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Register a new userIntegral Eisenstein cocycles on GLn, II : Shintanis method
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We define a cocycle on Gln using Shintanis method. It is closely related to cocycles defined earlier by Solomon and Hill, but differs in that the cocycle property is achieved through the introduction of an auxiliary perturbation vector Q. As a corollary of our result we obtain a new proof of a theorem of Diaz y Diaz and Friedman on signed fundamental domains, and give a cohomological reformulation of Shintanis proof of the Klingen-Siegel rationality theorem on partial zeta functions of totally real fields. Next we prove that the cohomology class represented by our Shintani cocycle is essentially equal to that represented by the Eisenstein cocycle introduced by Sczech. This generalizes a result of Sczech and Solomon in the case n=2. Finally we introduce an integral version of our Shintani cocycle by smoothing at an auxiliary prime ell. Applying the formalism of the first paper in this series, we prove that certain specializations of the smoothed class yield the p-adic L-functions of totally real fields. Combining our cohomological construction with a theorem of Spiess, we show that the order of vanishing of these p-adic L-functions is at least as large as the one predicted by a conjecture of Gross.
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We define an integral version of Sczechs Eisenstein cocycle on GLn by smoothing at a prime ell. As a result we obtain a new proof of the integrality of the values at nonpositive integers of the smoothed partial zeta functions associated to ray class extensions of totally real fields. We also obtain a new construction of the p-adic L-functions associated to these extensions. Our cohomological construction allows for a study of the leading term of these p-adic L-functions at s=0. We apply Spiesss formalism to prove that the order of vanishing at s=0 is at least equal to the expected one, as conjectured by Gross. This result was already known from Wiles proof of the Iwasawa Main Conjecture.
In this paper, we study top Fourier coefficients of certain automorphic representations of $mathrm{GL}_n(mathbb{A})$. In particular, we prove a conjecture of Jiang on top Fourier coefficients of isobaric automorphic representations of $mathrm{GL}_n(mathbb{A})$ of form $$ Delta(tau_1, b_1) boxplus Delta(tau_2, b_2) boxplus cdots boxplus Delta(tau_r, b_r),, $$ where $Delta(tau_i,b_i)$s are Speh representations in the discrete spectrum of $mathrm{GL}_{a_ib_i}(mathbb{A})$ with $tau_i$s being unitary cuspidal representations of $mathrm{GL}_{a_i}(mathbb{A})$, and $n = sum_{i=1}^r a_ib_i$. Endoscopic lifting images of the discrete spectrum of classical groups form a special class of such representations. The result of this paper will facilitate the study of automorphic forms of classical groups occurring in the discrete spectrum.
Let $Gamma$ be a Fuchsian group of the first kind acting on the hyperbolic upper half plane $mathbb H$, and let $M = Gamma backslash mathbb H$ be the associated finite volume hyperbolic Riemann surface. If $gamma$ is parabolic, there is an associated (parabolic) Eisenstein series, which, by now, is a classical part of mathematical literature. If $gamma$ is hyperbolic, then, following ideas due to Kudla-Millson, there is a corresponding hyperbolic Eisenstein series. In this article, we study the limiting behavior of parabolic and hyperbolic Eisenstein series on a degenerating family of finite volume hyperbolic Riemann surfaces. In particular, we prove the following result. If $gamma in Gamma$ corresponds to a degenerating hyperbolic element, then a multiple of the associated hyperbolic Eisenstein series converges to parabolic Eisenstein series on the limit surface.
We provide an introduction to the theory of Eisenstein series and automorphic forms on real simple Lie groups G, emphasising the role of representation theory. It is useful to take a slightly wider view and define all objects over the (rational) adeles A, thereby also paving the way for connections to number theory, representation theory and the Langlands program. Most of the results we present are already scattered throughout the mathematics literature but our exposition collects them together and is driven by examples. Many interesting aspects of these functions are hidden in their Fourier coefficients with respect to unipotent subgroups and a large part of our focus is to explain and derive general theorems on these Fourier expansions. Specifically, we give complete proofs of the Langlands constant term formula for Eisenstein series on adelic groups G(A) as well as the Casselman--Shalika formula for the p-adic spherical Whittaker function associated to unramified automorphic representations of G(Q_p). In addition, we explain how the classical theory of Hecke operators fits into the modern theory of automorphic representations of adelic groups, thereby providing a connection with some key elements in the Langlands program, such as the Langlands dual group LG and automorphic L-functions. Somewhat surprisingly, all these results have natural interpretations as encoding physical effects in string theory. We therefore also introduce some basic concepts of string theory, aimed toward mathematicians, emphasising the role of automorphic forms. In particular, we provide a detailed treatment of supersymmetry constraints on string amplitudes which enforce differential equations of the same type that are satisfied by automorphic forms. Our treatise concludes with a detailed list of interesting open questions and pointers to additional topics which go beyond the scope of this book.
We carry out Hecke summation for the classical Eisenstein series $E_k$ in an adelic setting. The connection between classical and adelic functions is made by explicit calculations of local and global intertwining operators and Whittaker functions. In the process we determine the automorphic representations generated by the $E_k$, in particular for $k=2$, where the representation is neither a pure tensor nor has finite length. We also consider Eisenstein series of weight $2$ with level, and Eisenstein series with character.
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