D. R. Heath-Brown
This is an exposition of work on Artin's Conjecture on the zeros of $p$-adic forms. A variety of lines of attack are described, going back to 1945. However there is particular emphasis on recent developments concerning quartic forms on the one hand, and systems of quadratic forms on the other.
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For a global function field K of positive characteristic p, we show that Artin conjecture for L-functions of geometric p-adic Galois representations of K is true in a non-trivial p-adic disk but is false in the full p-adic plane. In particular, we prove the non-rationality of the geometric unit root L-functions.
We define and study a $p$-adic analogue of the incomplete gamma function related to Morita's $p$-adic gamma function. We also discuss a combinatorial identity related to the Artin-Hasse series, which is a special case of the exponential principle in combinatorics. From this we deduce a curious $p$-adic property of $|\mathrm{Hom} (G,S_n)|$ for a topologically finitely generated group $G$, using a characterization of $p$-adic continuity for certain functions $f \colon \mathbb Z_{>0} \to \mathbb Q_p$ due to O'Desky-Richman. In the end, we give an exposition of some standard properties of the Arti…
An equivariant $p$-adic Artin conjecture
本文提出 Greenberg $p$-进 Artin 猜想的等变形式，并借助 Iwasawa 代数全商环的 Wedderburn 分解证明其在多类一维容许 $p$-进 Lie 扩张情形成立。
Artin formalism for $p$-adic $L$-functions of modular forms at non-ordinary primes
本文研究非普通素数处模形式的双变量带符号p进L函数在虚二次域上的Artin形式化，并为Castella–Ciperiani–Skinner–Sprung的预测提供证据。
On the Artin formalism for triple product $p$-adic $L$-functions
研究三重积\(p\)-adic \(L\)-函数因式分解问题，在相关定义性质无关时，以ETNC理念为指导，通过重塑问题为对角循环等的比较来进行分析。
Let $K$ be an imaginary quadratic field. In this article, we study the eigenvariety for $\mathrm{GL}_2/K$, proving an étaleness result for the weight map at non-critical classical points and a smoothness result at base-change classical points. We give three main applications of this; let $f$ be a $p$-stabilised newform of weight $k \geq 2$ without CM by $K$. Suppose $f$ has finite slope at $p$ and its base-change $f_{/K}$ to $K$ is $p$-regular. Then: (1) We construct a two-variable $p$-adic $L$-function attached to $f_{/K}$ under assumptions on $f$ that conjecturally always hold, in particular…
This article is the first of a pair of articles dealing with the Iwasawa theory of modular forms of weight 1 and, more generally, of Artin representations satisfying certain conditions. The main results in this part analyze the structure of certain Selmer groups for the Artin representation. In particular, it is shown that the Selmer groups are co-torsion as $Λ$-modules. For each Selmer group, we consider a generator of the characteristic ideal of its Pontrjagin dual. We call that the algebraic p-adic L-function. We also construct an analytic $p$-adic L-function via deformation to higher weigh…
On the Artin formalism for triple product $p$-adic $L$-functions: Chow--Heegner points vs. Heegner points
本文为综述性研究，围绕Artin形式体系探索三重乘积$p$进$L$-函数分解公式的证明步骤，核心依托显式互反律与Gross--Zagier型公式下的Chow--Heegner点和扭Heegner点比较。
On the Artin formalism for triple product $p$-adic $L$-functions: Super-factorization
该研究针对三个因子中两个具复乘的特殊情形，证明了姊妹篇提出的三重乘积$p$-进$L$-函数因子分解猜想，推进了相关数论问题的研究。
Wall-crossing and $p$-adic Artin formalism for ${\rm GSp}_4 \times {\rm GL}_2 \times {\rm GL}_2$
本文在 ${\ m GSp}_4 \ imes {\ m GL}_2 \ imes {\ m GL}_2$ 自守形式 $p$-adic 族中发展 $p$-adic Artin 形式主义，基于双重壁穿越原理，提出猜想并给出代数 $p$-adic $L$-函数及内窥镜情形下的证据。
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。
We construct a notion of p-adic measure on Artin n-stacks which are strongly of finite type over the ring of p-adic integers. We also prove the rationality of of the Poincare series and the Serre series associated with such stacks. Finally, we use motivic integration to show that the rationality of the Serre series is uniform with respect to p.
The Artin-Hasse $p$-section: weighted convolutions and $p$-adic recovery
本文研究 Artin-Hasse 指数的加权 $p$-截断卷积，证明并推广 Avitabile-Mattarei 猜想，得到基 $p$ 数字公式和 $p$-自动性，并通过 $p$-进不动点迭代与 Big-Witt 重构实现系数的精确恢复。
We show that Artin's conjecture concerning p-adic solubility of Diophantine equations fails for infinitely many systems of r homogeneous diagonal equations whenever r>1.
We give a new formula for the Artin conductor of an $\ell$-adic representation of the Weil group of a local field of residue characteristic $p\neq\ell$.
(Algebraic) $ p $-adic Artin formalism of twisted triple product Galois representations over real quadratic fields
本文将Büyükboduk等人的代数分解框架推广到实二次域，研究扭三元积伽罗瓦表示的分解问题，并首次在秩(0,2)情形下取得分解结果，推动了全实域上三元积动机的研究。
On the $p$-adic $L$-function and Iwasawa Main Conjecture for an Artin motive over a CM field
本文在若干技术条件下推广 Katz 的结果，构造了 CM 域上含 $d+1+δ_{F,p}$ 个变量的 $p$-adic Artin $L$-函数，插值相关 Artin $L$-函数临界值，是 Greenberg 全实域构造的类比且面临新困难。
G-companions on algebraic stacks and applications to canonical $\ell$-adic local systems on Shimura stacks
本文扩展了Drinfeld的定理至一般reductive群上的光滑Artin堆，并应用于研究Shimura堆上canonical ℓ-adic局部系统的兼容性。
It is established that for every pair of additive forms $f=\sum_{i=1}^s a_i x_i^k, g=\sum_{i=1}^s b_i x_i^k$ of degree $k$ in $s>2k^2$ variables the equations $f=g=0$ have a non-trivial $p$-adic solution for all odd primes $p$.
This work is devoted to the study of integral $p$-adic Hodge theory in the context of Artin stacks. For a Hodge-proper stack, using the formalism of prismatic cohomology, we establish a version of $p$-adic Hodge theory with the étale cohomology of the Raynaud generic fiber as an input. In particular, we show that the corresponding Galois representation is crystalline and that the associated Breuil-Kisin module is given by the prismatic cohomology. An interesting new feature of the stacky setting is that the natural map between étale cohomology of the algebraic and the Raynaud generic fibers is…
Artin's conjecture is established for all forms that can be realised as a diagonal form on an hyperplane.
We conjecture that the p-adic L-function of a non-trivial irreducible even Artin character over a totally real field is non-zero at all non-zero integers. This implies that a conjecture formulated by Coates and Lichtenbaum at negative integers extends in a suitable way to all positive integers. We also state a conjecture that for certain characters the Iwasawa series underlying the p-adic L-series have no multiple roots except for those corresponding to the zero at s=0 of the p-adic L-function. We provide some theoretical evidence for our first conjecture, and prove both conjectures by means o…
We introduce a new class of exponentials of Artin-Hasse type, called $\boldsymbolπ$-exponentials. These exponentials depends on the choice of a generator $\boldsymbolπ$ of the Tate module of a Lubin-Tate group $\mathfrak{G}$ over $\mathbb{Z}_p$. They arise naturally as solutions of solvable differential modules over the Robba ring. If $\mathfrak{G}$ is isomorphic to $\hat{\mathbb{G}}_m$ over $\mathbb{Z}_p$, we develop methods to test their over-convergence, and get in this way a stronger version of the Frobenius structure theorem for differential equations. We define a natural transformation o…
The author reviews results and conjectures of Selberg on a class of Dirichlet series functions which share properties with the Riemann zeta function, and he relates this work to the theory of Artin L-functions.
The purpose of this article is to prove a "Newton over Hodge" result for finite characters on curves. Let $X$ be a smooth proper curve over a finite field $\mathbb{F}_q$ of characteristic $p\geq 3$ and let $V \subset X$ be an affine curve. Consider a nontrivial finite character $ρ:π_1^{et}(V) \to \mathbb{C}^\times$. In this article, we prove a lower bound on the Newton polygon of the $L$-function $L(ρ,s)$. The estimate depends on monodromy invariants of $ρ$: the Swan conductor and the local exponents. Under certain nondegeneracy assumptions this lower bound agrees with the irregular Hodge filt…
Strong Artin Conjecture for Generalized Octahedral Representations in $\mathrm{GL}_3$
本文证明了三维最大可解阿廷表示（射影像为 $C_3^2\rtimes \mathrm{SL}(2,3)$）的强阿廷猜想，利用非伽罗瓦四次扩张的基变换与自守诱导，并推广至所有6维本原可解情形。
E. Artin conjectured that any integer $a >1$ which is not a perfect square is a primitive root modulo $p$ for infinitely many primes $p.$ Let $f_a(p)$ be the multiplicative order of the non-square integer $a$ modulo the prime $p.$ M. R. Murty and S. Srinivasan [10] showed that if $\sum_{p<x} \frac 1 {f_a(p)}= O(x^{1/4})$ then Artin's conjecture is true for $a.$ We relate the Murty-Srinivasan condition to sums involving the cyclotomic periods from the subfields of $\mathbb Q(e^{2πi/p})$ corresponding to the subgroups $<a> \subseteq \mathbb F*_p.$
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…
This is an expanded version of a write-up of a talk given in the fall of 2000 in Oberwolfach. A large part of it is intended to be understandable by non-number theorists with a mathematical background. The talk covered some of the history, results and ideas connected with Artin's celebrated primitive root conjecture dating from 1927. In the update several new results established after 2000 are also discussed.
A non-abelian large sieve and Artin's primitive root conjecture
本文研究经典阿贝尔大筛法不等式的非阿贝尔推广，假设该推广成立可证明阿廷原根猜想，还通过对偶技术得到针对该推广的无条件结果。
Refinements of Artin's primitive root conjecture
本文提出并证明了关于整数模素数阶分布的精细猜想，在广义黎曼假设下成立，无条件时证明较弱版本，推广了Artin原始根猜想。
We discuss how one could study asymptotics of cyclotomic quantities via the mean values of certain multiplicative functions and their Dirichlet series using a theorem of Delange. We show how this could provide a new approach to Artin's conjecture on primitive roots. We focus on whether a fixed prime has a certain order modulo infinitely many other primes. We also give an estimate for the mean value of one such Dirichlet series.
Théorie d'Iwasawa des motifs d'Artin et des formes modulaires de poids 1
本文研究 Artin 模式与权 1 模形式的 Iwasawa 理论，证明 cyclotomic Greenberg-Selmer 组的 torsion 性，并提出 Iwasawa 主猜想。
A question of Erdös on $3$-powerful numbers and an elliptic curve analogue of the Ankeny-Artin-Chowla conjecture
该研究利用椭圆曲线族的 Mordell-Weil 群构造 Erdös 关于 $3$-powerful numbers 猜想的解，并提出对应于 Ankeny-Artin-Chowla 猜想的椭圆曲线类比。
A counterexample to the Conjecture of Ankeny, Artin and Chowla
本文提供了一个反驳安肯耶、阿廷和楚拉猜想的反例，展示了存在一个素数p使得p整除y。
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…
Fermat quotients and the Ankeny-Artin-Chowla conjecture
本文为 Ankeny-Artin-Chowla 关于实二次域基本单位数的结论提供了精简证明，推广了该猜想，并据此将模 p 的二次非剩余的费马商与调和数之和联系起来。
In this article, we introduce a systematic and uniform construction of non-singular plane curves of odd degrees $n \geq 5$ which violate the local-global principle. Our construction works unconditionally for $n$ divisible by $p^2$ for some odd prime number $p$. Moreover, our construction also works for $n$ divisible by some $p \geq 5$ which satisfies a conjecture on $p$-adic properties of the fundamental units of $\mathbb{Q}(p^{1/3})$ and $\mathbb{Q}((2p)^{1/3})$. This conjecture is a natural cubic analogue of the classical Ankeny-Artin-Chowla-Mordell conjecture for $\mathbb{Q}(p^{1/2})$ and e…
Let K/F be an arbitrary Galois extension of number fields and r be a representation of Gal(K/F) into GSp(4,C). Let E_16 be the elemetary abelian group of order 16 and C_5 the cyclic group of order 5. If the image of r in the projective space PGSp(4,C) is isomorphic to the semidirect product of E_16 by C_5, then we show r satisfies Artin's conjecture by proving r corresponds to an automorphic representation. A specific case is given where r is primitive, so Artin's conjecture does not follow from previous results.
We prove that (under the assumption of the generalized Riemann hypothesis) a totally real multiquadratic number field $K$ has a positive density of primes $p \in \mathbb{Z}$ for which the image of the unit group $(\mathcal{O}_K)^{\times})$ in $(\mathcal{O}_K/p\mathcal{O}_K)^{\times})$ has minimal index $(p-1)/2$ if and only if $K$ contains a unit of norm $-1$.
We examine the sums $S(k,\,n)$ of the $k-$th powers of the $ϕ(n)$ integers $α_1<α_2<\cdots<α_{ϕ(n)}$ less than and prime to $n$ (Euler set) and prove a formula (new) for $S(3,\,n)$. If $n$ equals a prime $p$, we prove a theorem showing a connection of $S(3,\,p)$ with Artin's conjectural constant for primitive roots and with other functions involving primes.
We establish some new cases of Artin's conjecture. Our results apply to Galois representations over $\Q$ with image $S_5$ satisfying certain local hypotheses, the most important of which is that complex conjugation is conjugate to $(12)(34)$. In fact, we prove the stronger claim conjectured by Langlands that these representations are automorphic. For the irreducible representations of dimensions 4 and 6, our result follows from known 2-dimensional cases of Artin's conjecture (proved by Sasaki) as well as the functorial properties of the Asai transfer proved by Ramakrishnan. For the irreducible…
Suppose $ m,n\geq 2 $ are co prime integers. We prove certain new symmetries of the base $ n $ representation of $ 1/m $, and in particular characterize the subgroup generated by $ n $ inside $ (\mathbb{Z}/m\mathbb{Z})^\times $. As an application we give a sufficient condition for a prime $ p $ such that a non square number $ n $ is a primitive root modulo $ p $.
Factorization of Algebraic $p$-adic $L$-functions Attached to Adjoint Representations of Coleman Families: Non-critical Case
该研究针对模形式Coleman族伴随表示的代数$p$-进$L$-函数，借助多类数论工具证明含Selmer复形行列式的分解公式，建立$p$-进Artin形式体系并推广了前人结果至$p$-非平凡情形。
We formulate a conjectural p-adic analogue of Borel's theorem relating regulators for higher K-groups of number fields to special values of the corresponding zeta-functions, using syntomic regulators and p-adic L-functions. We also formulate a corresponding conjecture for Artin motives, and state a conjecture about the precise relation between the p-adic and classical situations. Parts of he conjectures are proved when the number field (or Artin motive) is Abelian over the rationals, and all conjectures are verified numerically in some other cases.
We prove Haynes' version of the Duffin--Schaeffer conjecture for the $p$-adic numbers. In addition, we prove several results about an associated related but false conjecture, related to $p$-adic approximation in the spirit of Jarník and Lutz.
Since Hooley's seminal 1967 resolution of Artin's primitive root conjecture under the Generalized Riemann Hypothesis, numerous variations of the conjecture have been considered. We present a framework generalizing and unifying many previously considered variants, and prove results in this full generality (under GRH).
In this paper, we survey some recent results on the Artin conjecture and discuss some aspects for the Artin 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.
The primary goal of this article is to study $p$-adic Beilinson conjectures in the presence of exceptional zeros for Artin motives over CM fields. In more precise terms, we address a question raised by Hida and Tilouine on the order of vanishing of Katz $p$-adic $L$-functions associated to CM fields, by means of a leading term formula we prove in terms of Rubin-Stark elements. In the particular case when the CM field in question is imaginary quadratic, our leading term formula and its consequences we record herein are unconditional.
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. …
On Artin's conjecture on average and short character sums
本文证明了在特定条件下，关于素数中原根分布的渐进行为符合Artin猜想，并改进了相关短特征和估计的范围。
The concept of Artin transfer pattern $((\ker(T_{K,N_i}))_i,(\mathrm{Cl}_p(N_i))_i)$ for homogeneous multiplets $(N_1,\ldots,N_m)$ of unramified cyclic prime degree p extensions $N_i/K$ of a base field K with p-class transfer homomorphisms$T_{K,N_i}:\,\mathrm{Cl}_p(K)\to\mathrm{Cl}_p(N_i)$ is generalized for heterogeneous multiplets of ramified extensions. By application to quadratic subfields K of dihedral fields N of degree 2p with an odd prime p, a conjecture of Scholz concerning the index of subfield units, $(U_N:U_0)$, for ramified extensions N/K with conductor f>1 is verified computation…
We prove the conjectured compatibility of $p$-adic fundamental lines with specializations at motivic points for a wide class of $p$-adic families of $p$-adic Galois representations (for instance, the families which arise from $p$-adic families of automorphic representations of the unit group of a quaternion algebra or of a totally definite unitary groups) and deduce the compatibility of the Equivariant Tamagawa Number Conjectures for them. However, we also show that fundamental lines are not compatible with arbitrary characteristic zero specializations with values in a domain in general. This …
Restricting ourselves to elliptic curves over $\mathbb{Q}$, we reformulate the $p$-adic Beilinson conjecture due to Perrin-Riou, which is customized to our computational approach. We then develop a new algorithm for numerical verifications of the $p$-adic Beilinson conjecture, which is based on the theory of rigid cohomology and $F$-isocrystals.
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.
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 show that the sum of the traces of Frobenius elements of Artin $L$-functions in a family of $G$-fields satisfies the Gaussian distribution under certain counting conjectures. We prove the counting conjectures for $S_4$ and $S_5$-fields. We also show central limit theorem for modular form $L$-functions with the trivial central character with respect to congruence subgroups as the level goes to infinity.
Recently Ritter and Weiss introduced an equivariant "main conjecture" than generalizes and refines the Main Conjecture of Iwasawa theory. In this paper, we show that, for the prime 2 and a dihedral extension of order 8 over Q, this conjecture is equivalent to a congruence condition on the coefficients of a power series with 2-adic integral coefficients constructed using the 2-adic L-series associated to the extension. We then verify that this congruence condition holds for the first coefficients in a large number of examples.
The purpose of this article is to give an overview of the series of papers [BK1], [BK2] concerning the $p$-adic Beilinson conjecture of motives associated to Hecke characters of an imaginary quadratic field $K$, for a prime $p$ which splits in $K$.
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