arXiv subjects

Rob de Jeu

Publications and source records attributed to Rob de Jeu.

17 recordsLinked to original sources

arXiv2026

The 2-part of the Bloch-Kato conjecture, and indivisibility results, for $K_2$ of some elliptic curves

For certain integers $u$, we investigate the 2-part of the Bloch-Kato conjecture for $L(E_u,2)$, where $E_u: y^2=x(x+1)(x+u^2)$ is part of a (twisted) Legendre family that is 2-isogenous to a family studied by Boyd. For this, we first work out the corresponding 2-parts of the Tamagawa factors and Galois invariants. Then we give an explicit description of the 2-torsion in the Selmer group $H_f^1(\mathbb{Q},E_u[2^\infty](-1))$. We construct a specific element in the kernel of the tame symbol for $K_2$ on an integral model of $E_u$, with non-vanishing real and 2-adic regulators. Using techniques involving the norm residue isomorphism of Merkur'ev-Suslin, we prove indivisibility of this element by 2 in that kernel, even modulo torsion, even though it is explicitly divisible by 2 in the kernel of the tame symbol for $K_2$ on $E_u$. We also bound the 2-divisibility of the images of these elements under the 2-adic regulator map. Finally, in many cases we investigate numerically the validity of the 2-part of the Bloch-Kato conjecture.

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arXiv2024

Bounding generators for the kernel and cokernel of the tame symbol for curves

Let $C$ be a regular, irreducible curve that is projective over a field. We obtain bounds in terms of the arithmetic genus of $C$ for the generators that are required for the cokernel of the tame symbol, as well as, under a simplifying assumption, its kernel. We briefly discuss a potential application to Chow groups.

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arXiv2024

$K_2$ of families of elliptic curves over non-Abelian cubic and quartic fields

We give two constructions of families of elliptic curves over cubic or quartic fields with three, respectively four, `integral' elements in the kernel of the tame symbol on the curves. The fields are in general non-Abelian, and the elements linearly independent. For their integrality, we prove a new criterion that does not ignore any torsion. We also verify Beilinson's conjecture numerically for just over 90 of the curves.

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arXiv2020

Hyperbolic tessellations and generators of K_3 for imaginary quadratic fields

We develop methods for constructing explicit generators, modulo torsion, of the K_3-groups of imaginary quadratic number fields. These methods are based on either tessellations of hyperbolic 3-space or on direct calculations in suitable pre-Bloch groups, and lead to the very first proven examples of explicit generators, modulo torsion, of any infinite K_3-group of a number field. As part of this approach, we make several improvements to the theory of Bloch groups for K_3 of any field, predict the precise power of 2 that should occur in the Lichtenbaum conjecture at -1 and prove that the latter prediction is valid for all abelian number fields.

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arXiv2020

Describing all multivariable functional equations of dilogarithms

We prove quite general statements about functional equations in any number of variables for the dilogarithms defined by Bloch-Wigner, Rogers, and Coleman, showing that they follow from certain 5-term and 2-term relations in a precise way. Unlike many other references, we use arbitrary coefficients, do not ignore any torsion, and get sharp results. For the Bloch-Wigner dilogarithm, we also consider complex conjugation, and for Coleman's p-adic dilogarithm we show independence of the branch.

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arXiv2019

Conjectures, consequences, and numerical experiments for p-adic Artin L-functions

We conjecture that the p-adic L-function of a non-trivial irreducible even Artin character over a totally real field is non-zero at all non-zero integers. This implies that a conjecture formulated by Coates and Lichtenbaum at negative integers extends in a suitable way to all positive integers. We also state a conjecture that for certain characters the Iwasawa series underlying the p-adic L-series have no multiple roots except for those corresponding to the zero at s=0 of the p-adic L-function. We provide some theoretical evidence for our first conjecture, and prove both conjectures by means of computer calculations for a large set of characters (and integers where appropriate) over the rationals and over real quadratic fields, thus proving many instances of conjectures by Coates and Lichtenbaum and by Schneider. The calculations and the theoretical evidence also prove that certain p-adic regulators corresponding to 1-dimensional characters for the rational numbers are units in many cases. We also verify Gross' conjecture for the order of the zero of the p-adic L-function at s=0 in many cases. We gather substantial statistical data on the constant term of the underlying Iwasawa series, and propose a model for its behaviour for certain characters.

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arXiv2015

Etale cohomology, cofinite generation, and p-adic L-functions

For a prime number p and a number field k, we first study certain etale cohomology groups with coefficients associated to a p-adic Artin representation of its Galois group, where we twist the coefficients using a modified Tate twist with a p-adic index. We show that those groups are cofinitely generated and explicitly compute an additive Euler characteristic. When k is totally real and the representation is even, we relate the order of vanishing of the p-adic L-function at a point of its domain and the corank of such a cohomology group with a suitable p-adic twist. If the groups are finite, then the value of the p-adic L-function is non-zero and its p-adic absolute value is related to a multiplicative Euler characteristic. For a negative integer n (and for 0 in certain cases), this gives a proof of a conjecture by Coates and Lichtenbaum, and a short proof of the equivariant Tamagawa number conjecture for classical L-functions that do not vanish at n. For p=2 our results involving p-adic L-functions depend on a conjecture in Iwasawa theory.

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arXiv2014

On K_2 of certain families of curves

We construct families of smooth, proper, algebraic curves in characteristic 0, of arbitrary genus g, together with g elements in the kernel of the tame symbol. We show that those elements are in general independent by a limit calculation of the regulator. Working over a number field, we show that in some of those families the elements are integral. We determine when those curves are hyperelliptic, finding, in particular, that over any number field we have non-hyperelliptic curves of all composite genera g with g independent integral elements in the kernel of the tame symbol.

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arXiv2014

A relative version of the Beilinson-Hodge conjecture

Let k be an algebraically closed subfield of the complex numbers, and X a variety defined over k. One version of the Beilinson-Hodge conjecture that seems to survive scrutiny is the statement that the Betti cycle class map cl_{r,m} : H_M^{2r-m}(k(X),Q(r)) -> hom_{MHS}(Q(0),H^{2r-m}(k(X)(C),Q(r))) is surjective, that being equivalent to the Hodge conjecture in the case m=0. Now consider a smooth and proper map ρ: X -> S of smooth quasi-projective varieties over k. We formulate a version of this conjecture for the generic fibre, expecting the corresponding cycle class map to be surjective. We provide some evidence in support of this in the case where X is a product, the map is the projection to one factor, and m=1.

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arXiv2013

Frobenius lifts and point counting for smooth curves

We describe an algorithm to compute the zeta-function of a proper, smooth curve over a finite field, when the curve is given together with some auxiliary data. Our method is based on computing the matrix of the action of a semi-linear Frobenius on the first cohomology group of the curve by means of Serre duality. The cup product involved can be computed locally, after first computing local expansions of a globally defined lift of Frobenius. The resulting algorithm's complexity is softly cubic in the field degree, which is also the case with Kedlaya's algorithm in the hyperelliptic case.

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arXiv2012

The syntomic regulator for $K_4$ of curves

Let $C$ be a curve defined over a discrete valuation field of characteristic zero where the residue field has positive characteristic. Assuming that $C$ has good reduction over the residue field, we compute the syntomic regulator on a certain part of $K_4^[(3)}(C)$. The result can be expressed in terms of $p$-adic polylogarithms and Coleman integration. We also compute the syntomic regulator on a certain part of $K_4^[(3)}(F)$ for the function field $F$ of $C$. The result can be expressed in terms of $p$-adic polylogarithms and Coleman integration, or by using a trilinear map ("triple index") on certain functions.

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arXiv2011

Beilinson's Hodge conjecture for smooth varieties

Consider the cycle class map cl_{r,m} : CH^r(U,m;\Q) \to ΓH^{2r-m}(U,\Q(r)), where CH^r(U,m;\Q) is Bloch's higher Chow group (tensored with \Q) of a smooth complex quasi-projective variety U, and H^{2r-m}(U,\Q(r)) is singular cohomology. We study the image of cl_{r,m} in terms of kernels of Abel-Jacobi maps. When r=m, we deduce from the Bloch-Kato theorem that the cokernel of cl_{r,m} at the generic point of U is the same for integral or rational coefficients.

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arXiv2011

On special elements in higher algebraic K-theory and the Lichtenbaum-Gross Conjecture

We conjecture the existence of special elements in odd degree higher algebraic K-groups of number fields that are related in a precise way to the values at strictly negative integers of the derivatives of Artin L-functions of finite dimensional complex representations. We prove this conjecture in certain important cases and also provide other evidence (both theoretical and numerical) in its support.

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arXiv2007

On the p-adic Beilinson conjecture for number fields

We formulate a conjectural p-adic analogue of Borel's theorem relating regulators for higher K-groups of number fields to special values of the corresponding zeta-functions, using syntomic regulators and p-adic L-functions. We also formulate a corresponding conjecture for Artin motives, and state a conjecture about the precise relation between the p-adic and classical situations. Parts of he conjectures are proved when the number field (or Artin motive) is Abelian over the rationals, and all conjectures are verified numerically in some other cases.

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arXiv2005

Numerical verification of Beilinson's conjecture for K2 of hyperelliptic curves

We construct families of hyperelliptic curves over Q of arbitrary genus g with (at least) g integral elements in K_2. We also verify the Beilinson conjectures about K_2 numerically for several curves with g=2, 3, 4 and 5. The paper is essentially self-contained and may serve as an elementary introduction to Beilinson's conjectures for K_2 of curves.

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arXiv2001

The syntomic regulator for K-theory of fields

We define complexes analogous to Goncharov's complexes for the K-theory of discrete valuation rings of characteristic zero. Under suitable assumptions in K-theory, there is a map from the cohomology of those complexes to the K-theory of the ring. In case the ring is the localization of the ring of integers in a number field, there are no assumptions necessary. We compute the composition of our map to the K-theory with the syntomic regulator. The result can be described in terms of a p-adic polylogarithm. Finally, we apply our theory in order to compute the regulator to syntomic cohomology on Beilinson's cyclotomic elements. The result is again given by the p-adic polylogarithm. This last result is related to one by Somekawa and generalizes work by Gros.

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arXiv1998

Towards regulator formulae for curves over number fields

In this paper we study the group K_{2n}^{(n+1)}(F) where F is the function field of a complete, smooth, geometrically irreducible curve C over a number field, assuming the Beilinson--Soulé conjecture on weights. In particular, we compute the Beilinson regulator on a subgroup of K_{2n}^{(n+1)}(F), using the complexes constructed in previous work by the author. We study the boundary map in the localization sequence for n = 3 (the case n = 2 was done in a previous paper). We combine our results with some results of Goncharov in order to obtain a complete description of the image of the regulator map on K_4^{(3)}(C) and K_6^{(4)}(C), independent of any conjectures.

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