M. Ram Murty
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.
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Let $Γ\backslash\mathbb H^3$ be a finite-volume quotient of the upper-half space, where $Γ\subset {\rm SL}(2,\mathbb C)$ is a discrete subgroup. To a finite dimensional unitary representation $χ$ of $Γ$ one associates the Selberg zeta function $Z(s;Γ;χ)$. In this paper we prove the Artin formalism for the Selberg zeta function. Namely, if $\tildeΓ$ is a finite index group extension of $Γ$ in ${\rm SL}(2,\mathbb C)$, and $π={\rm Ind}_Γ^{\tildeΓ}χ$ is the induced representation, then $Z(s;Γ;χ)=Z(s;\tildeΓ;π)$. In the second part of the paper we prove by a direct method the analogous identity for…
For Kleinian groups acting on hyperbolic three-space, we prove factorization formulas for both the Selberg zeta-function and the automorphic scattering matrix. We extend results of Venkov and Zograf from Fuchsian groups, to Kleinian groups, and we give a proof that is simple and extendable to more general groups.
We derive a fully explicit version of the Selberg trace formula for twist-minimal Maass forms of weight 0 and arbitrary conductor and nebentypus character, and apply it to prove two theorems. First, conditional on Artin's conjecture, we classify the even 2-dimensional Artin representations of small conductor; in particular, we show that the even icosahedral representation of smallest conductor is the one found by Doud and Moore, of conductor 1951. Second, we verify the Selberg eigenvalue conjecture for groups of small level, improving on a result of Huxley from 1985.
We generalise the work of Sarnak-Tsimerman to twisted sums of Kloosterman sums and thus give evidence towards the twisted Linnik-Selberg Conjecture.
In the paper we introduce the new approach how to use an orthonormality relation of coefficients of Dirichlet series defining given L-functions from the Selberg class to prove joint universality.
The paper is withdrawn. The reviewer pointed out that the assertion that the quantity $D_Γ$ is finite (used by the author) is still unproved for the general group, and perhaps is false.
As the reviewer have pointed out, the proof of Roelke Conjecture contains an error. For cofinite groups, we obtain a formula connecting the discrete spectrum of Laplace operator and the resonance spectrum. Using this formula, we give a proof of the Roelke conjecture. In this version, we make several corrections in Theorem 1 and Lemma 5.
We construct motivic cohomology classes attached to Rankin--Selberg convolutions of modular forms of weights $\ge 2$, show that these vary analytically in p-adic families, and relate their image under the p-adic regulator map to values of L-functions. As consequences, we prove new cases of Perrin-Riou's conjecture on motivic L-values; we prove finiteness results for Tate--Shafarevich groups for twists of elliptic curves by dihedral Artin characters; and we prove one inclusion in the Iwasawa main conjecture for a single modular form over an imaginary quadratic field.
In this paper we prove that the p-adic L-function that interpolates the Rankin-Selberg product of a general modular form and a CM form of higher weight divides the characteristic ideal of the corresponding Selmer group. This is one divisibility of the Iwasawa main conjecture for the p-adic L-function. We prove this conjecture using the congruences between Klingen Eisensteinseries and cusp forms on the group GU(3; 1), following the strategy of a recent work of C. Skinner and E. Urban. This theorem can be used to deduce a converse of Gross-Zagier-Kolyvagin theorem and the precise BSD formula in …
In this paper we prove that the $p$-adic $L$-function that interpolates the Rankin-Selberg product of a general weight two modular form which is unramified and non-ordinary at $p$, and an ordinary CM form of higher weight contains the characteristic ideal of the corresponding Selmer group. This is one divisibility of the Iwasawa-Greenberg main conjecture for the $p$-adic $L$-function. This generalizes an earlier work of the author to the non-ordinary case. The result of this paper plays a crucial role in the proof of Iwasawa main conjecture and refined Birch-Swinnerton-Dyer formula for supersi…
We prove a version of the Extra-zero conjecture formulated by the first named author for p-adic L-functions associated to Rankin-Selberg convolutions of modular forms of the same weight. The novelty of this result is to provide strong evidence in support of this conjecture in the non-critical case, which remained essentially unstudied.
Algebraicity of ratios of Rankin-Selberg $L$-functions and applications to Deligne's conjecture
本文证明了权至少5的模形式对称幂$L$-函数临界值的Deligne猜想，并建立了Blasius猜想的新情形，以及$\GL_n \times \GL_2$的Rankin-Selberg $L$-函数在非平衡情形下的代数性结果。
We study the Fitting ideals over the finite layers of the cyclotomic $\mathbb{Z}_p$-extension of $\mathbb{Q}$ of Selmer groups attached to the Rankin--Selberg convolution of two modular forms $f$ and $g$. Inspired by the Theta elements for modular forms defined by Mazur and Tate in ``Refined conjectures of the Birch and Swinnerton-Dyer type'', we define new Theta elements for Rankin--Selberg convolutions of $f$ and $g$ using Loeffler--Zerbes' geometric $p$-adic $L$-functions attached to $f$ and $g$. Under certain technical hypotheses, we generalize a recent work of Kim--Kurihara on elliptic cu…
Let $π$ be an irreducible admissible (complex) representation of $GL(2)$ over a non-archimedean characteristic zero local field with odd residual characteristic. In this paper we prove the equality between the local symmetric square $L$-function associated to $π$ arising from integral representations and the corresponding Artin $L$-function for its Langlands parameter through the local Langlands correspondence. With this in hand, we show the stability of local symmetric $γ$-factors attached to $π$ under highly ramified twists.
Artin's conjecture is established for all forms that can be realised as a diagonal form on an hyperplane.
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.
The conjectural newform theory for generic representations of $p$-adic ${\rm SO}_{2n+1}$ was formulated by P.-Y. Tsai in her thesis in which Tsai also verified the conjecture when the representations are supercuspidal. The main purpose of this work is to compute the Rankin-Selberg integrals for ${\rm SO}_{2n+1}\times{\rm GL}_r$ with $1\le r\le n$ attached to newforms and also oldforms under the validity of the conjecture.
We show that Selberg's eigenvalue conjecture concerning small eigenvalues of the automorphic Laplacian for congruence groups is equivalent to a conjecture about the non-existence of residual eigenvalues for a perturbed system. We prove this using a combination of methods from asymptotic perturbation theory and number theory.
In a 2005 paper, Yang constructed families of Hilbert Eisenstein series, which when restricted to the diagonal are conjectured to span the underlying space of elliptic modular forms. One approach to these conjectures is to show the non-vanishing of an inner product of elliptic eigenforms with the restrictions of Eisenstein series. In this paper, we compute this inner product locally by using explicit values of new vectors in the Waldspurger model.
Anticyclotomic main conjecture and the non-triviality of Rankin-Selberg $L$-values in Hida families
本文通过双变量反圆控制定理及p进L-函数特殊化的自对偶扭转关系，证明了Hida族的双变量反圆Iwasawa主猜想和水平非消失猜想的确定版本。
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。
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.
We explore an idea of Conrey and Li of expressing the Selberg trace formula as a Dirichlet series. We describe two applications, including an interpretation of the Selberg eigenvalue conjecture in terms of quadratic twists of certain Dirichlet series, and a formula for an arithmetically weighted sum of the complete symmetric square L-functions associated to cuspidal Maass newforms of squarefree level N>1.
The subconvexity problem for Rankin-Selberg and triple product L-functions
本文扩展 Michel-Venkatesh 方法，研究 Rankin-Selberg 与三重积 L-函数的次凸性问题，在允许联合分歧和导子下降的条件下验证局部猜想，获得新的次凸界。
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.
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 $<a> \subseteq \mathbb F*_p.$
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。
Iwasawa's main conjecture for Rankin-Selberg motives in the anticyclotomic case
本文在酉群的Gan--Gross--Prasad猜想框架下，研究了CM域上GL(n)×GL(n+1)尖点自守表示沿反循环方向的岩泽理论，并证明了相应岩泽主猜想的单侧整除性。
We give a new non-trivial upper bound for the Selberg integral of the three-divisor function $d_3(n)$. Our method applies our recent conjecture together with Laporta, for the modified Selberg integral of $d_3(n)$, and a kind of modified Gallagher Lemma for the exponential sums.
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.
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。
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 $p$.
The Ramanujan-Petersson and Selberg conjectures for Maass forms
本文通过自守分布理论和伪微分分析，证明了马ass形式的拉马努金-彼得森猜想，并推导了海格尔群Γ₀(M)的塞尔伯格猜想。
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。
Anticyclotomic $p$-adic $L$-functions for Rankin--Selberg product
本文在两种根数下沿反分圆方向为 hermitian 型自守形式的 Rankin--Selberg 乘积构造了 $p$-adic $L$-functions，并提出相应的非消失猜想和主猜想。
In this article, we study the Beilinson-Bloch-Kato conjecture for motives corresponding to the Rankin-Selberg product of conjugate self-dual automorphic representations, within the framework of the Gan-Gross-Prasad conjecture. We show that if the central critical value of the Rankin-Selberg $L$-function does not vanish, then the Bloch-Kato Selmer group with coefficients in a favorable field of the corresponding motive vanishes. We also show that if the class in the Bloch-Kato Selmer group constructed from certain diagonal cycle does not vanish, which is conjecturally equivalent to the nonvanis…
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。
On the Argument of the Lerch, Chowla--Selberg Formula and CM Values of $η(τ)$
本文给出了不含绝对值的完全显式Lerch、Chowla--Selberg公式，并应用于η函数的个别CM值的参数，同时提出了关于这些CM值的精确猜想并提供了大量数据支持。
Congruences for the ratios of Rankin--Selberg $L$-functions
本文通过计算研究Rankin-Selberg L-函数比值的同余关系，提出一个普遍的精确猜想。
J.-C. Yoccoz proposed a natural extension of Selberg's Eigenvalue Conjecture to moduli spaces of abelian differentials. We prove an approximation to this conjecture. This gives a qualitative generalization of Selberg's $\frac{3}{16}$ Theorem to moduli spaces of abelian differentials on surfaces of genus $\geq 2$.
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 $-1$.
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 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 $.
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).
Residues of Rankin-Selberg Zeta integrals and the split non-tempered Gan-Gross-Prasad conjectures
本文通过 Zeta 积分残差正则化方法，解决了非温控 Gan-Gross-Prasad 猜想及其细化版本在局部域上的问题。
Let $f$ and $g$ be two modular forms which are non-ordinary at $p$. The theory of Beilinson-Flach elements gives rise to four rank-one non-integral Euler systems for the Rankin-Selberg convolution $f \otimes g$, one for each choice of $p$-stabilisations of $f$ and $g$. We prove (modulo a hypothesis on non-vanishing of $p$-adic $L$-fuctions) that the $p$-parts of these four objects arise as the images under appropriate projection maps of a single class in the wedge square of Iwasawa cohomology, confirming a conjecture of Lei-Loeffler-Zerbes. Furthermore, we define an explicit logarithmic matrix…
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…
In this paper, we survey some recent results on the Artin conjecture and discuss some aspects for the Artin conjecture.
A proof of Newman's conjecture for the extended Selberg class
本文通过新方法证明了扩展塞尔伯格类中L函数的布里尤恩-新曼常数非负性，从而验证了新曼猜想。
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.
We formulate and for the most part prove a conjecture in the style of Mazur-Greenberg for the nonvanishing of central values of Rankin-Selberg $L$-functions attached to elliptic curves in abelian extensions of imaginary quadratic fields. This in particular generalizes the theorem of Rohrlich on $L$-functions of elliptic curves in cyclotomic towers to the setting of abelian extensions of imaginary quadratic fields, corresponding to families of degree-four $L$-functions given by $\operatorname{GL}(2)\times\operatorname{GL}(2)$ Rankin-Selberg $L$-functions. It also generalizes the theorems of Roh…
We establish the equivalence of conjectures concerning the pair correlation of zeros of $L$-functions in the Selberg class and the variances of sums of a related class of arithmetic functions over primes in short intervals. This extends the results of Goldston & Montgomery [7] and Montgomery & Soundararajan [11] for the Riemann zeta-function to other $L$-functions in the Selberg class. Our approach is based on the statistics of the zeros because the analogue of the Hardy-Littlewood conjecture for the auto-correlation of the arithmetic functions we consider is not available in general. One of o…
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