Alex Kontorovich
We present an elementary derivation of the Jacquet-Shalika construction for the exterior square L-function on GL(n), as a classical Dirichlet series in the Fourier coefficients $A(m_1,...,m_{n-1})$.
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Let $G={\rm GL}_{2n}$ over a totally real number field $F$ and $n\geq 2$. Let $Π$ be a cuspidal automorphic representation of $G(\mathbb A)$, which is cohomological and a functorial lift from SO$(2n+1)$. The latter condition can be equivalently reformulated that the exterior square $L$-function of $Π$ has a pole at $s=1$. In this paper, we prove a rationality result for the residue of the exterior square $L$-function at $s=1$ and also for the holomorphic value of the symmetric square $L$-function at $s=1$ attached to $Π$. On the way, we also show a rationality result for the residue of the Ran…
We establish the existence and uniqueness of twisted exterior and symmetric square $γ$-factors in positive characteristic by studying the Siegel Levi case of generalized spinor groups. The corresponding theory in characteristic zero is due to Shahidi. In addition, in characteristic $p$ we prove that these twisted local factors are compatible with the local Langlands correspondence. As a consequence, still in characteristic $p$, we obtain a proof of the stability property of $γ$-factors under twists by highly ramified characters. Next we use the results on the compatibility of the Langlands-Sha…
On the exterior square $\varepsilon$-factors of $GL_n$
本文通过全局方法证明了GL_r(F)上外平方表示的ε因子在Jacquet-Shalika积分和Langlands-Shahidi方法下一致，同时给出了p-adic情况下的局部函数方程新证明。
In this paper, we show that the twisted partial exterior-square $L$-function has a meromorphic continuation to the whole complex plane with only two possible simple poles at $s=1$ and $s=0$. We do this by establishing the nonvanishing of the local zeta integrals defined by Jacquet and Shalika for any fixed $s_0$. The even case is treated in detail. The odd case is treated briefly, in which case, the $L$-function is shown to be entire.
We prove the functional equation of the non archimedean exterior-square L-function of irreducible representations of GL(n), when n is odd.
Let F be a non-archimedean local field of characteristic zero. Jacquet and Shalika attached a family of zeta integrals to unitary irreducible generic representations $π$ of GL_n(F). In this paper, we show that Jacquet-Shalika integral attains a certain L-function, so called the formal exterior square L-function, when the Whittaker function is associated to a newform for $π$. By consideration on the Galois side, formal exterior square L-functions are equal to exterior square L-functions for some principal series representations.
For a cuspidal automorphic representation Πof GL(4,A), H. Kim proved that the exterior square transfer \wedge^2Πis an isobaric automorphic representation of GL(6,A). In this paper we characterize those representations Πfor which \wedge^2Πis cuspidal.
We establish an analogue of a conjecture of Balasubramanian, Conrey, and Heath-Brown for the family of all Dirichlet characters with conductor up to $Q$. This forms another application of our work in developing an asymptotic large sieve.
Exterior square gamma factors for cuspidal representations of $\mathrm{GL}_n$: simple supercuspidal representations
本文计算了简单超尖点表示的局部扭曲外平方 gamma 因子，并据此证明了该类表示的局部逆定理。
Let $F$ be a non-Archimedean local field. Let $\mathcal{A}_n(F)$ be the set of equivalence classes of irreducible admissible representations of $\textrm{GL}_n(F)$, and $\mathcal{G}_n(F)$ be the set of equivalence classes of n-dimensional Frobenius semisimple Weil-Deligne representations of $W'_F$. The local Langlands correspondence(LLC) establishes the reciprocity maps $\textrm{Rec}_{n,F}: \mathcal{A}_n(F)\longrightarrow \mathcal{G}_n(F)$ , satisfying some nice properties. An important invariant under this correspondence is the L- and $ε$-factors. This is also expected to be true under paralle…
We verify the conjecture of [CFKRS] for the mean square near the critical point of Dirichlet L-functions for a composite modulus q. We also prove a kind of reciprocity formula when the second moment for a prime modulus is twisted by a character evaluated at a different prime.
We follow Jacquet-Shalika, Matringe and Cogdell-Matringe to define exterior square gamma factors for irreducible cuspidal representations of $\mathrm{GL}_n(\mathbb{F}_q)$. These exterior square gamma factors are expressed in terms of Bessel functions, or in terms of the regular characters associated with the cuspidal representations. We also relate our exterior square gamma factors over finite fields to those over local fields through level zero representations.
Finding the mean square averages of the Dirichlet $L$-functions over Dirichlet characters $χ$ of same parity is an active problem in number theory. Here we explicitly evaluate such averages of $L(3,χ)$ and $L(4,χ)$ using certain trigonometric sums and Bernoulli polynomials and express them in terms of the Euler totient function $ϕ$ and the Jordan totient function $J_s$.
We show that the local exterior square L-functions of GL_n constructed via the theory of integral representations by Jacquet and Shalika coincide with those constructed by the Langlands-Shahidi method for square integrable representations (and for all irreducible representations when n is even). We also deduce several local and global consequences.
Let $F$ be a $p$--adic field, i.e., a finite extension of $\mathbb Q_p$ for some prime $p$. The local Langlands correspondence attaches to each continuous $n$--dimensional $Φ$-semisimple representation $ρ$ of $W'_F$, the Weil--Deligne group for $\bar F/F$, an irreducible admissible representation $π(ρ)$ of $GL_n(F)$ such that, among other things, the local $L$- and $\varepsilon$-factors of pairs are preserved. This correspondence should be robust and preserve various parallel operations on the arithmetic and analytic sides, such as taking the exterior square or symmetric square. In this paper,…
In this work, we obtain an asymptotic formula for the twisted mean square of a Dirichlet $L$-function with a longer mollifier, whose coefficients are also more general than before. As an application we obtain that, for every Dirichlet $L$-function, more than 41.72\% of zeros are on the critical line and more than 40.74\% of zeros are simple and on the critical line. These proportions also improve previous results which were proved only for the Riemann zeta-function.
Arithmetic Progressions of Squares and Multiple Dirichlet Series
本文研究计数平方数本原三项等差数列的二元多重 Dirichlet 级数，证明其在 $\mathbb{C}^2$ 上的亚纯延拓，并借助 Tauberian 方法获得相关平方数等差数列与圆锥曲线有理点的计数。
Let $π$ be an irreducible admissible representation of $GL_m(F)$, where $F$ is a non-archimedean local field of characteristic zero. We follow the method developed by Cogdell and Piatetski-Shapiro to complete the computation of the local exterior square $L$-function $L(s,π,\wedge^2)$ in terms of $L$-functions of supercuspidal representations via an integral representation established by Jacquet and Shalika in $1990$. We analyze the local exterior square $L$-functions via exceptional poles and Bernstein and Zelevinsky derivatives. With this result, we show the equality of the local analytic $L$…
Improving earlier work of Balasubramanian, Conrey and Heath-Brown, we obtain an asymptotic formula for the mean-square of the Riemann zeta-function times an arbitrary Dirichlet polynomial of length $T^{1/2 + δ}$, with $δ= 0.01515....$ As an application we obtain an upper bound of the correct order of magnitude for the third moment of the Riemann zeta-function. We also refine previous work of Deshouillers and Iwaniec, obtaining asymptotic estimates in place of bounds. Using the work of Watt, we compute the mean-square of the Riemann zeta-function times a Dirichlet polynomial of length going up …
The mean square of the product of the Riemann zeta-function and a Dirichlet polynomial in the critical strip
本文改进了Matsumoto和Ishikawa的工作，通过获得Atkinson型误差项显式公式，给出了临界带内黎曼zeta函数与Dirichlet多项式乘积均方值的渐近公式，与广义Dirichlet除数问题紧密相关。
Mean square of inverses of Dirichlet $L$-functions involving conductors
本文研究Dirichlet $L$-函数的负平方矩，得到涉及导子的负二阶矩渐近公式，并应用于改进分圆数域中恢复短生成元算法的成功概率下界。
Finite-Sum Realization of Archimedean Asai and Exterior-Square $L$-Factors
该研究在数论领域，将 Archimedean Asai L-因子与外平方 L-因子分别表示为特定局部 ζ 积分的有限和，完善了 L-因子的有限和实现相关理论。
The analytic properties of automorphic L-functions have historically been obtained either through integral representations (the "Rankin-Selberg method"), or properties of the Fourier expansions of Eisenstein series (the "Langlands-Shahidi method"). We introduce a method based on pairings of automorphic distributions, that appears to be applicable to a wide variety of L-functions, including all which have integral representations. In some sense our method could be considered a completion of the Rankin-Selberg method because of its common features. We consider a particular but representative exa…
In this paper we give Rankin-Selberg integrals for the quasisplit unitary group on four variables, $\mathrm{GU}(2,2)$, and a closely-related quasisplit form of $\mathrm{GSpin}_6$. First, we give a two-variable Rankin-Selberg integral on $\mathrm{GU}(2,2)$. This integral applies to generic cusp forms, and represents the product of the exterior square (degree six) $L$-function and the standard (degree eight) $L$-function. Then we give a set of integral representations for just the degree six $L$-function on the quasisplit $\mathrm{GSpin}_6$. The $\mathrm{GSpin}_6$ integrals are reinterpretations…
We define epsilon factors for irreducible representations of finite general linear groups using Macdonald's correspondence. These epsilon factors satisfy multiplicativity, and are expressible as products of Gauss sums. The tensor product epsilon factors are related to the Rankin-Selberg gamma factors, by which we prove that the Rankin-Selberg gamma factors can be written as products of Gauss sums. The exterior square epsilon factors relate the Jacquet-Shalika exterior square gamma factors and the Langlands-Shahidi exterior square gamma factors for level zero supercuspidal representations. We p…
On Swan exponents of symmetric and exterior square Galois representations
本文研究了局部非Archimedes域上Galois表示经张量运算后的Swan指数关系，在剩余特征奇偶不同时分别确定或界定了对称与外平方表示的Swan指数，并应用于局部Langlands对应。
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