We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.
To send content items to your account,
please confirm that you agree to abide by our usage policies.
If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account.
Find out more about sending content to .
To send content items to your Kindle, first ensure no-reply@cambridge.org
is added to your Approved Personal Document E-mail List under your Personal Document Settings
on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part
of your Kindle email address below.
Find out more about sending to your Kindle.
Note you can select to send to either the @free.kindle.com or @kindle.com variations.
‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi.
‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.
We prove a functional equation for a vector valued real analytic Eisenstein series transforming with the Weil representation of $\operatorname{Sp}(n,\mathbb{Z})$ on $\mathbb{C}[(L^{\prime }/L)^{n}]$. By relating such an Eisenstein series with a real analytic Jacobi Eisenstein series of degree $n$, a functional equation for such an Eisenstein series is proved. Employing a doubling method for Jacobi forms of higher degree established by Arakawa, we transfer the aforementioned functional equation to a zeta function defined by the eigenvalues of a Jacobi eigenform. Finally, we obtain the analytic continuation and a functional equation of the standard $L$-function attached to a Jacobi eigenform, which was already proved by Murase, however in a different way.
We improve upon the local bound in the depth aspect for sup-norms of newforms on $D^\times$, where $D$ is an indefinite quaternion division algebra over ${\mathbb {Q}}$. Our sup-norm bound implies a depth-aspect subconvexity bound for $L(1/2, f \times \theta _\chi )$, where $f$ is a (varying) newform on $D^\times$ of level $p^n$, and $\theta _\chi$ is an (essentially fixed) automorphic form on $\textrm {GL}_2$ obtained as the theta lift of a Hecke character $\chi$ on a quadratic field. For the proof, we augment the amplification method with a novel filtration argument and a recent counting result proved by the second-named author to reduce to showing strong quantitative decay of matrix coefficients of local newvectors along compact subsets, which we establish via $p$-adic stationary phase analysis. Furthermore, we prove a general upper bound in the level aspect for sup-norms of automorphic forms belonging to any family whose associated matrix coefficients have such a decay property.
This paper completes the construction of
$p$
-adic
$L$
-functions for unitary groups. More precisely, in Harris, Li and Skinner [‘
$p$
-adic
$L$
-functions for unitary Shimura varieties. I. Construction of the Eisenstein measure’, Doc. Math.Extra Vol. (2006), 393–464 (electronic)], three of the authors proposed an approach to constructing such
$p$
-adic
$L$
-functions (Part I). Building on more recent results, including the first named author’s construction of Eisenstein measures and
$p$
-adic differential operators [Eischen, ‘A
$p$
-adic Eisenstein measure for unitary groups’, J. Reine Angew. Math.699 (2015), 111–142; ‘
$p$
-adic differential operators on automorphic forms on unitary groups’, Ann. Inst. Fourier (Grenoble)62(1) (2012), 177–243], Part II of the present paper provides the calculations of local
$\unicode[STIX]{x1D701}$
-integrals occurring in the Euler product (including at
$p$
). Part III of the present paper develops the formalism needed to pair Eisenstein measures with Hida families in the setting of the doubling method.
Let $d_{3}(n)$ be the divisor function of order three. Let $g$ be a Hecke–Maass form for $\unicode[STIX]{x1D6E4}$ with $\unicode[STIX]{x1D6E5}g=(1/4+t^{2})g$. Suppose that $\unicode[STIX]{x1D706}_{g}(n)$ is the $n$th Hecke eigenvalue of $g$. Using the Voronoi summation formula for $\unicode[STIX]{x1D706}_{g}(n)$ and the Kuznetsov trace formula, we estimate a shifted convolution sum of $d_{3}(n)$ and $\unicode[STIX]{x1D706}_{g}(n)$ and show that
This corrects and improves the result of the author [‘Shifted convolution sum of $d_{3}$ and the Fourier coefficients of Hecke–Maass forms’, Bull. Aust. Math. Soc.92 (2015), 195–204].
We prove that the complete
$L$
-function associated to any cuspidal automorphic representation of
$\operatorname{GL}_{2}(\mathbb{A}_{\mathbb{Q}})$
has infinitely many simple zeros.
We characterize the cuspidal representations of $G_{2}$ whose standard ${\mathcal{L}}$-function admits a pole at $s=2$ as the image of the Rallis–Schiffmann lift for the commuting pair ($\widetilde{\text{SL}}_{2}$, $G_{2}$) in $\widetilde{\text{Sp}}_{14}$. The image consists of non-tempered representations. The main tool is the recent construction, by the second author, of a family of Rankin–Selberg integrals representing the standard ${\mathcal{L}}$-function.
We answer a challenge posed in Booker [
$L$
-functions as distributions. Math. Ann.363(1–2) (2015), 423–454, §1.3] by proving a version of Weil’s converse theorem [Über die Bestimmung Dirichletscher Reihen durch Funktionalgleichungen, Math. Ann.168 (1967), 149–156] that assumes a functional equation for character twists but allows their root numbers to vary arbitrarily.
We generalize a method of Conrey and Ghosh [Simple zeros of the Ramanujan
$\unicode[STIX]{x1D70F}$
-Dirichlet series. Invent. Math.94(2) (1988), 403–419] to prove quantitative estimates for simple zeros of modular form
$L$
-functions of arbitrary conductor.
The standard twist
$F(s,\unicode[STIX]{x1D6FC})$
of
$L$
-functions
$F(s)$
in the Selberg class has several interesting properties and plays a central role in the Selberg class theory. It is therefore natural to study its finer analytic properties, for example the functional equation. Here we deal with a special case, where
$F(s)$
satisfies a functional equation with the same
$\unicode[STIX]{x1D6E4}$
-factor of the
$L$
-functions associated with the cusp forms of half-integral weight; for simplicity we present our results directly for such
$L$
-functions. We show that the standard twist
$F(s,\unicode[STIX]{x1D6FC})$
satisfies a functional equation reflecting
$s$
to
$1-s$
, whose shape is not far from a Riemann-type functional equation of degree 2 and may be regarded as a degree 2 analog of the Hurwitz–Lerch functional equation. We also deduce some results on the growth on vertical strips and on the distribution of zeros of
$F(s,\unicode[STIX]{x1D6FC})$
.
In this paper we prove a conjecture relating the Whittaker function of a certain generating function with the Whittaker function of the theta representation
$\unicode[STIX]{x1D6E9}_{n}^{(n)}$
. This enables us to establish that a certain global integral is factorizable and hence deduce the meromorphic continuation of the standard partial
$L$
function
$L^{S}(s,\unicode[STIX]{x1D70B}^{(n)})$
. In fact we prove that this partial
$L$
function has at most a simple pole at
$s=1$
. Here,
$\unicode[STIX]{x1D70B}^{(n)}$
is a genuine irreducible cuspidal representation of the group
$\text{GL}_{r}^{(n)}(\mathbf{A})$
.
We prove an exact formula for the second moment of Rankin–Selberg
$L$
-functions
$L(\frac{1}{2},f\times g)$
twisted by
$\unicode[STIX]{x1D706}_{f}(p)$
, where
$g$
is a fixed holomorphic cusp form and
$f$
is summed over automorphic forms of a given level
$q$
. The formula is a reciprocity relation that exchanges the twist parameter
$p$
and the level
$q$
. The method involves the Bruggeman–Kuznetsov trace formula on both ends; finally the reciprocity relation is established by an identity of sums of Kloosterman sums.
We give a Rankin–Selberg integral representation for the Spin (degree eight)
$L$
-function on
$\operatorname{PGSp}_{6}$
that applies to the cuspidal automorphic representations associated to Siegel modular forms. If
$\unicode[STIX]{x1D70B}$
corresponds to a level-one Siegel modular form
$f$
of even weight, and if
$f$
has a nonvanishing maximal Fourier coefficient (defined below), then we deduce the functional equation and finiteness of poles of the completed Spin
$L$
-function
$\unicode[STIX]{x1D6EC}(\unicode[STIX]{x1D70B},\text{Spin},s)$
of
$\unicode[STIX]{x1D70B}$
.
for $m,n$ positive integers, to all $s\in \mathbb{C}$. There are poles of the function corresponding to zeros of the Riemann zeta function and the spectral parameters of Maass forms. The analytic properties of this function are rather delicate. It turns out that the spectral expansion of the zeta function converges only in a left half-plane, disjoint from the region of absolute convergence of the Dirichlet series, even though they both are analytic expressions of the same meromorphic function on the entire complex plane.
Let $\{{\it\phi}_{j}(z):j\geq 1\}$ be an orthonormal basis of Hecke–Maass cusp forms with Laplace eigenvalue $1/4+t_{j}^{2}$. Let ${\it\lambda}_{j}(n)$ be the $n$th Fourier coefficient of ${\it\phi}_{j}$ and $d_{3}(n)$ the divisor function of order three. In this paper, by the circle method and the Voronoi summation formula, the average value of the shifted convolution sum for $d_{3}(n)$ and ${\it\lambda}_{j}(n)$ is considered, leading to the estimate
We address the problem of evaluating an $L$-function when only a small number of its Dirichlet coefficients are known. We use the approximate functional equation in a new way and find that it is possible to evaluate the $L$-function more precisely than one would expect from the standard approach. The method, however, requires considerably more computational effort to achieve a given accuracy than would be needed if more Dirichlet coefficients were available.
As the simplest case of Langlands functoriality, one expects the existence of the symmetric power $S^n(\pi )$, where $\pi $ is an automorphic representation of ${\rm GL}(2,{\mathbb{A}})$ and ${\mathbb{A}}$ denotes the adeles of a number field $F$. This should be an automorphic representation of ${\rm GL}(N,{\mathbb{A}})$ ($N=n+1)$. This is known for $n=2,3$ and $4$. In this paper we show how to deduce the general case from a recent result of J.T. on deformation theory for ‘Schur representations’, combined with expected results on level-raising, as well as another case (a particular tensor product) of Langlands functoriality. Our methods assume $F$ totally real, and the initial representation $\pi $ of classical type.
We study the distribution, in the space of Satake parameters, of local components of Siegel cusp forms of genus 2 and growing weight k, subject to a specific weighting which allows us to apply results concerning Bessel models and a variant of Petersson’s formula. We obtain for this family a quantitative local equidistribution result, and derive a number of consequences. In particular, we show that the computation of the density of low-lying zeros of the spinor L-functions (for restricted test functions) gives global evidence for a well-known conjecture of Böcherer concerning the arithmetic nature of Fourier coefficients of Siegel cusp forms.
We generalize the method of A. R. Booker (Poles of Artin L-functions and the strong Artin conjecture, Ann. of Math. (2) 158 (2003), 1089–1098; MR 2031863(2004k:11082)) to prove a version of the converse theorem of Jacquet and Langlands with relaxed conditions on the twists by ramified idèle class characters.
We break the convexity bound in the t-aspect for L-functions attached to cusp forms f for GL2(k) over arbitrary number fields k. The argument uses asymptotics with error term with a power saving, for second integral moments over spectral families of twists L(s,f⊗χ) by Grossencharacters χ, from our previous paper on integral moments.
We obtain second integral moments of automorphic L-functions on adele groups GL2 over arbitrary number fields, by a spectral decomposition using the structure and representation theory of adele groups GL1 and GL2. This requires reformulation of the notion of Poincaré series, replacing the collection of classical Poincaré series over GL2(ℚ) or GL2(ℚ(i)) with a single, coherent, global object that makes sense over a number field. This is the first expression of integral moments in adele-group terms, distinguishing global and local issues, and allowing uniform application to number fields. When specialized to the field of rational numbers ℚ, we recover the classical results on moments.
Recommend this
Email your librarian or administrator to recommend adding this to your organisation's collection.