We partially generalize the results of Kudla and Rallis on the poles of degenerate, Siegel-parabolic Eisenstein series to residual-data Eisenstein series. In particular, for $a,b$ integers greater than 1, we show that poles of the Eisenstein series induced from the Speh representation $Δ(τ,b)$ on the Levi $\mathrm{GL}_{ab}$ of $\mathrm{Sp}_{2ab}$ are located in the "segment" of half integers $X_{b}$ between a "right endpoint" and its negative, inclusive of endpoints. The right endpoint is $\pm b/2$, or $(b-1)/2$, depending on the analytic properties of the automorphic $L$-functions attached to $τ$. We study the automorphic forms $Φ_{i}^{(b)}$ obtained as residues at the points $s_i^{(b)}$ (defined precisely in the paper) by calculating their cuspidal exponents in certain cases. In the case of the "endpoint" $s_0^{(b)}$ and `first interior point' $s_1^{(b)}$ in the segment of singularity points, we are able to determine a set containing \textit{all possible} cuspidal exponents of $Φ_0^{(b)}$ and $Φ_1^{(b)}$ precisely for all $a$ and $b$. In these cases, we use the result of the calculation to deduce that the residual automorphic forms lie in $L^2(G(k)\backslash G(\mathbf{A}))$. In a more precise sense, our result establishes a relationship between, on the one hand, the actually occurring cuspidal exponents of $Φ_i^{(b)}$, residues at interior points which lie to the right of the origin, and, on the other hand, the "analytic properties" of the original residual-data Eisenstein series at the origin. This preprint is a longer version of the paper "Analytic Properties of Residual Eisenstein Series, I", with the details of some proofs added and some additional examples adduced in support of the main conjecture.
Let $G$ be a split group of type $F_4$ defined over a number field. We study the square-integrable automorphic representations of $G$ that can be realized as leading terms of degenerate Eisenstein series associated to various maximal parabolic subgroups. These representations appear in the residual spectrum. The local representation theory over finite places plays a central role in our work.
We prove a sup norm bound on the real-analytic Eisenstein series, of the form $E(z, 1/2 + iT) \ll T^{3/8 + \varepsilon}$, uniformly for $z$ in a fixed compact subset of $\mathbb{H}$.
In this work, we define a new type of Eisenstein-like series by using Pell-Lucas numbers and call them the Pell-Lucas-Eisenstein Series. Firstly, we show that the Pell-Lucas-Eisenstein series are convergent on their domain. Afterwards we prove that they satisfy some certain functional equations. Proofs follows from some on calculations on Pell-Lucas numbers.
The present paper provides the details omitted from the more concise study "On the zeros of Eisenstein series for $Γ_0^* (5)$ and $Γ_0^* (7)$." We locate almost all of the zeros of the Eisenstein series associated with the Fricke groups of level 5 and 7 in their fundamental domains by applying and extending the method of F. K. C. Rankin and H. P. F. Swinnerton-Dyer (1970). We also use the arguments of some terms of the Eisenstein series in order to improve existing error bounds.
In this note, we give a criteria whether given two Eisenstein polynomials over a padic field define the same extension (Proposition 1.6). In particular, we completely identify Eisenstein polynomials of degree p (Theorem 1.16). This note is an English translation of a part of the original paper [12].
We prove a formula for the coefficients of a weight $3/2$ Cohen-Eisenstein series of square-free level $N$. This formula generalizes a result of Gross and in particular, it proves a conjecture of Quattrini. Let $l$ be an odd prime number. For any elliptic curve $E$ defined over $\mathbb{Q}$ of rank zero and square-free conductor $N$, if $l \mid |E(\mathbb{Q})|$, under certain conditions on the Shafarevich-Tate group $III_D$, we show that $l$ divides $|III_D|$ if and only if $l$ divides the class number $h(-D)$ of $\mathbb{Q}(\sqrt{-D}).$
Let $f$ a smooth function with compact support defined on $PSL(2,\mathbb{Z}[i]) \backslash PSL(2,\mathbb{C})$, we prove a formula for the Mellin transform of $f$, then we can define the micro-local lift $dε_{it}$ to $SL(2,\mathbb{C})$. We calculate $(f,dε_{it})$ for $f$ a cuspidal form and for $f$ an incomplete Eisenstein series. We also establish asymptotic estimates when $t$ tends to $\infty$. We conjecture that a new positive distribution $dε_{it}^F$, constructed with the Friedrichs' symmetrization technique, satisfies the same asymptotic estimates that $dε_{it}$. This would imply the quant…
In this short note, we will construct a harmonic Eisenstein series of weight one, whose image under the $ξ$-operator is a weight one Eisenstein series studied by Hecke.
For a totally real field $F$, a finite extension $\mathbf{F}$ of $\mathbf{F}_p$ and a Galois character $χ: G_F \to \mathbf{F}^{\times}$ unramified away from a finite set of places $Σ\supset \{\mathfrak{p} \mid p\}$ consider the Bloch-Kato Selmer group $H:=H^1_Σ(F, χ^{-1})$. In an earlier paper of the authors it was proved that the number $d$ of isomorphism classes of (non-semisimple, reducible) residual representations $\overlineρ$ giving rise to lines in $H$ which are modular by some $ρ_f$ (also unramified outside $Σ$) satisfies $d \geq n:= \dim_{\mathbf{F}} H$. This was proved under the assu…
We firstly discuss properties of the L-group for Brylinski-Deligne (BD) extensions constructed by M. Weissman. Secondly, the Gindikin-Karpelevich (GK) formula for arbitrary BD extensions is computed and expressed in terms of naturally defined elements of the group. We show that the GK formula can be interpreted as Langlands-Shahidi type L-functions associated with the ajoint action of the L-group of the Levi on certain Lie algebras. As a consequence, the constant term of Eisenstein series for BD extensions could be expressed in terms of global (partial) Langlands-Shahidi type L-functions. In t…
This note points out that for any odd prime $p$, Zagier's weight $3/2$ mock Eisenstein series can be completed to a $p$-adic modular form in a way that bears some resemblance to its completion to a harmonic Maass form.
Let $τ_1^{(r)}$, $τ_2^{(r)}$ be two genuine cuspidal automorphic representations on $r$-fold covers of the adelic points of the general linear groups $GL_{n_1}$, $GL_{n_2}$, resp., and let $E(g,s)$ be the associated Eisenstein series on an $r$-fold cover of $GL_{n_1+n_2}$. Then the value or residue at any point $s=s_0$ of $E(g,s)$ is an automorphic form, and generates an automorphic representation. In this note we show that if $n_1\neq n_2$ these automorphic representations (when not identically zero) are generic, while if $n_1=n_2:=n$ they are generic except for residues at $s=\frac{n\pm1}{2n…
We compute the residual spectrum of the global metaplectic group Mp_4(A_k), by using the theory of Eisenstein series. The residual spectra obtained are interpreted as near equivalence classes in the framework of the Arthur conjecture.
Multiple Eisenstein series are holomorphic functions in the complex upper-half plane, which can be seen as a crossbreed between multiple zeta values and classical Eisenstein series. They were originally defined by Gangl-Kaneko-Zagier in 2006, and since then, many variants and regularizations of them have been studied. They give a natural bridge between the world of modular forms and multiple zeta values. In this note, we give a new algebraic interpretation of stuffle regularized multiple Eisenstein series based on the Hopf algebra structure of the harmonic algebra introduced by Hoffman.
We study the (mock) Eisenstein series $E_k$ of weight $k \in \{1,3/2,2\}$ for the Weil representation on an even lattice, defined as the result of Bruinier and Kuss's coefficient formula for the Eisenstein series naively evaluated at $k$. We describe the transformation law of $E_k$ in general. Most of this note is dedicated to collecting examples where the coefficients of $E_k$ contain interesting arithmetic information. Finally we make a few remarks about the case $k=1/2$.
In the workshop of the July 2016 Building Bridges 3 conference in Sarajevo, I presented the results from a joint article with W. Raji (Mathematische Annalen 2017, preprint arXiv:1402.1854). That article gave a proof of various linear relations between products of two Eisenstein series on $Γ(N)$, including an interesting identity related to the action of a Hecke operator on such a product. The real proofs involve some care to deal with issues of convergence. In this note we give "fake" proofs for these identities, ignoring the convergence issues; some of these fake proofs appeared in the worksh…
We provide a new and simple automorphic method using Eisenstein series to study the equidistribution of modular symbols modulo primes, which we apply to prove an average version of a conjecture of Mazur and Rubin. More precisely, we prove that modular symbols corresponding to a Hecke basis of weight 2 cusp forms are asymptotically jointly equidistributed mod $p$ while we allow restrictions on the location of the cusps. As an application, we obtain a residual equidistribution result for Dedekind sums. Furthermore, we calculate the variance of the distribution and show a surprising bias with con…
In \cite{MR3284482} and \cite{MR3658191}, the twisted standard $\mathcal{L}$-function $\mathcal{L}(s,π,χ,st)$ of a cuspidal representation $ π$ of the exceptional group of type $G_2$ was shown to be represented by a family of new-way Rankin-Selberg integrals. These integrals connect the analytic behaviour of $\mathcal{L}(s,π,χ,st)$ with that of a family of degenerate Eisenstein series $\mathcal{E}_E(χ, f_s, s, g)$ on quasi-split forms $H_E$ of $Spin_8$, induced from Heisenberg parabolic subgroups. The analytic behaviour of the series $\mathcal{E}_E(χ, f_s, s, g)$ in the right half-plane $Re(s)…
