for a split singular del Pezzo surface of degree four, with singularity type ${\mathbf{A}}_4$.

We prove Manin’s conjecture for a del Pezzo surface of degree six which has one singularity of type ${\mathbf{A}}_2$. Moreover, we achieve a meromorphic continuation and explicit expression of the associated height zeta function.

The class of linear (resp. quadratic) mappings over a commutative ring is determined by a set of equation-type relations. For the class of homogeneous polynomial mappings of degree m ≥ 3 it is so over a field, and over a ring there exists a smallest equationally definable class of mappings containing the preceding one. It is proved that generating relations determining that class can be chosen to be strong relations (that is, of the same form over all commutative rings) if{f} m ≤ 5. These relations...

Ce travail consiste à étudier les comportements des marches sur les arbres homogènes suivant la suite engendrée par une substitution. Dans la première partie, on étudie d’abord les marches sans orientation sur $\mathbb{Z}$ et on détermine complètement, d’après les propriétés combinatoires de la substitution, les conditions assurant que les marches sont bornées, récurrentes ou transientes. Comme corollaire, on obtient le comportement asymptotique des sommes partielles des coefficients de la suite substitutive....

The Markoff conjecture states that given a positive integer $c$, there is at most one triple $(a,b,c)$ of positive integers with $a\le b\le c$ that satisfies the equation $a^2+b^2+c^2=3abc$. The conjecture is known to be true when $c$ is a prime power or two times a prime power. We present an elementary proof of this result. We also show that if in the class group of forms of discriminant $d=9c^2-4$, every ambiguous form in the principal genus corresponds to a divisor of $3c-2$, then the conjecture is true. As a result, we obtain criteria in terms of...

The starting point of this note is the observation that the local condition used in the notion of a Hilbert-symbol equivalence and a quaternion-symbol equivalence — once it is expressed in terms of the Witt invariant — admits a natural generalisation. In this paper we show that for global function fields as well as the formally real function fields over a real closed field all the resulting equivalences coincide.

We give a graph theoretic interpretation of $r$-Lah numbers, namely, we show that the $r$-Lah number ${⌊\genfrac{}{}{0pt}{}{n}{k}⌋}_r$ counting the number of $r$-partitions of an $(n+r)$-element set into $k+r$ ordered blocks is just equal to the number of matchings consisting of $n-k$ edges in the complete bipartite graph with partite sets of cardinality $n$ and $n+2r-1$ ($0\le k\le n$, $r\ge 1$). We present five independent proofs including a direct, bijective one. Finally, we close our work with a similar result for $r$-Stirling numbers of the second kind.

In this paper, we study groupoid actions acting on arithmetic functions. In particular, we are interested in the cases where groupoids are generated by directed graphs. By defining an injective map α from the graph groupoid G of a directed graph G to the algebra A of all arithmetic functions, we establish a corresponding subalgebra AG = C*[α(G)]︀ of A. We construct a suitable representation of AG, determined both by G and by an arbitrarily fixed prime p. And then based on this representation, we...