Questions tagged [trace-formula]
Theoretical issues and applications of the Selberg, Arthur and relative trace formulas
54 questions
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Measure normalizations in the adelic hyperbolic Poisson summation formula
Let $\mathsf{A}$ be the diagonal torus of $\mathsf{G}=\mathsf{GL}_2$. We can consider nice functions $f$ on
\begin{equation}
\mathsf{A}(\mathbb{Q})\backslash(\mathsf{A}(\mathbb{A})\cap \mathsf{G}(\...
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Estimate of the normalized intertwining operator in the continuous part of the trace formula for GL(2)
I am now reading Langlands' Beyond Endoscopy. In Section 2.3, Langlands claimed that in Arthur's paper: On the Fourier transforms of weighted orbital integrals, one has the following estimate
\begin{...
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Trace inequality for symmetrized tensor products
Let $B=M_k(\mathbb{C})$ be the matrix algebra of $(k\times k)$ matrices, and let $B^{\otimes N}$ be the $N$-fold tensor product algebra. Consider $B_{\text{sym}}^{\otimes N}$, the symmetric subalgebra ...
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Number of rational points of a connected reductive group in a compact subset
Let $G$ be a connected reductive $\mathbb{Q}$-group. Let $\mathbb{A}$ denote the ring of adèles of $\mathbb{Q}$. Let $B \subset G(\mathbb{A})$ be a compact, let $x \in G(\mathbb{A})$ and consider the ...
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Using the Eichler-Selberg Trace formula to compute class numbers?
The Eichler-Selberg trace formula (Theorem 2.2 here) gives a relation between the trace of a Hecke operator acting on the space of cusp forms and sums of weighted class numbers of imaginary quadratic ...
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Divergence of integrals in the trace formula
I am trying to understand the following situation for $G=GL(2)$, when going from the compact trace formula to the non-compact case.
The integral over $G(\mathbb{A})^1_\gamma \backslash G(\mathbb{A})^1$...
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Grothendieck trace formula for schemes with étale fundamental groups that have no dense cyclic subgroup
This question may be more of a philosophical rather than mathematical nature.
Assume I have a scheme $X$ and an endomorphism $F:X\longrightarrow X$. For instance, $X$ might be of finite type over $\...
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Comparing Selberg and Eichler-Selberg trace formulas
The trace formula of Selberg gives an equality between trace of Hecke operators (a spectral sum) on spaces of Maass forms and sums over closed geodesics mostly. The Eichler-Selberg trace formula, ...
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An application of Koike's Trace Formula
Koike's Trace Formula states that
\begin{equation}
\mbox{Tr}((U_p^{\kappa})^n) = - \sum_{0 \leq u < \sqrt{p^n}\\
(u,p)=1}H(u^2-4p^n)\frac{\gamma(u)^\kappa}{\gamma(u)^2 - p^n}-1,
\end{equation}
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The meaning of $L_{\chi}^2(G(\mathbb Q) \backslash G(\mathbb A)^1)$
I'm reading James Arthur's notes on the trace formula and am confused on a point on pages 65 and 66. For $G$ a reductive group over $\mathbb Q$ we are going over the decomposition of the space $L^2(G(...
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Variants of Selberg trace formula
I am familiar with a basic case of Selberg's trace formula, in the case of quotients of upper half plane (for example, see Sections 5.1 - 5.3 of Bergeron's book). Section 5.1 describes a general setup ...
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$\DeclareMathOperator\SL{SL}$Multiplicities of irreducible representations in discrete part of $L^2(\SL(2,\mathbb{Z})\backslash{\SL(2,\mathbb R)})$
$\DeclareMathOperator\SL{SL}$It is well-known that the cuspidal (or discrete) part of $L^2(\SL(2,\mathbb{Z})\backslash{\SL(2,\mathbb{R})})$ decomposes into irreducible representations of $\SL(2,\...
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Sum of Kloosterman sums with oscillating factor
Denote by $S(c;n,m)$ Kloosterman's sum. Take $X>0$ and take $n,m\in \mathbb Z$ smaller than a small power of $X$ in modulus. It is known that essentially
\[ \sum _{c\sim X}\frac {S(c;n,m)}{c}\ll ...
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An inequality regarding operator concave function
Crossposted from math.SE
Let $\mathbb P_n$ be the space of all $n \times n$ self-adjoint positive definite matrices. Consider the function $\varphi: \mathbb P_n \longrightarrow \mathbb R$ defined by $...
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The specific connection between the Hecke operator and the t'Hooft Operator
As I was reading some articles concern about the Selberg trace formula and its general form, I have noticed that the Selberg trace formula and its general form can be understand as the energy spectrum ...
