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Hermitian vector bundles and extension groups on arithmetic schemes II. THE ARITHMETIC ATIYAH EXTENSION
, 2008
"... In a previous paper, we have defined arithmetic extension groups in the context of Arakelov geometry. In the present one, we introduce an arithmetic analogue of the Atiyah extension that defines an element — the arithmetic Atiyah class — in a suitable arithmetic extension group. Namely, if E is a ..."
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In a previous paper, we have defined arithmetic extension groups in the context of Arakelov geometry. In the present one, we introduce an arithmetic analogue of the Atiyah extension that defines an element — the arithmetic Atiyah class — in a suitable arithmetic extension group. Namely, if E is a hermitian vector bundle on an arithmetic scheme X, its arithmetic Atiyah class bat X/Z(E) lies in the group d Ext 1 X(E, E ⊗ Ω1), and is an obstruction to the algebraicity over X of the X/Z unitary connection on the vector bundle EC over the complex manifold X(C) that is compatible with its holomorphic structure. In the first sections of this article, we develop the basic properties of the arithmetic Atiyah class which can be used to define characteristic classes in arithmetic Hodge cohomology. Then we study the vanishing of the first Chern class ĉH 1 (L) of a hermitian line bundle L in the arithmetic Hodge cohomology group d Ext 1
ROOT NUMBERS AND RANKS IN POSITIVE CHARACTERISTIC
"... Abstract. For a global field K and an elliptic curve Eη over K(T), Silverman’s specialization theorem implies rank(Eη(K(T))) ≤ rank(Et(K)) for all but finitely many t ∈ P 1 (K). If this inequality is strict for all but finitely many t, the elliptic curve Eη is said to have elevated rank. All known ..."
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Abstract. For a global field K and an elliptic curve Eη over K(T), Silverman’s specialization theorem implies rank(Eη(K(T))) ≤ rank(Et(K)) for all but finitely many t ∈ P 1 (K). If this inequality is strict for all but finitely many t, the elliptic curve Eη is said to have elevated rank. All known examples of elevated rank for K = Q rest on the parity conjecture for elliptic curves over Q, and the examples are all isotrivial. Some additional standard conjectures over Q imply that there does not exist a nonisotrivial elliptic curve over Q(T) with elevated rank. In positive characteristic, an analogue of one of these additional conjectures is false. Inspired by this, for the rational function field K = κ(u) over any finite field κ with characteristic ̸ = 2, we construct an explicit 2parameter family Ec,d of nonisotrivial elliptic curves over K(T) (depending on arbitrary c, d ∈ κ × ) such that, under the parity conjecture, each Ec,d has elevated rank. To Mike Artin on his 70th birthday 1.
Non–simple abelian varieties in a family: geometric and analytic approaches. arXiv:0804.2166 [math.NT
"... Abstract. Let At be a family of abelian varieties over a number field k parametrized by a rational coordinate t, and suppose the generic fiber of At is geometrically simple. For example, we may take At to be the Jacobian of the hyperelliptic curve y 2 = f(x)(x − t) for some polynomial f. We give two ..."
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Abstract. Let At be a family of abelian varieties over a number field k parametrized by a rational coordinate t, and suppose the generic fiber of At is geometrically simple. For example, we may take At to be the Jacobian of the hyperelliptic curve y 2 = f(x)(x − t) for some polynomial f. We give two upper bounds for the number of t ∈ k of height at most B such that the fiber At is geometrically nonsimple. One bound comes from arithmetic geometry, and shows that there are only finitely many such t; but one has very little control over how this finite number varies as f changes. Another bound, from analytic number theory, shows that the number of geometrically nonsimple fibers grows quite slowly with B; this bound, by contrast with the arithmetic one, is effective, and is uniform in the coefficients of f. We hope that the paper, besides proving the particular theorems we address, will serve as a good example of the strengths and weaknesses of the two complementary approaches.
Lfunctions with large analytic rank and abelian varieties with large algebraic rank over function fields
"... The goal of this paper is to explain how a simple but apparently new fact of linear algebra together with the cohomological interpretation of Lfunctions allows one to produce many examples of Lfunctions over function fields vanishing to high order at the center point of their functional equation. ..."
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The goal of this paper is to explain how a simple but apparently new fact of linear algebra together with the cohomological interpretation of Lfunctions allows one to produce many examples of Lfunctions over function fields vanishing to high order at the center point of their functional equation. Conjectures of Birch
JACOBI SUMS, FERMAT JACOBIANS, AND RANKS OF ABELIAN VARIETIES OVER TOWERS OF FUNCTION FIELDS
, 2006
"... 1.1. Given an abelian variety A over a function field K = k(C) with C an absolutely irreducible, smooth, proper curve over a field k, it is natural to ask about the behavior of the MordellWeil group of A in the layers of a tower of fields over K. The simplest case, which is already very interesting ..."
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1.1. Given an abelian variety A over a function field K = k(C) with C an absolutely irreducible, smooth, proper curve over a field k, it is natural to ask about the behavior of the MordellWeil group of A in the layers of a tower of fields over K. The simplest case, which is already very interesting, is when A is an elliptic curve,
FINITENESS THEOREMS FOR ALGEBRAIC CYCLES OF SMALL CODIMENSION ON QUADRIC FIBRATIONS OVER CURVES
, 809
"... Abstract. We obtain finiteness theorems for algebraic cycles of small codimension on quadric fibrations X over curves over perfect fields k. For example, if k is finitely generated over Q and the fibration has relative dimension at least 11, then CH i (X) is finitely generated for i ≤ 4. 1. Introduc ..."
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Abstract. We obtain finiteness theorems for algebraic cycles of small codimension on quadric fibrations X over curves over perfect fields k. For example, if k is finitely generated over Q and the fibration has relative dimension at least 11, then CH i (X) is finitely generated for i ≤ 4. 1. Introduction. A wellknown conjecture of S.Bloch asserts that the Chow ring of a smooth projective variety over a number field is a finitely generated abelian group. In connection with this conjecture, a number of authors have studied 0cycles (i.e., cycles of maximal codimension) on quadric
HODGE THEORY AND THE MORDELLWEIL RANK OF ELLIPTIC CURVES OVER EXTENSIONS OF FUNCTION
"... Abstract. We use Hodge theory to prove a new upper bound on the ranks of MordellWeil groups for elliptic curves over function fields after regular geometrically Galois extensions of the base field, improving on previous results of Silverman and Ellenberg, when the base field has characteristic zero ..."
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Abstract. We use Hodge theory to prove a new upper bound on the ranks of MordellWeil groups for elliptic curves over function fields after regular geometrically Galois extensions of the base field, improving on previous results of Silverman and Ellenberg, when the base field has characteristic zero and the supports of the conductor of the elliptic curve and of the ramification divisor of the extension are disjoint. 1.
Documenta Math. 23 On a Positive Equicharacteristic Variant of the pCurvature Conjecture
, 2012
"... Abstract. Our aim is to formulate and prove a weak form in equal characteristic p>0 of the pcurvature conjecture. We also show the existence of a counterexample to a strong form of it. ..."
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Abstract. Our aim is to formulate and prove a weak form in equal characteristic p>0 of the pcurvature conjecture. We also show the existence of a counterexample to a strong form of it.