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Equations in simple matrix groups: algebra, geometry, arithmetic, dynamics

Tatiana Bandman, Shelly Garion, Boris Kunyavskiĭ (2014)

Open Mathematics

We present a survey of results on word equations in simple groups, as well as their analogues and generalizations, which were obtained over the past decade using various methods: group-theoretic and coming from algebraic and arithmetic geometry, number theory, dynamical systems and computer algebra. Our focus is on interrelations of these machineries which led to numerous spectacular achievements, including solutions of several long-standing problems.

Evaluation of the convolution sum involving the sum of divisors function for 22, 44 and 52

Ebénézer Ntienjem (2017)

Open Mathematics

The convolution sum, [...] ∑(l,m)∈N02αl+βm=nσ(l)σ(m), ∑ ( l , m ) ∈ ℕ 0 2 α l + β m = n σ ( l ) σ ( m ) , where αβ = 22, 44, 52, is evaluated for all natural numbers n. Modular forms are used to achieve these evaluations. Since the modular space of level 22 is contained in that of level 44, we almost completely use the basis elements of the modular space of level 44 to carry out the evaluation of the convolution sums for αβ = 22. We then use these convolution sums to determine formulae for the number of representations of a positive integer by...

Excellent connections in the motives of quadrics

Alexander Vishik (2011)

Annales scientifiques de l'École Normale Supérieure

In this article we prove the conjecture claiming that the motive of a real quadric is the “most decomposable” among anisotropic quadrics of given dimension over all fields. This imposes severe restrictions on the motive of arbitrary anisotropic quadric. As a corollary we estimate from below the rank of indecomposable direct summand in the motive of a quadric in terms of its dimension. This generalizes the well-known Binary Motive Theorem. Moreover, we have the description of the Tate motives involved....

Existence et équidistribution des matrices de dénominateur n dans les groupes unitaires et orthogonaux

Antonin Guilloux (2008)

Annales de l’institut Fourier

Soit G un groupe défini sur les rationnels, simplement connexe, ℚ -quasisimple et compact sur ℝ . On étudie des suites de sous-ensembles des points rationnels de G définis par des conditions sur leur projection dans le groupe des adèles finies de G . Nous montrons dans ce cadre un résultat d’équirépartition vers la probabilité de Haar sur le groupe des points réels. On utilise pour cela des propriétés de mélange de l’action du groupe des points adéliques G ( 𝔸 ) sur l’espace L 2 ( G ( 𝔸 ) / G ( ℚ ) ) . Pour illustrer ce résultat,...

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