Stark Units Research Articles

Brumer–Stark units and explicit class field theory

Brumer–Stark units and explicit class field theory

On the root of unity ambiguity in a formula for the Brumer–Stark units

Abstract We prove a conjectural formula for the Brumer–Stark units. Dasgupta and Kakde have shown the formula is correct up to a bounded root of unity. In this paper, we resolve the ambiguity in their result. We also remove an assumption from Dasgupta–Kakde’s result on the formula.

Comparing two formulas for the Gross–Stark units

Let F be a totally real number field. Dasgupta conjectured an explicit p-adic analytic formula for the Gross–Stark units of F. In a later paper, Dasgupta–Spieß conjectured a cohomological formula for the principal minors and the characteristic polynomial of the Gross regulator matrix associated to a totally odd character of F. Dasgupta–Spieß conjectured that these conjectural formulas coincide for the diagonal entries of Gross regulator matrix. In this paper, we prove this conjecture when F is a cubic field. For a video summary of this paper, please visit https://youtu.be/hlBRUIOke04.

SIC-POVMs from Stark units: Prime dimensions n2 + 3

We propose a recipe for constructing a fiducial vector for a symmetric informationally complete positive operator valued measure (SIC-POVM) in a complex Hilbert space of dimension of the form d = n2 + 3, focusing on prime dimensions d = p. Such structures are shown to exist in 13 prime dimensions of this kind, the highest being p = 19 603. The real quadratic base field K (in the standard SIC-POVM terminology) attached to such dimensions has fundamental units uK of norm −1. Let ZK denote the ring of integers of K; then, pZK splits into two ideals: p and p′. The initial entry of the fiducial is the square ξ2 of a geometric scaling factor ξ, which lies in one of the fields K(uK). Strikingly, each of the other p − 1 entries of the fiducial vector is a product of ξ and the square root of a Stark unit. These Stark units are obtained via the Stark conjectures from the value at s = 0 of the first derivatives of partial L-functions attached to the characters of the ray class group of ZK with modulus p∞1, where ∞1 is one of the real places of K.

The Derived Hecke Algebra for Dihedral Weight One Forms

We study the action of the derived Hecke algebra in the setting of dihedral weight one forms and prove a conjecture of the second- and fourth- named authors relating this action to certain Stark units associated to the symmetric square L-function. The proof exploits the theta correspondence between various Hecke modules as well as the ideas of Merel and Lecouturier on higher Eisenstein elements.

On Log-Algebraic Identities for Anderson t-Modules and Characteristic p Multiple Zeta Values

Abstract Based on the notion of Stark units, we present a new approach that obtains refinements of log-algebraic identities for Anderson $t$-modules. As a consequence, we use our techniques to recover many earlier results and prove stronger results in some cases. Further, we devise a direct and conceptual way to get logarithmic interpretations for multiple zeta values in positive characteristic. This generalizes the work of Anderson and Thakur for Carlitz zeta values.

A class formula for admissible Anderson modules

In 2012 Taelman proved a class formula for Drinfeld \({\mathbb {F}}_q[t]\)-modules. For an arbitrary coefficient ring A, several deep but partial results in the direction of a class formula for Drinfeld A-modules have been obtained. In this paper, using a completely new approach based on the notion of Stark units, we establish the class formula for Drinfeld A-modules in full generality. Further we extend this class formula for A-finite or abelian Anderson modules and more generally for admissible Anderson modules.

Special values of triple-product p-adic L-functions and non-crystalline diagonal classes

The main purpose of this note is to understand the arithmetic encoded in the special value of the $p$-adic $L$-function $\mathcal{L}_p^g(\mathbf{f},\mathbf{g},\mathbf{h})$ associated to a triple of modular forms $(f,g,h)$ of weights $(2,1,1)$, in the case where the classical $L$-function $L(f\otimes g\otimes h,s)$ - which typically has sign $+1$ - does not vanish at its central critical point $s=1$. When $f$ corresponds to an elliptic curve $E/\mathbb{Q}$ and the classical $L$-function vanishes, the Elliptic Stark Conjecture of Darmon-Lauder-Rotger predicts that $\mathcal{L}_p^g(\mathbf{f},\mathbf{g},\mathbf{h})(2,1,1)$ is either $0$ (when the order of vanishing of the complex $L$-function is $>2$) or related to logarithms of global points on $E$ and a certain Gross--Stark unit associated to $g$. We complete the picture proposed by the Elliptic Stark Conjecture by providing a formula for the value $\mathcal{L}_p^g(\mathbf{f},\mathbf{g},\mathbf{h})(2,1,1)$ in the case where $L(f\otimes g\otimes h,1)\neq 0$.

Diagonal restrictions of p-adic Eisenstein families

We compute the diagonal restriction of the first derivative with respect to the weight of a p-adic family of Hilbert modular Eisenstein series attached to a general (odd) character of the narrow class group of a real quadratic field, and express the Fourier coefficients of its ordinary projection in terms of the values of a distinguished rigid analytic cocycle in the sense of Darmon and Vonk (Duke Math J, to appear, 2020) at appropriate real quadratic points of Drinfeld’s p-adic upper half-plane. This can be viewed as the p-adic counterpart of a seminal calculation of Gross and Zagier (J Reine Angew Math 355:191–220, 1985, §7) which arose in their “analytic proof” of the factorisation of differences of singular moduli, and whose inspiration can be traced to Siegel’s proof of the rationality of the values at negative integers of the Dedekind zeta function of a totally real field. Our main identity enriches the dictionary between the classical theory of complex multiplication and its extension to real quadratic fields based on RM values of rigid meromorphic cocycles, and leads to an expression for the p-adic logarithms of Gross–Stark units and Stark–Heegner points in terms of the first derivatives of certain twisted Rankin triple product p-adic L-functions.

Beilinson–Flach elements, Stark units and 𝑝-adic iterated integrals

Abstract We study weight one specializations of the Euler system of Beilinson–Flach elements introduced by Kings, Loeffler and Zerbes, with a view towards a conjecture of Darmon, Lauder and Rotger relating logarithms of units in suitable number fields to special values of the Hida–Rankin p-adic L-function. We show that the latter conjecture follows from expected properties of Beilinson–Flach elements and prove the analogue of the main theorem of Castella and Hsieh about generalized Kato classes.

P-Adic measures associated with zeta values and p-adic log multiple gamma functions

We study a relation between two refinements of the rank one abelian Gross–Stark conjecture. For a suitable abelian extension [Formula: see text] of number fields, a Gross–Stark unit is defined as a [Formula: see text]-unit of [Formula: see text] satisfying certain properties. Let [Formula: see text]. Yoshida and the author constructed the symbol [Formula: see text] by using [Formula: see text]-adic [Formula: see text] multiple gamma functions, and conjectured that the [Formula: see text] of a Gross–Stark unit can be expressed by [Formula: see text]. Dasgupta constructed the symbol [Formula: see text] by using the [Formula: see text]-adic multiplicative integration, and conjectured that a Gross–Stark unit can be expressed by [Formula: see text]. In this paper, we give an explicit relation between [Formula: see text] and [Formula: see text] and prove that two refinements are consistent.

On the ratios of Barnes' multiple gamma functions to the p-adic analogues

We consider the partial zeta function ζ(s,c) associated with a narrow ray class c of a totally real field F. We have a canonical decomposition ζ′(0,c)=∑ιX(c,ι) where ι runs over all real embeddings of F. The symbol X(c,ι) denotes Hiroyuki Yoshida's class invariant defined as a finite sum of log of Barnes' multiple gamma functions and some correction terms. Yoshida studied the relation between the values of exp⁡(X(c,ι)), Stark units, and Shimura's period symbol. Yoshida and the author also defined and studied the p-adic analogue Xp(c,ι), and obtained an explicit relation between the “ratios” [exp⁡(X(c,ι)):expp⁡(Xp(c,ι))] and Gross–Stark units. In a previous paper, the author proved the algebraicity of some products of exp⁡(X(c,ι))'s. In this paper, we prove its p-adic analogue. As an application, we obtain an explicit relation between the same “ratios” and Stark units. As a result, we can discuss the rank 1 abelian Stark conjecture with respect to real places and Gross' p-adic analogue uniformly.

ON IWASAWA THEORY OF RUBIN–STARK UNITS AND NARROW CLASS GROUPS

AbstractLet K be a totally real number field of degree r. Let K∞ denote the cyclotomic -extension of K, and let L∞ be a finite extension of K∞, 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 rth exterior power of totally positive units modulo a subgroup of Rubin–Stark units, for some $\overline{\mathbb{Q}_{2}}$-irreducible characters χ of Gal(L∞/K∞).

Partial zeta values, Gross's tower of fields conjecture, and Gross–Stark units

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. This is achieved by introducing an integral version of the Eisenstein cocycle.

Derived Hecke Algebra for Weight One Forms

ABSTRACTWe study the action of the derived Hecke algebra on the space of weight one forms. By analogy with the topological case, we formulate a conjecture relating this to a certain Stark unit. We verify the truth of the conjecture numerically, for the weight one forms of level 23 and 31, and many derived Hecke operators at primes less than 200. Our computation depends in an essential way on Merel’s evaluation of the pairing between the Shimura and cuspidal subgroups of J0(q).

On the algebraicity of some products of special values of Barnes' multiple gamma function

We consider partial zeta functions $\zeta(s,c)$ associated with ray classes $c$'s of a totally real field. Stark's conjecture implies that an appropriate product of $\exp(\zeta'(0,c))$'s is an algebraic number which is called a Stark unit. Shintani gave an explicit formula for $\exp(\zeta'(0,c))$ in terms of Barnes' multiple gamma function. Yoshida ``decomposed'' Shintani's formula: he defined the symbol $X(c,\iota)$ satisfying that $\exp(\zeta'(0,c))=\prod_{\iota} \exp(X(c,\iota))$ where $\iota$ 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 is to show that $([F:\mathbb Q]-1)$ of them are algebraic numbers. We also study a relation between Yoshida's conjecture on CM-periods and Stark's conjecture.

Stark units in positive characteristic

We show that the module of Stark units associated to a sign-normalized rank one Drinfeld module can be obtained from Anderson's equivariant A-harmonic series. We apply this to obtain a class formula a la Taelman and to prove a several variable log-algebraicity theorem, generalizing Anderson's log-algebraicity theorem. We also give another proof of Anderson's log-algebraicity theorem using shtukas and obtain various results concerning the module of Stark units for Drinfeld modules of arbitrary rank.

Théorie d’Iwasawa des unités de Stark et groupe de classes

Let [Formula: see text] be a number field and let [Formula: see text] be an odd rational prime. Let [Formula: see text] be a [Formula: see text]-extension of [Formula: see text] and let [Formula: see text] be a finite extension of [Formula: see text], abelian over [Formula: see text]. In this paper we extend the classical results, e.g. [16], relating characteristic ideal of the [Formula: see text]-quotient of the projective limit of the ideal class groups to the [Formula: see text]-quotient of the projective limit of units modulo Stark units, in the non-semi-simple case, for some [Formula: see text]-irreductible characters [Formula: see text] of [Formula: see text]. The proof essentially uses the theory of Euler systems.

Arithmetic of function field units

We introduce the module of Stark units associated to a Drinfeld module defined over the ring of integers of a global function field. We investigate the arithmetic properties of these objects, and as an application, we prove a “discrete analogue” for Taelman’s class modules of certain Conjectures formulated by Greenberg for cyclotomic fields.

Gross–Stark units and p-adic iterated integrals attached to modular forms of weight one

This article can be read as a companion and sequel to the authors’ earlier article on Stark points and p-adic iterated integrals attached to modular forms of weight one, which proposes a conjectural expression for the so-called p-adic iterated integrals attached to a triple (f, g, h) of classical eigenforms of weights (2, 1, 1). When f is a cusp form, this expression involves the p-adic logarithms of so-called Stark points: distinguished points on the modular abelian variety attached to f, defined over the number field cut out by the Artin representations attached to g and h. The goal of this paper is to formulate an analogous conjecture when f is a weight two Eisenstein series rather than a cusp form. The resulting formula involves the p-adic logarithms of units and p-units in suitable number fields, and can be seen as a new variant of Gross’s p-adic analogue of Stark’s conjecture on Artin L-series at \(s=0\).