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Effective bounds for Faltings’s delta function

Jay Jorgenson, Jürg Kramer (2014)

Annales de la faculté des sciences de Toulouse Mathématiques

In his seminal paper on arithmetic surfaces Faltings introduced a new invariant associated to compact Riemann surfaces X , nowadays called Faltings’s delta function and here denoted by δ Fal ( X ) . For a given compact Riemann surface X of genus g X = g , the invariant δ Fal ( X ) is roughly given as minus the logarithm of the distance with respect to the Weil-Petersson metric of the point in the moduli space ℳ g of genus g curves determined by X to its boundary ∂ ℳ g . In this paper we begin by revisiting a formula derived in [14],...

Effective equidistribution of S-integral points on symmetric varieties

Yves Benoist, Hee Oh (2012)

Annales de l’institut Fourier

Let K be a global field of characteristic not 2. Let Z = H ∖ G be a symmetric variety defined over K and S a finite set of places of K . We obtain counting and equidistribution results for the S-integral points of Z . Our results are effective when K is a number field.

Elements of large order on varieties over prime finite fields

Mei-Chu Chang, Bryce Kerr, Igor E. Shparlinski, Umberto Zannier (2014)

Journal de Théorie des Nombres de Bordeaux

Let 𝒱 be a fixed algebraic variety defined by m polynomials in n variables with integer coefficients. We show that there exists a constant C ( 𝒱 ) such that for almost all primes p for all but at most C ( 𝒱 ) points on the reduction of 𝒱 modulo p at least one of the components has a large multiplicative order. This generalises several previous results and is a step towards a conjecture of B. Poonen.

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...

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