Stark Units Research Articles

Refined class number formulas for \mathbb{G}_m

We formulate a generalization of a `refined class number formula' of Darmon. Our conjecture deals with Stickelberger-type elements formed from generalized Stark units, and has two parts: the `order of vanishing' and the `leading term'. Using the theory of Kolyvagin systems we prove a large part of this conjecture when the order of vanishing of the corresponding complex $L$-function is $1$.

Some new maps and ideals in classical Iwasawa theory with applications

We introduce a new ideal {\mathfrak D} of the p-adic Galois group-ring associated to a real abelian field and a related ideal {\mathfrak J} for imaginary abelian fields. Both result from an equivariant, Kummer-type pairing applied to Stark units in a Z_p-tower of abelian fields and {\mathfrak J} is linked by explicit reciprocity to a third ideal {\mathfrak S} studied more generally in a previous work. This leads to a new and unifying framework for the Iwasawa Theory of such fields including a real analogue of Stickelberger's Theorem, links with certain Fitting ideals and \Lambda-torsion submodules, and a new exact sequence related to the Main Conjecture.

Index formulae for Stark units and their solutions

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 result we deduce a weak version of the conjecture (“up to absolute values”) in these cases and precise results on when the Stark unit, if it exists, is a square.

Explicit computation of Gross–Stark units over real quadratic fields

We present an effective and practical algorithm for computing Gross–Stark units over a real quadratic base field F . Our algorithm allows us to explicitly construct certain relative abelian extensions of F where these units lie, using only information from the base field. These units were recently proved to always exist within the correct extension fields of F by Dasgupta, Darmon, and Pollack, without directly producing them. For a video summary of this paper, please click here or visit http://www.youtube.com/watch?v=8h1-GW-sTNc . Author Video Watch what authors say about their articles

A conjectural product formula for Brumer–Stark units over real quadratic fields

Following methods of Hayes, we state a conjectural product formula for ratios of Brumer–Stark units over real quadratic fields.

The χ-part of the Analytic Class Number Formula, for Global Function Fields

Let $F/k$ be a finite abelian extension of global function fields, totally split at a distinguished place $\infty $ of $k$. We show that a complex Gras conjecture holds for Stark units, and we derive a refined analytic class number formula.

Stark units in $\mathbb{Z}_p$-extensions

We study the connection between the projective limit $\overline{St}_\infty$ of Stark units in $\mathbb{Z}_p$-extensions and a certain Iwasawa module already appeared in the particular cases of circular units or elliptic units. This connection is used to investigate the structure of $\overline{St}_\infty$.

ANOTHER LOOK AT GROSS–STARK UNITS OVER THE NUMBER FIELD ℚ

We provide another description of the Gross–Stark units over the rational field ℚ (studied in [B. Gross, p-Adic L-series at s = 0, J. Fac. Sci. Univ. Tokyo28(3) (1981) 979–994]) which is essentially a Gauss sum, using a p-adic multiplicative integral of the p-adic Kubota–Leopoldt distribution, and give a simplified proof of the Ferrero–Greenberg theorem (see [B. Ferrero and R. Greenberg, On the behavior of p-adic L-functions at s = 0, Invent. Math.50(1) (1978/79) 91–102]) for p-adic Hurwitz zeta functions. This is a precise analog for ℚ of Darmon–Dasgupta's work on elliptic units for real quadratic fields (see [H. Darmon and S. Dasgupta, Elliptic units for real quadratic fields, Ann. of Math. (2)163(1) (2006) 301–346]).

Index-modules and applications

Let K be a commutative field, \({A\subseteq K}\) be a Dedekind ring and V be a K-vector space. For any pair of A-lattices R ≠ 0 and S of V, we define an A-submodule \({\left[R : S\right]^{\prime}_{A}}\) of K, their A-index-module. Once the basic properties of these modules are stated, we show that this notion can be used to recover more usual ones: the group-index, the relative invariant, the Fitting ideal of R/S when \({S\subseteq R}\), and the generalized index of Sinnott. As an example, we consider the following situation. Let F/k be a finite abelian extension of global function fields, with Galois group G, and degree g. Let ∞ be a place of k which splits completely in F/k. Let \({{\mathcal O}_{F}}\) be the ring of functions of F, which are regular outside the places of F sitting over ∞. Then one may use Stark units to define a subgroup \({\mathcal E_F}\) of \({{\mathcal O}_{F}^{\times}}\), the group of units of \({\mathcal O_F}\). We use the notion of index-module to prove that for every nontrivial irreducible rational character ψ of G, the ψ-part of \({\mathbb Z\left[g^{-1}\right]\otimes_\mathbb Z\left(\mathcal O_F^\times/\mathcal E_F\right)}\) and the ψ-part of \({\mathbb Z\left[g^{-1}\right]\otimes_\mathbb ZCl(\mathcal O_F)}\) have the same order.

The Gras conjecture in function fields by Euler systems

We use Euler systems to prove the Gras conjecture for groups generated by Stark units in global function fields. The techniques applied here are classical and go back to Thaine, Kolyvagin and Rubin. We obtain our Euler systems from the torsion points of sign-normalized Drinfel'd modules.

Stark units and the main conjectures for totally real fields

AbstractThe main theorem of the author’s thesis suggests that it should be possible to lift the Kolyvagin systems of Stark units, constructed by the author in an earlier paper, to a Kolyvagin system over the cyclotomic Iwasawa algebra. In this paper, we verify that this is indeed the case. This construction of Kolyvagin systems over the cyclotomic Iwasawa algebra from Stark units provides the first example towards a more systematic study of Kolyvagin system theory over an Iwasawa algebra when the core Selmer rank (in the sense of Mazur and Rubin) 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 in the context of local Iwasawa theory, assuming the truth of the Rubin–Stark conjecture and Leopoldt’s conjecture. This statement in the local Iwasawa theory context turns out to be interesting in its own right, as it suggests a relation between the solutions to p-adic and complex Stark conjectures.

On Shintani’s ray class invariant for totally real number fields

We introduce a ray class invariant $${X(\mathfrak{C})}$$ for a totally real field, following Shintani’s work in the real quadratic case. We prove a factorization formula $${X(\mathfrak{C})=X_1(\mathfrak{C})\cdots X_n(\mathfrak{C})}$$ where each $${X_i(\mathfrak{C})}$$ corresponds to a real place. Although this factorization depends a priori on some choices (especially on a cone decomposition), we can show that it is actually independent of these choices. Finally, we describe the behavior of $${X_i(\mathfrak{C})}$$ when the signature of $${\mathfrak{C}}$$ at a real place is changed. This last result is also interpreted in terms of the derivatives L′(0, χ) of the L-functions and certain Stark units.

The canonical fractional Galois ideal at [formula omitted

The Stickelberger elements attached to an abelian extension of number fields conjecturally participate, under certain conditions, in annihilator relations involving higher algebraic K-groups. In [Victor P. Snaith, Stark's conjecture and new Stickelberger phenomena, Canad. J. Math. 58 (2) (2006) 419–448], Snaith introduces canonical Galois modules hoped to appear in annihilator relations generalising and improving those involving Stickelberger elements. In this paper we study the first of these modules, corresponding to the classical Stickelberger element, and prove a connection with the Stark units in a special case.

Continued fractions, special values of the double sine function, and Stark units over real quadratic fields

Let F be a real quadratic field and m an integral ideal of F . Two Stark units, ε m , 1 and ε m , 2 , are conjectured to exist corresponding to the two different embeddings of F into R . We define new ray class invariants U m ( 1 ) ( C + ) and U m ( 2 ) ( C + ) associated to each class C + of the narrow ray class group modulo m and dependent separately on the two different embeddings of F into R . These invariants are defined as a product of special values of the double sine function in a compact and canonical form using a continued fraction approach due to Zagier and Hayes. We prove that both Stark units ε m , 1 and ε m , 2 , assuming they exist, can be expressed simultaneously and symmetrically in terms of U m ( 1 ) ( C + ) and U m ( 2 ) ( C + ) , thus giving a canonical expression for every existent Stark unit over F as a product of double sine function values. We prove that Stark units do exist as predicted in certain special cases.

Stark's conjecture over complex cubic number fields

jjSystematic computation of Stark units over nontotally real base fields is carried out for the first time. Since the information provided by Stark's conjecture is significantly less in this situation than the information provided over totally real base fields, new techniques are required. Precomputing Stark units in relative quadratic extensions (where the conjecture is already known to hold) and coupling this information with the Fincke-Pohst algorithm applied to certain quadratic forms leads to a significant reduction in search time for finding Stark units in larger extensions (where the conjecture is still unproven). Stark's conjecture is verified in each case for these Stark units in larger extensions and explicit generating polynomials for abelian extensions over complex cubic base fields, including Hilbert class fields, are obtained from the minimal polynomials of these new Stark units.

Computation of Stark-Tamagawa units

Let K be a totally real number field and let l denote an odd prime number. We design an algorithm which computes strong numerical evidence for the validity of the Equivariant Tamagawa Number Conjecture for the Q[G]-equivariant motive h0(Spec(L)), where L/K is a cyclic extension of degree l and group G. This conjecture is a very deep refinement of the classical analytic class number formula. In the course of the algorithm, we compute a set of special units which must be considered as a generalization of the (conjecturally existing) Stark units associated to first order vanishing Dirichlet L-functions.

The Local Stark Conjecture at a Real Place

A refinement of the rank 1 Abelian Stark conjecture has been formulated by B. Gross. This conjecture predicts some is a real place, and interpret it in terms of 2-adic properties of special values of L-functions. We prove the conjecture for CM extensions; here the original Stark conjecture is uninteresting, but the refined conjecture is nontrivial. In more generality, we show that, under mild hypotheses, if the subgroup of the Galois group generated by complex conjugations has less than full rank, then the refined conjecture implies that the Stark unit should be a square. This phenomenon has been discovered by Dummit and Hayes in a particular type of situation. We show that it should hold in much greater generality.

Elliptic Curves and Class Fields of Real Quadratic Fields: Algorithms and Evidence

The article [Darmon 02] proposes a conjectural p-adic analytic construction of points on (modular) elliptic curves, points which are defined over ring class fields of real quadratic fields. These points are related to classical Heegner points in the same way as Stark units to circular or elliptic units. For this reason they are called “Stark-Heegner points,” following a terminology introduced in [Darmon 98]. If K is a real quadratic field, the Stark-Heegner points attached to K are conjectured to satisfy an analogue of the Shimura reciprocity law, so that they can in principle be used to find explicit generators for the ring class fields of K. It is also expected that their heights can be expressed in terms of derivatives of the Rankin L-series attached to E and K, in analogy with the Gross-Zagier formula . The main goal of this paper is to describe algorithms for calculating Stark-Heegner points and supply numerical evidence for the Shimura reciprocity and Gross-Zagier conjectures, focussing primarily on elliptic curves of prime conductor.

Stark's Conjectures and Hilbert's Twelfth Problem

We give a constructive proof of a theorem of Tate, which states that (under Stark's Conjecture) the field generated over a totally real field K by the Stark units contains the maximal real abelian extension of K. As a direct application of this proof, we show how one can compute explicitly real abelian extensions of K. We give two examples.