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We study the -adic valuation of the values of normalised Hecke eigenforms attached to nonisotrivial elliptic curves defined over function fields of transcendence degree one over finite fields of characteristic . We derive upper bounds on the smallest attained valuation in terms of the minimal discriminant under a certain assumption on the function field, and provide examples to show that our estimates are optimal. As an application of our results, we prove the analogue of the degree conjecture unconditionally for strong Weil curves with square-free conductor defined over function fields satisfying the assumption mentioned above.
On obtient une nouvelle minoration du minimum essentiel en petite codimension sur les variétés abéliennes, sous une conjecture concernant leurs idéaux premiers ordinaires. Cette minoration, déjà connue dans le cas torique depuis les travaux d’Amoroso et David, est optimale « à près » en le degré de la sous-variété. La preuve suit la méthode des pentes et est basée sur les propriétés -adiques des points de torsion des variétés abéliennes.
We give a new lower bound for the essential minimum of subvarieties of abelian varieties with small codimension, under a conjecture about ordinary primes in abelian varieties. This lower bound, known in the toric case through the work of Amoroso and David, is best “up to an ” in the degree of the subvariety. The proof follows the slope method and is based on the -adic properties of torsion points in abelian varieties.
Sherman, Zelevinsky and Cerulli constructed canonically positive bases in cluster algebras associated to affine quivers having at most three vertices. Their constructions involve cluster monomials and normalized Chebyshev polynomials of the first kind evaluated at a certain “imaginary” element in the cluster algebra. Using this combinatorial description, it is possible to define for any affine quiver a set , which is conjectured to be the canonically positive basis of the acyclic cluster algebra .
In this article, we provide a geometric realization of the elements in in terms of the representation theory of . This is done by introducing an analogue of the Caldero–Chapoton cluster character, where the usual quiver Grassmannian is replaced by a constructible subset called the transverse quiver Grassmannian.
The main purpose of this paper is to provide explicit computations of the fundamental groups of several algebras. For this purpose, given a -algebra , we consider the category of all connected gradings of by a group and we study the relation between gradings and Galois coverings. This theoretical tool gives information about the fundamental group of , which allows its computation using complete lists of gradings.