Xavier-Francois Roblot, David Solomon
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
In this note we describe our personal encounters with the $p$-adic Stark conjecture. Gross describes the period between $1977$ and $1986$ when he came to formulate these conjectures (§1-8), and Dasgupta describes the period between $1998$ and $2021$, when he worked with others to finally prove them (§9-12).
We develop a detailed arithmetic theory related to special values at arbitrary integers of the Artin $L$-series of linear characters. To do so we define canonical generalized Stark elements of arbitrary `rank' and `weight', thereby extending the classical theory of Rubin-Stark elements. We then formulate an extension to arbitrary weight of the refined version of the Rubin-Stark Conjecture that we studied in an earlier article and also show that generalized Stark elements constitute a $p$-adic family by formulating precise conjectural congruence relations between elements of differing weights. …
We give a new definition of a $p$-adic $L$-function for a mixed signature character of a real quadratic field and for a nontrivial ray class character of an imaginary quadratic field. We then state a $p$-adic Stark conjecture for this $p$-adic $L$-function. We prove our conjecture in the case when $p$ is split in the imaginary quadratic field by relating our construction to Katz's $p$-adic $L$-function. We also provide numerical evidence for our conjecture in three examples.
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.
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…
On Selmer complexes, Stark systems and derived $p$-adic heights
论文发展了Nekovář的塞勒复形理论，证明在温和假设下，塞勒复形与Poitou-Tate复形准同构，并应用此结果构建了控制塞勒群的Heegner点斯塔克系统，同时证明Bertolini-Darmon导出p-进高度配对与Nekovář的相同。
On generalized Iwasawa main conjectures and $p$-adic Stark conjectures for Artin motives
本文提出关于Artin动机的广义Iwasawa主猜想和$p$-adic Stark猜想，并证明其与Rubin-Stark元素的等价性及虚二次域中相关猜想的无条件成立。
In a recent work of Darmon, Pozzi and Vonk, the authors consider a particular $p$-adic family of Hilbert Eisenstein series $E_k(1,\brch)$ associated with an odd character $\brch$ of the narrow ideal class group of a real quadratic field $F$ and compute the first derivative of a certain one-variable twisted triple product $p$-adic $L$-series attached to $E_k(1,\brch)$ and an elliptic newform $f$ of weight $2$ on $Γ_0(p)$. In this paper, we generalize their construction to include the cyclotomic variable and thus obtain a two-variable twisted triple product $p$-adic $L$-series. Moreover, when $f…
Balanced triple product $p$-adic $L$-functions and Stark points
本研究推广Hsieh的三变量平衡三重积$p$-进$L$-函数构造，针对秩一情形提出联系其插值域外值与Stark点、Stark单位的椭圆Stark猜想，并对同一虚二次域的二面体表示证明该猜想。
This article is devoted to the elliptic Stark conjecture formulated by Darmon, Lauder and Rotger [DLR], which proposes a formula for the transcendental part of a $p$-adic avatar of the leading term at $s=1$ of the Hasse-Weil-Artin $L$-series $L(E,\varrho_1\otimes \varrho_2,s)$ of an elliptic curve $E$ twisted by the tensor product $\varrho_1\otimes \varrho_2$ of two odd $2$-dimensional Artin representations, when the order of vanishing is two. The main ingredient of this formula is a $2\times 2$ $p$-adic regulator involving the $p$-adic formal group logarithm of suitable Stark points on $E$. T…
The purpose of this paper is to formulate and study a common refinement of a version of Stark's conjecture and its $p$-adic analogue, in terms of Fontaine's $p$-adic period ring and $p$-adic Hodge theory. We construct period-ring-valued functions under a generalization of Yoshida's conjecture on the transcendental parts of CM-periods. Then we conjecture a reciprocity law on their special values concerning the absolute Frobenius action. We show that our conjecture implies a part of Stark's conjecture when the base field is an arbitrary real field and the splitting place is its real place. It al…
We propose a conjectural construction of global points on modular elliptic curves over arbitrary number fields, generalizing both the p-adic construction of Heegner points via Cerednik-Drinfeld uniformization and the definition of classical Stark-Heegner points. In alignment with Nekovar and Scholl's plectic conjectures, we expect the non-triviality of these plectic Stark-Heegner points to control the Mordell-Weil group of higher rank elliptic curves. We provide some indirect evidence for our conjectures by showing that higher order derivatives of anticyclotomic p-adic L-functions compute plec…
$p$-adic Higher Green's Functions for Stark-Heegner Cycles
本文将 Darmon-Vonk 的刚性亚纯上闭链理论推广至高权，构造了实二次除子的高阶 Green 配对，为代数 Stark-Heegner 闭链的存在性提供了间接证据。
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.
Mazur and Rubin have recently developed a theory of higher rank Kolyvagin and Stark systems over principal artinian rings and discrete valuation rings. In this article we describe a natural extension of (a slightly modified version of) their theory to systems over more general coefficient rings. We also construct unconditionally, and for general $p$-adic representations, a canonical, and typically large, module of higher rank Euler systems and show that for $p$-adic representations satisfying standard hypotheses the image under a natural higher rank Kolyvagin-derivative type homomorphism of ea…
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 Conjecture.
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 a…
Let $F$ be a totally real field. For each ideal class $c$ of $F$ and each real embedding $ι$ of $F$, Hiroyuki Yoshida defined an invariant $X(c,ι)$ as a finite sum of log of Barnes' multiple gamma functions with some correction terms. Then the derivative value of the partial zeta function $ζ(s,c)$ has a canonical decomposition $ζ'(0,c)=\sum_ιX(c,ι)$, where $ι$ runs over all real embeddings of $F$. Yoshida studied the relation between $\exp(X(c,ι))$'s, Stark units, and Shimura's period symbol. Yoshida and the author also defined and studied the $p$-adic analogue $X_p(c,ι)$: In particular, we di…
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.
The eigencurve at crystalline points with scalar Frobenius and Gross-Stark regulators
本文研究了R=T方法失效时p-adic eigencurve在p-irregular权一尖点形式处的局部几何，并证明了椭圆模曲线塔的普通p-adic étale上同调群在局部化时不是Hecke代数上的自由模。
We study the special values of the triple product $p$-adic $L$-function constructed by Darmon and Rotger at all classical points outside the region of interpolation. We propose conjectural formulas for these values that can be seen as extending the Elliptic Stark Conjecture, and we provide theoretical evidence for them by proving some particular cases.
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},\ma…
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 conjecture that special elements associated with rank-one motives are obtained $p$-adically from Rubin-Stark elements by means of a precise `higher-rank Soulé twist' construction. We show this conjecture incorporates a variety of known results and existing predictions and also gives rise to a concrete strategy for proving the equivariant Tamagawa Number Conjecture for rank-one motives. We then use this approach to obtain new evidence in support of the equivariant Tamagawa Number Conjecture in the setting of CM abelian varieties.
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 prove formulas for the p-adic logarithm of quaternionic Darmon points on p-adic tori and modular abelian varieties over Q having purely multiplicative reduction at p. These formulas are amenable to explicit computations and are the first to treat Stark-Heegner type points on higher-dimensional abelian varieties.
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…
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…
Brumer-Stark Units and Explicit Class Field Theory
本文证明了全实域CM阿贝尔扩张中Brumer-Stark p-单位的整Gross-Stark猜想的p部分，结合n-1个F中元素的平方根生成全实域极大阿贝尔扩张，给出其无条件构造，并推广Ribet方法结合新伽罗瓦模完成证明。
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…
Towards the $p$-adic derived Hecke algebra for weight one forms
本文提出在模曲线完备上同调中定义 $p$-进 Shimura 类与导出 Hecke 算子的路径，并猜想其作用关联权一尖点形式与伴随 Deligne–Serre 表示的 Stark 单位对数。
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…
Our first goal in this note is to explain that a weak form of Perrin-Riou's conjecture on the non-triviality of Beilinson-Kato classes follows as an easy consequence of the Iwasawa main conjectures, and deduce its refined versions in the supersingular case from this fact and a variety of Gross-Zagier formulae. Our second goal is to set up a conceptual framework in the context of $Λ$-adic Kolyvagin systems to treat analogues of Perrin-Riou's conjectures for higher motives of higher rank. We apply this general discussion in order to establish a link between Heegner points on a general class of C…
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) \…
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 give a new proof of a conjecture of Darmon, Lauder and Rotger regarding the computation of the $\mathcal L$-invariant of the adjoint of a weight one modular form in terms of units and $p$-units. While in our previous work with Rotger the essential ingredient was the use of Galois deformations techniques following the computations of Bellaïche and Dimitrov, we propose a new approach exclusively using the properties of Beilinson--Flach classes. One of the key ingredients is the computation of a cyclotomic derivative of a cohomology class in the framework of Perrin-Riou theory, which can be se…
Let f be a cuspidal newform with complex multiplication (CM) and let p be an odd prime at which f is non-ordinary. We construct admissible p-adic L-functions for the symmetric powers of f, thus verifying general conjectures of Dabrowski and Panchishkin in this special case. We also construct their "mixed" plus and minus counterparts and prove an analogue of Pollack's decomposition of the admissible p-adic L-functions into mixed plus and minus p-adic L-functions. On the arithmetic side, we define corresponding mixed plus and minus Selmer groups. We unite the arithmetic with the analytic by firs…
On non-commutative Iwasawa theory and derivatives of Euler systems
本文利用约化行列式函子描述非交换代数的相对 $K_0$-群，结合 zeta element 推广，提出了非交换 $p$-进岩泽理论的主猜想及 Rubin-Stark 欧拉系统的高阶导数公式，为等变 Tamagawa 数猜想提供了广泛推广的证据。
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 reduction at $p$ and the arithmetic phenomena are specially rich. For that purpose, we resort to the principle of improved $p$-adic $L$-functions and study their $\mathcal L$-invariants. We further interpret these results in terms of derived cohomology classes coming from the setting of diagonal cycles, showing that the same $\mathcal L$-invariant which arises in …
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 Λ-torsion submodules, …
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。