We study the behaviour of the Stark conjecture for an abelian extension K/k of totally real number fields as K varies in a cyclotomic Z_p-tower. We consider possible strengthenings of the natural norm-coherence in the tower of putative solution of the complex conjecture and,especially, the consequences for the analogous p-adic conjecture. More precisely, under two principal assumptions - (i) that these solutions are given by exterior powers of norm-coherent sequences of global (or p-semilocal) units and (ii) that p splits in k - we show how the values of p-adic twisted zeta-functions for K/k are determined at any integer by a group-ring-valued regulator formed from the appropriate Coates-Wiles homomorphisms.
Main theorem of [Buyukboduk, arXiv:0706.0377v1] suggests that it should be possible to lift the Kolyvagin systems of Stark units constructed in [Buyukboduk, arXiv:math/0703426v1] to a Kolyvagin system over the cyclotomic Iwasawa algebra. This is what we prove in this paper. This construction gives the first example towards a more systematic study of Kolyvagin system theory over an Iwasawa algebra when the core Selmer rank is greater than one. As a result of this construction, we reduce the main conjectures of Iwasawa theory for totally real fields to a statement of local Iwasawa theory. This s…
For an abelian extension of number fields we show that the Stark conjecture for all Artin L-functions with zero of order r is equivalent to existence of a special element in the rational span of the r-th exterior power of the Galois module of units of the bigger field.
In 1980, Gross conjectured a formula for the expected leading term at $s=0$ of the Deligne--Ribet $p$-adic $L$-function associated to a totally even character $ψ$ of a totally real field $F$. The conjecture states that after scaling by $L(ψω^{-1}, 0)$, this value is equal to a $p$-adic regulator of units in the abelian extension of $F$ cut out by $ψω^{-1}$. In this paper, we prove Gross's conjecture.
In a previous paper the second author developed a new approach to the abelian p-adic Stark Conjecture at s=1 and stated some related conjectures. This paper develops and applies techniques using p-adic measures and continued fractions to numerically investigate one of these conjectures to high precision in fifteen cases with a real quadratic base field.
Let $K$ be a totally real number field of degree $r$. Let $K_{\infty}$ denote the cyclotomic $\mathbb{Z}_{2}$-extension of $K$ and let $L_{\infty}$ be a finite extension of $K_{\infty}$, abelian over $K$. The goal of this paper is to compare the characteristic ideal of the $χ$-quotient of the projective limit of the narrow class groups to the $χ$-quotient of the projective limit of the $r$-th exterior power of totally positive units modulo a subgroup of Rubin-Stark units, for some $\overline{\mathbb{Q}_{2}}$-irreducible characters $χ$ of $\mathrm{Gal}(L_{\infty}/K_{\infty})$.
We describe a conjectural construction (in the spirit of Hilbert's 12th problem) of units in abelian extensions of certain base fields which are neither totally real nor CM. These base fields are quadratic extensions with exactly one complex place of a totally real number field F, and are referred to as Almost Totally real (ATR) extensions. Our construction involves certain null-homologous topological cycles on the Hilbert modular variety attached to F. The special units are the images of these cycles under a map defined by integration of weight two Eisenstein series on GL_2(F). This map is fo…
We show that for an odd prime p, the p-primary parts of refinements of the (imprimitive) non-abelian Brumer and Brumer-Stark conjectures are implied by the equivariant Iwasawa main conjecture (EIMC) for totally real fields. Crucially, this result does not depend on the vanishing of the relevant Iwasawa mu-invariant. In combination with the authors' previous work on the EIMC, this leads to unconditional proofs of the non-abelian Brumer and Brumer-Stark conjectures in many new cases.
We formulate a conjecture which generalizes Darmon's "refined class number formula". We discuss relations between our conjecture and the equivariant leading term conjecture of Burns. As an application, we give another proof of the "except 2-part" of Darmon's conjecture, which was first proved by Mazur and Rubin.
Given an abelian, CM extension K of any totally real number field k, we consider two conjectures `of Stark type'. The `Integrality Conjecture' concerns the image of a p-adic map `\mathfrak{s}_{K/k,S}' determined by the minus-part of the S-truncated equivariant L-function for K/k at s=1. It is connected to the Equivariant Tamagawa Number Conjecture of Burns and Flach. The `Congruence Conjecture' says that \mathfrak{s}_{K/k,S} gives an explicit reciprocity law for the element predicted by the corresponding Rubin-Stark Conjecture for K^+/k. We study the general properties of these conjectures and…
We discuss three different formulations of the equivariant Iwasawa main conjecture attached to an extension K/k of totally real fields with Galois group G, where k is a number field and G is a p-adic Lie group of dimension 1 for an odd prime p. All these formulations are equivalent and hold if Iwasawa's μ-invariant vanishes. Under mild hypotheses, we use this to prove non-abelian generalizations of Brumer's conjecture, the Brumer-Stark conjecture and a strong version of the Coates-Sinnott conjecture provided that μ= 0.
We give an introduction to generalisations of conjectures of Brumer and Stark on the annihilator of the class group of a number field. We review the relation to the equivariant Tamagawa number conjecture, the main conjecture of Iwasawa theory for totally real fields, and a conjecture of Gross on the behaviour of $p$-adic Artin $L$-functions at zero.
This version improves the old version entitled "On the modularity of elliptic curves with a residually irreducible representation". Let $E$ be an elliptic curve over an abelian totally real field $K$ unramified at 3,5, and 7. We prove that $E$ is modular.
Let $F$ be a totally real number field, $p$ a rational prime, and $χ$ a finite order totally odd abelian character of Gal$(\bar{F}/F)$ such that $χ(\mathfrak{p})=1$ for some $\mathfrak{p}|p$. Motivated by a conjecture of Stark, Gross conjectured a relation between the derivative of the $p$-adic $L$-function associated to $χ$ at its exceptional zero and the $\mathfrak{p}$-adic logarithm of a $p$-unit in the $χ$ component of $F_χ^\times$. In a recent work, Dasgupta, Darmon, and Pollack have proven this conjecture assuming two conditions: that Leopoldt's conjecture holds for $F$ and $p$, and that…
We show that abelian surfaces (and consequently curves of genus 2) over totally real fields are potentially modular. As a consequence, we obtain the expected meromorphic continuation and functional equations of their Hasse--Weil zeta functions. We furthermore show the modularity of infinitely many abelian surfaces A over Q with End_C(A)=Z. We also deduce modularity and potential modularity results for genus one curves over (not necessarily CM) quadratic extensions of totally real fields.
In this paper, we construct (many) Kolyvagin systems out of Stickelberger elements, utilizing ideas borrowed from our previous work on Kolyvagin systems of Rubin-Stark elements. We show how to apply this construction to prove results on the odd parts of the ideal class groups of CM fields which are abelian over a totally real field, and deduce the main conjecture of Iwasawa theory for totally real fields (for totally odd characters). Although the main results of this paper have already been established by Wiles, our approach provides another example (which slightly differs from the case of Sta…
We show that up to potential isogeny, there are only finitely many abelian varieties of dimension $d$ defined over a number field $K$, such that for any finite place $v$ outside a fixed finite set $S$ of places of $K$ containing the archimedean places, it has either good reduction at $v$, or totally bad reduction at $v$ and good reduction over a quadratic extension of the completion of $K$ at $v$.
Let $K/k$ be a finite Galois CM-extension of number fields whose Galois group $G$ is monomial and $S$ a finite set of places of $k$.\ Then the "Stickelberger element" $θ_{K/k,S}$ is defined.\ Concerning this element,\ Andreas Nickel formulated the non-abelian Brumer and Brumer-Stark conjectures and their "weak" versions.\ In this paper,\ when $G$ is a monomial group,\ we prove that the weak non-abelian conjectures are reduced to the weak conjectures for abelian subextensions.\ We write $D_{4p},\ Q_{2^{n+2}}$ and $A_4$ for the dihedral group of order $4p$ for any odd prime $p$,\ the generalized…
The `Congruence Conjecture' was developed by the second author in a previous paper. It provides a conjectural explicit reciprocity law for a certain element associated to an abelian extension of a totally real number field whose existence is predicted by earlier conjectures of Rubin and Stark. The first aim of the present paper is to design and apply techniques to investigate the Congruence Conjecture numerically. We then present complete verifications of the conjecture in 48 varied cases with real quadratic base fields.
Let $K/k$ be an abelian extension of number fields with a distinguished place of $k$ that splits totally in $K$. In that situation, the abelian rank one Stark conjecture predicts the existence of a unit in $K$, called the Stark unit, constructed from the values of the $L$-functions attached to the extension. In this paper, assuming the Stark unit exists, we prove index formulae for it. In a second part, we study the solutions of the index formulae and prove that they admit solutions unconditionally for quadratic, quartic and sextic (with some additional conditions) cyclic extensions. As a resu…
We introduce an integral version of the Eisenstein cocycle. As applications we prove a conjecture of Gross regarding the "order of vanishing" of Stickelberger elements relative to an abelian tower of fields and give a cohomological construction of the conjectural Gross--Stark units.
In this paper we determine the number of endomorphism rings of superspecial abelian surfaces over a field $\mathbb{F}_q$ of odd degree over $\mathbb{F}_p$ in the isogeny class corresponding to the Weil $q$-number $\pm\sqrt{q}$. This extends earlier works of T.-C. Yang and the present authors on the isomorphism classes of these abelian surfaces, and also generalizes the classical formula of Deuring for the number of endomorphism rings of supersingular elliptic curves. Our method is to explore the relationship between the type and class numbers of the quaternion orders concerned. We study the Pi…
We prove that a refinement of Stark's Conjecture formulated by Rubin is true up to primes dividing the order of the Galois group, for finite, abelian extensions of function fields over finite fields. We also show that in the case of constant field extensions a statement stronger than Rubin's holds true.
Let $K$ be a totally real number field of degree $r=[K:\mathbb{Q}]$ and let $p$ be an odd rational prime. Let $K_{\infty}$ denote the cyclotomic $\mathbb{Z}_{p}$-extension of $K$ and let $L_{\infty}$ be a finite extension of $K_{\infty}$, abelian over $K$. In this article, we extend results of \cite{Kazim109} relating characteristic ideal of the $χ$-quotient of the projective limit of the ideal class groups to the $χ$-quotient of the projective limit of the $r$-th exterior power of units modulo Rubin-Stark units, in the non semi-simple case, for some $\overline{\mathbb{Q}_{p}}$-irreductible ch…
In this article, we study the p-ordinary Iwasawa theory of the (conjectural) Rubin-Stark elements defined over abelian extensions of a CM field F and develop a rank-g Euler/Kolyvagin system machinery (where 2g is the degree of F), refining and generalizing Perrin-Riou's theory and the author's prior work. This has several important arithmetic consequences: Using the recent results of Hida and Hsieh on the CM main conjectures, we prove a natural extension of a theorem of Yager for the CM field F, where we relate the Rubin-Stark elements to the several-variable Katz p-adic L-function. Furthermor…
We prove the Bogomolov conjecture for an abelian variety A over a function field which is totally degenerate at a place v. We adapt Zhang's proof of the number field case replacing the complex analytic tools by tropical analytic geometry. A key step is the tropical equidistribution theorem for A at the totally degenerate place. As an application, we obtain finiteness of torsion points with coordinates in the maximal unramified algebraic extension over v.
We give a systematic method of providing numerical evidence for higher order Stark-type conjectures such as (in chronological order) Stark's conjecture over $\mathbb{Q}$, Rubin's conjecture, Popescu's conjecture, and a conjecture due to Burns that constitutes a generalization of Brumer's classical conjecture on annihilation of class groups. Our approach is general and could be used for any abelian extension of number fields, independent of the signature and type of places (finite or infinite) that split completely in the extension. We then employ our techniques in the situation where $K$ is a …
We study a relation between two refinements of the rank one abelian Gross-Stark conjecture: For a suitable abelian extension $H/F$ of number fields, a Gross-Stark unit is defined as a $p$-unit of $H$ satisfying some proporties. Let $τ\in \mathrm{Gal}(H/F)$. Yoshida and the author constructed the symbol $Y_p(τ)$ by using $p$-adic $\log$ multiple gamma functions, and conjectured that the $\log_p$ of a Gross-Stark unit can be expressed by $Y_p(τ)$. Dasgupta constructed the symbol $u_T(τ)$ by using the $p$-adic multiplicative integration, and conjectured that a Gross-Stark unit can be expressed by…
Let E/F be a finite Galois extension of totally real number fields and let p be a prime. The `p-adic Stark conjecture at s=1' relates the leading terms at s=1 of p-adic Artin L-functions to those of the complex Artin L-functions attached to E/F. We prove this conjecture unconditionally when E/Q is abelian. We also show that for certain non-abelian extensions E/F the p-adic Stark conjecture at s=1 is implied by Leopoldt's conjecture for E at p. Moreover, we prove that for a fixed prime p, the p-adic Stark conjecture at s=1 for E/F implies Stark's conjecture at s=1 for E/F. This leads to a `prim…
In this paper, we present a unifying approach to the general theory of abelian Stark conjectures. To do so we define natural notions of `zeta element', of `Weil-étale cohomology complexes' and of `integral Selmer groups' for the multiplicative group $\mathbb{G}_m$ over finite abelian extensions of number fields. We then conjecture a precise connection between zeta elements and Weil-étale cohomology complexes, we show this conjecture is equivalent to a special case of the equivariant Tamagawa number conjecture and we give an unconditional proof of the analogous statement for global function fie…
We consider partial zeta functions $ζ(s,c)$ associated with ray classes $c$'s of a totally real field. Stark's conjecture implies that an appropriate product of $\exp(ζ'(0,c))$'s is an algebraic number which is called a Stark unit. Shintani gave an explicit formula for $\exp(ζ'(0,c))$ in terms of Barnes' multiple gamma function. Yoshida ``decomposed'' Shintani's formula: he defined the symbol $X(c,ι)$ satisfying that $\exp(ζ'(0,c))=\prod_ι \exp(X(c,ι))$ where $ι$ runs over all real embeddings of $F$. Hence we can decompose a Stark unit into a product of $[F:\mathbb Q]$ terms. The main result i…
The goal of this article is two-fold: First, to prove a (two-variable) main conjecture for a CM field $F$ without assuming the $p$-ordinary hypothesis of Katz, making use of what we call the Rubin-Stark $\mathcal{L}$-restricted Kolyvagin systems which is constructed out of the conjectural Rubin-Stark Euler system of rank $g$. (We are also able to obtain weaker unconditional results in this direction.) Second objective is to prove the Park-Shahabi plus/minus main conjecture for a CM elliptic curve $E$ defined over a general totally real field again using (a twist of the) Rubin-Stark Kolyvagin s…
Let $F$ be a totally real number field and $A/F$ a principally polarized abelian variety with real multiplication by the ring of integers $\mathcal{O}$ of a totally real field. Assuming $A$ admits an $\mathcal{O}$-linear 3-isogeny over $F$, we prove that a positive proportion of the quadratic twists $A_d$ have rank 0. We also prove that a positive proportion of $A_d$ have rank $\dim A$, assuming the Tate-Shafarevich groups are finite. If $A$ is the Jacobian of a hyperelliptic curve $C$, we deduce that a positive proportion of twists $C_d$ have no rational points other than those fixed by the h…
Let $F$ be a totally real field with ring of integers $O_F$, and $D$ be a totally definite quaternion algebra over $F$. A well-known formula established by Eichler and then extended by Körner computes the class number of any $O_F$-order in $D$. In this paper we generalize the Eichler class number formula so that it works for arbitrary $\mathbb{Z}$-orders in $D$. The motivation is to count the isomorphism classes of supersingular abelian surfaces in a simple isogeny class over a prime finite field $\mathbb{F}_p$. We give explicit formulas for the number of these isomorphism classes for all prim…
We extend the main result of [Math. Res. Lett. 15 (2008), 715-725] to Galois extensions L/K of totally real number fields of arbitrary odd prime power degree, thereby offering support for the validity of the 'main conjecture' of equivariant Iwasawa theory.
We define Dedekind sums attached to a totally real number field of class number one. We prove that they satisfy some reciprocity law. Then we relate them to special values of Hecke $L$-functions. We conclude that they are ruled by Stark's conjecture.
We completely calculate the Fitting ideal of the classical $p$-ramified Iwasawa module for any abelian extension $K/k$ of totally real fields, using the shifted Fitting ideals recently developed by the second author. This generalizes former results by the first and third authors where we had to assume that only $p$-adic places may ramify in $K/k$. One of the important ingredients is the computation of some complexes in appropriate derived categories.
Let $H/F$ be a finite abelian extension of number fields with $F$ totally real and $H$ a CM field. Let $S$ and $T$ be disjoint finite sets of places of $F$ satisfying the standard conditions. The Brumer-Stark conjecture states that the Stickelberger element $Θ^{H/F}_{S, T}$ annihilates the $T$-smoothed class group $\text{Cl}^T(H)$. We prove this conjecture away from $p=2$, that is, after tensoring with $\mathbf{Z}[1/2]$. We prove a stronger version of this result conjectured by Kurihara that gives a formula for the 0th Fitting ideal of the minus part of the Pontryagin dual of $\text{Cl}^T(H) \…
In an earlier article we proved the existence of a canonical Kolyvagin derivative homomorphism between the modules of Euler and Kolyvagin systems (in any given rank) that are associated to $p$-adic representations over number fields. We now explain how the existence of such a homomorphism leads to new results on the structure of the Selmer modules of Galois representations over Gorenstein orders and to a strategy for verifying (refinements of) the Tamagawa number conjecture of Bloch and Kato. We describe concrete applications relating to the multiplicative group over arbitrary number fields an…
Let A be a CM abelian variety defined over a number field K. We compute congruence relations on units in fields generated by adjoining torsion points of A to K. For elliptic curves, congruence relations of the type we compute were used in the proof of the Coates-Wiles theorem. In general, we compute congruence relations on exterior products of units in division fields. This result seems to fit more naturally into the framework of the Stark/Rubin conjectures.
The paper starts out from pseudomeasures (in the sense of Serre) which hold the arithmetic properties of the abelian $l$-adic Artin $L$-functions over totally real number fields. In order to generalize to non-abelian $l$-adic $L$-functions, these abelian pseudomeasures must satisfy congruences which are introduced but not yet known to be true. The relation to the ``equivariant main conjecture'' of Iwasawa theory is discussed.
We state the Brumer-Stark conjecture and motivate it from two perspectives. Stark's perspective arose in his attempts to generalize the classical Dirichlet class number formula for the leading term of the Dedekind zeta function at $s=1$ (equivalently, $s=0$). Brumer's perspective arose by generalizing Stickelberger's work regarding the factorization of Gauss sums and the annihilation of class groups of cyclotomic fields. These viewpoints were synthesized by Tate, who stated the Brumer-Stark conjecture in its current form. The conjecture considers a totally real field $F$ and a finite abelian C…
We present a conjectural formula for the principal minors and the characteristic polynomial of Gross's regulator matrix associated to a totally odd character of a totally real field. The formula is given in terms of the Eisenstein cocycle, which was defined and studied earlier by the authors and collaborators. For the determinant of the regulator matrix, our conjecture follows from recent work of Kakde, Ventullo and the first author. For the diagonal entries, our conjecture overlaps with the conjectural formula presented in our prior work. The intermediate cases are new and provide a refinemen…
Fix an odd prime $p$. Let $G$ be a compact $p$-adic Lie group containing a closed, normal, pro-$p$ subgroup $H$ which is abelian and such that $G/H$ is isomorphic to the additive group of $p$-adic integers $\mathbbZ_p$ . First we assume that $H$ is finite and compute the Whitehead group of the Iwasawa algebra, $Λ(G)$, of $G$. We also prove some results about certain localisation of $Λ(G)$ needed in Iwasawa theory. Let $F$ be a totally real number field and let $F_{\infty}$ be an admissible $p$-adic Lie extension of $F$ with Galois group $G$. The computation of the Whitehead groups are used to …
The aim of this paper is to give upper bounds for the Euclidean minima of abelian fields of odd prime conductor. In particular, these bounds imply Minkowski's conjecture for totally real number fields of conductor p^r, where p is an odd prime and r is at least 2.
We give the complete proof of a conjecture of Georges Gras which claims that, for any extension $K/k$ of number fields in which at least one infinite place is totally split, every ideal $I$ of $K$ principalizes in the compositum $Kk^{ab}$ of $K$ with the maximal abelian extension $k^{ab}$ of $k$
In an earlier paper the author proved one divisibility of Perrin- Riou's Iwasawa main conjecture for Heegner points on elliptic curves. In the present paper, that result is generalized to abelian varieties of GL2-type (i.e. abelian varieties with real multiplication defined over totally real fields) under the hypothesis that the abelian variety is associated to a Hilbert modular form via a construction of Zhang.
Let F/k be a finite abelian extension of global function fields, totally split at a distinguished place \infty. We prove that a complex Gras conjecture holds for a suitable group of Stark units, and we derive a refined analytic class number formula.
Let G be a finite abelian group. A number field K is called a Hilbert-Speiser field of type G if for every tame G-Galois extension L/K has a normal integral basis, i.e., the ring of integers O_L is free as an O_K[G]-module. Let C_p denote the cyclic group of prime order p. We show that if p >= 7 (or p=5 and extra conditions are met) and K is totally real with K/Q ramified at p, then K is not Hilbert-Speiser of type C_p.
