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Elliptic curves with ℚ ( ℰ [ 3 ] ) = ℚ ( ζ 3 ) and counterexamples to local-global divisibility by 9

Laura Paladino (2010)

Journal de Théorie des Nombres de Bordeaux

We give a family ℱ h , β of elliptic curves, depending on two nonzero rational parameters β and h , such that the following statement holds: let ℰ be an elliptic curve and let ℰ [ 3 ] be its 3-torsion subgroup. This group verifies ℚ ( ℰ [ 3 ] ) = ℚ ( ζ 3 ) if and only if ℰ belongs to ℱ h , β .Furthermore, we consider the problem of the local-global divisibility by 9 for points of elliptic curves. The number 9 is one of the few exceptional powers of primes, for which an answer to the local-global divisibility is unknown in the case of such...

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.

Extension of Estermann’s theorem to Euler products associated to a multivariate polynomial

Ludovic Delabarre (2013)

Bulletin de la Société Mathématique de France

Given a multivariate polynomial h X 1 , ⋯ , X n with integral coefficients verifying an hypothesis of analytic regularity (and satisfying h ( 0 ) = 1 ), we determine the maximal domain of meromorphy of the Euler product ∏ p prime h p - s 1 , ⋯ , p - s n and the natural boundary is precisely described when it exists. In this way we extend a well known result for one variable polynomials due to Estermann from 1928. As an application, we calculate the natural boundary of the multivariate Euler products associated to a family of toric varieties.

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