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Register a new userThe Shafarevich conjecture for hypersurfaces in abelian varieties
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Faltings proved that there are finitely many abelian varieties of genus $g$ of a number field $K$, with good reduction outside a finite set of primes $S$. Fixing one of these abelian varieties $A$, we prove that there are finitely many smooth hypersurfaces in $A$, with good reduction outside $S$, representing a given ample class in the Neron-Severi group of $A$, up to translation, as long as the dimension of $A$ is at least $4$. Our approach builds on the approach of arXiv:1807.02721 which studies $p$-adic variations of Hodge structure to turn finiteness results for $p$-adic Galois representations into geometric finiteness statements. A key new ingredient is an approach to proving big monodromy for the variations of Hodge structure arising from the middle cohomology of these hypersurfaces using the Tannakian theory of sheaf convolution on abelian varieties.
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The Shafarevich-Tate group $W (mathscr{A})$ measures the failure of the Hasse principle for an abelian variety $mathscr{A}$. Using a correspondence between the abelian varieties and the higher dimensional non-commutative tori, we prove that $W (mathscr{A})cong Cl~(Lambda)oplus Cl~(Lambda)$ or $W (mathscr{A})cong left(mathbf{Z}/2^kmathbf{Z}right) oplus Cl_{~mathbf{odd}}~(Lambda)oplus Cl_{~mathbf{odd}}~(Lambda)$, where $Cl~(Lambda)$ is the ideal class group of a ring $Lambda$ associated to the K-theory of the non-commutative tori and $2^k $ divides the order of $Cl~(Lambda)$. The case of elliptic curves with complex multiplication is considered in detail.
We show that very general hypersurfaces in odd-dimensional simplicial projective toric varieties verifying a certain combinatorial property satisfy the Hodge conjecture (these include projective spaces). This gives a connection between the Oda conjecture and Hodge conjecture. We also give an explicit criterion which depends on the degree for very general hypersurfaces for the combinatorial condition to be verified.
We give a characterizaton of smooth ample Hypersurfaces in Abelian Varieties and also describe an irreducible connected component of their moduli space: it consists of the Hypersurfaces of a given polarization type, plus the iterated univariate coverings of normal type (of the same polarization type). The above manifolds yield also a connected component of the open set of Teichmuller space consisting of Kahler complex structures.
The Hecke orbit conjecture asserts that every prime-to-$p$ Hecke orbit in a Shimura variety is dense in the central leaf containing it. In this paper, we prove the conjecture for certain irreducible components of Newton strata in Shimura varieties of PEL type A and C, when $p$ is an unramified prime of good reduction. Our approach generalizes Chai and Oorts method for Siegel modular varieties.
We consider an abelian variety defined over a number field. We give conditional bounds for the order of its Tate-Shafarevich group, as well as conditional bounds for the Neron-Tate height of generators of its Mordell-Weil group. The bounds are implied by strong but nowadays classical conjectures, such as the Birch and Swinnerton-Dyer conjecture and the functional equation of the L-series. In particular, we improve and generalise a result by D. Goldfeld and L. Szpiro on the order of the Tate-Shafarevich group, and extends a conjecture of S. Lang on the canonical height of a system of generators of the free part of the Mordell-Weil group. The method is an extension of the algorithm proposed by Yu. Manin for finding a basis for the non-torsion rational points of an elliptic curve defined over the rationals.
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