We develop Milne’s theory of Lefschetz motives for general adequate equivalence relations and over a not necessarily algebraically closed base field. The corresponding categories turn out to enjoy ...all properties predicted by standard and less standard conjectures, in a stronger way: algebraic and numerical equivalences agree in this context. We also compute the Tannakian group associated to a Weil cohomology in a different and more conceptual way than Milne’s case-by-case approach.
This article is a survey of our work (joint with Davesh Maulik, Arul Shankar, and Salim Tayou) on arithmetic intersection theory on GSpin Shimura varieties with applications to abelian varieties, K3 ...surfaces, and the ordinary Hecke orbit conjecture.
Let p≥5 be a prime number, E/Q an elliptic curve with good supersingular reduction at p and K an imaginary quadratic field such that the root number of E over K is +1. When p is split in K, Darmon ...and Iovita formulated the plus and minus Iwasawa main conjectures for E over the anticyclotomic Zp-extension of K, and proved one-sided inclusion: an upper bound for plus and minus Selmer groups in terms of the associated p-adic L-functions. We generalize their results to two new settings:1.Under the assumption that p is split in K but without assuming ap(E)=0, we study Sprung-type Iwasawa main conjectures for abelian varieties of GL2-type, and prove an analogous inclusion.2.We formulate, relying on the recent work of the first named author with Kobayashi and Ota, plus and minus Iwasawa main conjectures for elliptic curves when p is inert in K, and prove an analogous inclusion.
Nous donnons une preuve nouvelle, élémentaire et rapide de la conjecture de Manin-Mumford. Nous obtenons même une forme renforcée en terme de topologie réelle plutôt que de Zariski, et n'utilisons ...guère cette dernière. La méthode part d'un résultat partiel sur une conjecture de Lang, qui permet d'appliquer un énoncé d'équidistribution dans Rn/Zn, lequel se réduit via le critère de Weyl au cas n=1, dû à Pólya et Vinogradov, vers 1915.
We provide a new, elementary and quick proof of Manin-Mumford conjecture. We even obtain a stronger variant involving the real topology instead of Zariski topology, and have no need of the latter. The method starts from a partial result of Serre towards a conjecture of Lang, allowing us to instantiate a statement about equidistribution in Rn/Zn, which can be reduced via Weyl's criterion to the case n=1, due to Pólya and Vinogradov, around 1915.
We prove that assuming the Colmez conjecture and the “no Siegel zeros” conjecture, the stable Faltings height of a CM abelian variety over a number field is less than or equal to the logarithm of the ...root discriminant of the field of definition of the abelian variety times an effective constant depending only on the dimension of the abelian variety. In view of the fact that the Colmez conjecture for abelian CM fields, the averaged Colmez conjecture, and the “no Siegel zeros” conjecture for CM fields with no complex quadratic subfields are already proved, we prove unconditional analogues of the result above. In addition, we also prove that the logarithm of the root discriminant of the field of everywhere good reduction of CM abelian varieties can be “small”.
We prove many non-generic cases of the Sato-Tate conjecture for abelian surfaces as formulated by Fité, Kedlaya, Rotger and Sutherland, using the potential automorphy theorems of Barnet-Lamb, Gee, ...Geraghty and Taylor.
Let O be a Dedekind domain whose field of fractions K is a global field. Let A be a finite-dimensional separable K-algebra and let Λ be an O-order in A. Suppose that X is a Λ-lattice such that K⊗OX ...is free of some finite rank n over A. Then X contains a (non-unique) free Λ-sublattice of rank n. The main result of the present article is to show there exists such a sublattice Y such that the generalised module index X:YO has explicit upper bounds with respect to division that are independent of X and can be chosen to satisfy certain conditions. We give examples of applications to the approximation of normal integral bases and strong Minkowski units, and to the Galois module structure of rational points over abelian varieties.
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