This is a revised version of ANT-0332: "A support problem for the intermediate Jacobians of l-adic representations", by G. Banaszak, W. Gajda & P. Krason, which was placed on these archives on the 29th of January 2002.
Following a suggestion of the referee we have subdivided the paper into two separate parts: "Support problem for the intermediate Jacobians of l-adic representations", and "On Galois representations for abelian varieties with complex and real multiplications".
Our results on the image of Galois and the Mumford-Tate conjecture for some RM abelian varieties are contained in the second paper. Both papers were accepted for publication.
We show that torsion points of certain orders are not on a theta divisor in the Jacobian variety of a hyperelliptic curve given by the equation $y^2=x^{2g+1}+x$ with $g \geq 2$. The proof employs a method of Anderson who proved an analogous result for a cyclic quotient of a Fermat curve of prime degree.
Let $C$ be a genus $2$ hyperelliptic curve over a number field $K$, with a Weierstrass point $\infty$ at infinity, let $J$ be its Jacobian, let $Θ$ be the theta divisor with respect to $\infty$, and let $p$ be any prime number. We give an explicit construction of a $p$-adic height $h_p\colon J(\overline{\mathbb{Q}})\to \mathbb{Q}_p$ by means of $p$-adic analogues of Néron functions of divisor $2Θ$. We define such Néron functions using division polynomials and a generalisation of Blakestad's $p$-adic sigma function on the formal group of $J$. We prove that our $p$-adic Néron function $λ_v$ at a…
We prove a p-adic analogue of Wüstholz's analytic subgroup theorem. We apply this result to show that a curve embedded in its Jacobian intersects the p-adic closure of the Mordell-Weil group transversely whenever the latter has rank equal to 1. This allows us to give some theoretical justification to Chabauty techniques applied to finding rational points on curves whose Jacobian has Mordel-Weil group of rank 1.
Results in $p$-adic transcendence theory are applied to two problems in the Chabauty-Coleman method. The first is a question of McCallum and Poonen regarding repeated roots of Coleman integrals. The second is to give lower bounds on the $p$-adic distance between rational points in terms of the heights of a set of Mordell-Weil generators of the Jacobian. We also explain how, in some cases, a conjecture on the 'Wieferich statistics' of Jacobians of curves implies a bound on the height of rational points of curves of small rank, in terms of the usual invariants of the curve and the height of Mord…
For an algebraic variety $X$ of dimension $d$ with totally degenerate reduction over a $p$-adic field (definition recalled below) and an integer $i$ with $1\leq i\leq d$, we define a rigid analytic torus $J^i(X)$ together with an Abel-Jacobi mapping to it from the Chow group $CH^i(X)_{hom}$ of codimension $i$ algebraic cycles that are homologically equivalent to zero modulo rational equivalence. These tori are analogous to those defined by Griffiths using Hodge theory over the complex numbers. We compare and contrast the complex and $p$-adic theories. Finally, we examine a special case of a $p…
Let $K$ be a number field, and let $C$ be a hyperelliptic curve over $K$ with Jacobian $J$. Suppose that $C$ is defined by an equation of the form $y^{2} = f(x)(x - λ)$ for some irreducible monic polynomial $f \in \mathcal{O}_{K}[x]$ of discriminant $Δ$ and some element $λ\in \mathcal{O}_{K}$. Our first main result says that if there is a prime $\mathfrak{p}$ of $K$ dividing $(f(λ))$ but not $(2Δ)$, then the image of the natural $2$-adic Galois representation is open in $\mathrm{GSp}(T_{2}(J))$ and contains a certain congruence subgroup of $\mathrm{Sp}(T_{2}(J))$ depending on the maximal power…
Anderson introduced a $p$-adic version of soliton theory. He then applied it to the Jacobian variety of a cyclic quotient of a Fermat curve and showed that torsion points of certain prime order lay outside of the theta divisor. In this paper, we evolve his theory further. As an application, we get a stronger result on the intersection of the theta divisor and torsion points on the Jacobian variety for more general curves. New examples are discussed as well. A key new ingredient is a map connecting the $p$-adic loop group and the formal group.
Let $k$ be a subfield of $\mathbb{C}$ which contains all $2$-power roots of unity, and let $K = k(α_{1}, α_{2}, ... , α_{2g + 1})$, where the $α_{i}$'s are independent and transcendental over $k$, and $g$ is a positive integer. We investigate the image of the $2$-adic Galois action associated to the Jacobian $J$ of the hyperelliptic curve over $K$ given by $y^{2} = \prod_{i = 1}^{2g + 1} (x - α_{i})$. Our main result states that the image of Galois in $\mathrm{Sp}(T_{2}(J))$ coincides with the principal congruence subgroup $Γ(2) \lhd \mathrm{Sp}(T_{2}(J))$. As an application, we find generator…
Let $X$ be a quotient of the modular curve $X_0(N)$ whose Jacobian $J_X$ is a simple factor of $J_0(N)^{new}$ over $\mathbb{Q}$. Let $f$ be the newform of level $N$ and weight 2 associated with $J_X$; assume $f$ has analytic rank 1. We give analytic methods for determining the rational points of $X$ using quadratic Chabauty by computing two $p$-adic Gross-Zagier formulas for $f$. Quadratic Chabauty requires a supply of rational points on the curve or its Jacobian; this new method eliminates this requirement. To achieve this, we give an algorithm to compute the special value of the anticyclotom…
Let $ρ$ be a mod $\ell$ Galois representation. We show how to compute $ρ$, given the characteristic polynomial of the image of the Frobenius at one prime $p$ and a curve $C$ whose Jacobian contains $ρ$ in its $\ell$-torsion. The main ingredient is a method to $p$-adically lift torsion points on a Jacobian in the framework of Makdisi's algorithms.
Let $K$ be a number field, $n>4$ an integer, $f(x)$ an irreducible polynomial over $K$ of degree $n$, whose Galois group is either the full symmetric group $S_n$ or the alternating group $A_n$. Suppose $C:y^2=f(x)$ is the corresponding hyperelliptic curve and $J$ its jacobian defined over $K$. For each prime $\ell$ we write $V_{\ell}(J)$ for the $Q_{\ell}$-Tate module of $J$ and $e_{\ell}$ for the Riemann form on $V_{\ell}(J)$ attached to the theta divisor. (Here $Q_{\ell}$ is the field of $\ell$-adic numbers.) We write $sp(V_{\ell}(J))$ for the $Q_{\ell}$-Lie algebra of the symplectic group o…
We construct Calabi-Yau threefolds defined over $\mathbb{Q}$ via quotients of abelian threefolds, and re-verify the rigid Calabi-Yau threefolds in this construction are modular by computing their L-series, without \cite{Dieulefait} or \cite{GouveaYui}. We compute the intermediate Jacobians of the rigid Calabi-Yau threefolds as complex tori, then compute a $\mathbb{Q}$-model for the 1-torus given a $\mathbb{Q}$-structure on the rigid Calabi-Yau threefolds, and find infinitely many examples and counterexamples for a conjecture of Yui about the relation between the $L$-series of the rigid Calabi-…
We show how our p-adic method to compute Galois representations occurring in the torsion of Jacobians of algebraic curves can be adapted to modular curves. The main ingredient is the use of "moduli-friendly" Eisenstein series introduced by Makdisi, which allow us to evaluate modular forms at p-adic of modular curves points and dispenses us of the need for equations of modular curves and for q-expansion computations. The resulting algorithm compares very favourably to the complex-analytic method.
We prove surjectivity criteria for $p$-adic representations and we apply them to abelian varieties over number fields. In particular, we provide examples of Jacobians over $\dbQ$ of dimension $d\in\{1,2,3\}$ whose 2-adic representations have as images $GSp_{2d}(\dbZ_2)$.
For a curve $X$ over a $p$-adic field $k$, using the class field theory of $X$ due to S. Bloch and S. Saito we study the abelian geometric fundamental group $π_1^{\mathrm{ab}}(X)^{\mathrm{geo}}$ of $X$. In particular, it is investigated a subgroup of $π_1^{\mathrm{ab}}(X)^{\mathrm{geo}}$ which classifies the geometric and abelian coverings of $X$ which allow possible ramification over the special fiber of the model of $X$. Under the assumptions that $X$ has a $k$-rational point, $X$ has good reduction and its Jacobian variety has good ordinary reduction, we give some upper and lower bounds of …
Let $k$ be a field of characteristic zero containing a primitive fifth root of unity. Let $X/k$ be a smooth cubic threefold with an automorphism of order five, then we observe that over a finite extension of the field actually the dihedral group $D_5$ is a subgroup of ${\rm Aut}(X)$. We find that the intermediate Jacobian $J(X)$ of $X$ is isogenous to the product of an elliptic curve $E$ and the self-product of an abelian surface $B$ with real multiplication by $\mathbb{Q}(\sqrt{5})$. We give explicit models of some algebraic curves related to the construction of $J(X)$ as a Prym variety. This…
Given a $p$-adic field $K$ and a prime number $\ell$, we count the total number of the isomorphism classes of $p^\ell$-extensions of $K$ having no intermediate fields. Moreover for each group that can appear as Galois group of the normal closure of such an extension, we count the number of isomorphism classes that contain extensions whose normal closure has Galois group isomorphic to the given group. Finally we determine the ramification groups and the discriminant of the composite of all $p^\ell$-extensions of K with no intermediate fields. The principal tool is a result, proved at the beginn…
In this article, we examine the arithmetic aspect of the Kummer-surface-type CY 3-folds $\hat{T/G}$, characterized by the crepant resolution of 3-torus-orbifold $T/G$ with only isolated singularities. Up to isomorphisms, there are only two such space $\hat{T/G}$ with $|G|=3, 7$, and both $T$ carrying the structure of triple-product structure of a CM elliptic curve. The (Griffiths) intermediate Jacobians of these $\hat{T/G}$ are identified explicitly as the corresponding elliptic curve appeared in the structure of $T$. We further provide the $\QZ$-structure of $\hat{T/G}$ and verify their modul…
In this article we give an algorithm for the computation of the number of rational points on the Jacobian variety of a generic ordinary hyperelliptic curve defined over a finite field of cardinality $q$ with time complexity $O(n^{2+o(1)})$ and space complexity $O(n^2)$, where $n=\log(q)$. In the latter complexity estimate the genus and the characteristic are assumed as fixed. Our algorithm forms a generalization of both, the AGM algorithm of J.-F. Mestre and the canonical lifting method of T. Satoh. We canonically lift a certain arithmetic invariant of the Jacobian of the hyperelliptic curve i…
In this paper we consider the problem of classifying the isomorphism classes of extensions of degree pk of a p-adic field, restricting to the case of extensions without intermediate fields. We establish a correspondence between the isomorphism classes of these extensions and some Kummer extensions of a suitable field F containing K. We then describe such classes in terms of the representations of Gal(F/K). Finally, for k = 2 and for each possible Galois group G, we count the number of isomorphism classes of the extensions whose normal closure has a Galois group isomorphic to G. As a byproduct,…
In this work we generalise the main result of arXiv:1812.05651 to the family of hyperelliptic curves with potentially good reduction over a $p$-adic field which have degree $p$ and the largest possible image of inertia under the $\ell$-adic Galois representation associated to its Jacobian. We will prove that this Galois representation factors as the tensor product of an unramified character and an irreducible representation of a finite group, which can be either equal to the inertia image (in which case the representation is easily determined) or a $C_2$-extension of it. In this second case, t…
There are two famous Abel Theorems. Most well-known is his description of abelian (analytic) functions on a one dimensional compact complex torus. The other collects together those complex tori, with their prime degree isogenies, into one space. Riemann's generalization of the first features his famous theta functions. His deepest work aimed at extending Abel's second theorem; he died before he fulfilled this. That extension is often pictured on complex higher dimension torii. For Riemann, though, it was to spaces of Jacobians of compact Riemann surfaces, W, of genus g, toward studying functio…
Let $L/K$ be a finite Galois extension of $p$-adic fields and let $L_{\infty}$ be the unramified $\mathbb Z_p$-extension of $L$. Then $L_{\infty}/K$ is a one-dimensional $p$-adic Lie extension. In the spirit of the main conjectures of equivariant Iwasawa theory, we formulate a conjecture which relates the equivariant local epsilon constants attached to the finite Galois intermediate extensions $M/K$ of $L_{\infty}/K$ to a natural arithmetic invariant arising from the étale cohomology of the constant sheaf $\mathbb Q_p/\mathbb Z_p$ on the spectrum of $L_{\infty}$. We give strong evidence of the…
Let k be a positive integer divisible by 4, l>k a prime, and f an elliptic cuspidal eigenform of weight k-1, level 4, and non-trivial character. Let ρ_f be the l-adic Galois representation attached to f. In this paper we provide evidence for the Bloch-Kato conjecture for a twist of the adjoint motif of ρ_f in the following way. Let L(f,s) denote the symmetric square L-function of f. We prove that (under certain conditions) the l-adic valuation of the algebraic part of L(f, k) is no greater than the l-adic valuation of the order of S, where S is (the Pontryagin dual of) the Selmer group attache…
