Related papers: Refined class number formulas for $\mathbb{G}_m$

We use the theory of Kolyvagin systems to prove (most of) a refined class number formula conjectured by Darmon. We show that for every odd prime $p$, each side of Darmon's conjectured formula (indexed by positive integers $n$) is "almost" a…

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…

In 1988, Gross proposed a conjectural congruence between Stickelberger elements and algebraic regulators, which is often referred to as the refined class number formula. In this paper, we prove this congruence.

We describe a refinement of the general theory of higher rank Euler, Kolyvagin and Stark systems in the setting of the multiplicative group over arbitrary number fields. We use the refined theory to prove new results concerning the Galois…

The primary objective of this paper is the study of different instances of the elliptic Stark conjectures of Darmon, Lauder and Rotger, in a situation where the elliptic curve attached to the modular form $f$ has split multiplicative…

We establish formulae of Stark type for the Stickelberger elements in the function field setting. Our result generalizes a work of Hayes and a conjecture of Gross. It is used to deduce a $p$-adic version of Rubin-Stark Conjecture and Burns…

Let $K/k$ be a finite Galois extension of global function fields. Let $E$ be a Drinfeld module over $k$. We state and prove an equivariant refinement of Taelman's analogue of the analytic class number formula for $(E,K/k)$, and derive…

In this paper we construct, using Stark elements of Rubin [Ann. Inst. Fourier (Grenoble) 46 (1996), no. 1, 33-62], Kolyvagin systems for certain modified Selmer structures (that are adjusted to have core rank one in the sense of [Mem. Amer.…

We propose a candidate, which we call the fractional Galois ideal after Snaith's fractional ideal, for replacing the classical Stickelberger ideal associated to an abelian extension of number fields. The Stickelberger ideal can be seen as…

We introduce new refinements of the Bell, factorial, and unsigned Stirling numbers of the first and second kind that unite the derangement, involution, associated factorial, associated Bell, incomplete Stirling, restricted factorial,…

Let T be Goedel's system of primitive recursive functionals of finite type in the lambda formulation. We define by constructive means using recursion on nested multisets a multivalued function I from the set of terms of T into the set of…

We show that the Generalized Vanishing Conjecture $$\forall_{m \ge 1} [\Lam^m f^m = 0] \Longrightarrow \forall_{m \gg 0} [\Lam^m (g f^m) = 0]$$ for a fixed differential operator $\Lam \in k[\partial]$ follows from a special case of it,…

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 $s=1$…

The theory of Weil-Stark elements is used to develop an axiomatic approach to the formulation of refined versions of Stark's Conjecture. This gives concrete new results concerning leading terms of Artin $L$-series and arithmetic properties…

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…

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…

In this paper, we formulate the notion of split elements of a unipotent class in a connected reductive group $G$. Generalized Green functions of $G$ can be computed by using Lusztig's algorithm, if split elements exist for any unipotent…

Repeated integration is a major topic of integral calculus. In this article, we study repeated integration. In particular, we study repeated integrals and recurrent integrals. For each of these integrals, we develop reduction formulae for…

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…

We construct Stickelberger elements for Hilbert modular cusp forms of parallel weight 2 and use recent results of Dasgupta and Spiess to bound their order of vanishing from below. As a special case the vanishing part of Mazur and Tate's…