# Questions tagged [trace-formula]

Theoretical issues and applications of the Selberg, Arthur and relative trace formulas

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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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### 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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### Combinatorial Skeleton of a Riemannian manifold

In Chung and Yau's paper: "A combinatorial trace formula" (MSN), they proved a combinatorial version of Selberg's trace formula for lattice graphs. I learned also in the setup that it makes sense to ...
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### Reduction to Lie algebra version of fundamental lemma?

Ngo famously proved the Langlands-Shelstad fundamental lemma for Lie algebras using the geometry of the Hitchin fibration. For the purposes of the trace formula, one actually needs the fundamental ...
1 vote
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### The number of rational semisimple conjugacy class/the Arthur-Selberg trace formula

I was trying to understand a statement in Theorem 1.5 of this where the author seems to imply that if $G$ is a reductive group over $\mathbb{Q}$ such that $G/Z(G)$ is anisotropic, then for any ...
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### What is the status of Arthur's book?

Arthur's long-awaited book project is now published (The endoscopic classification of representations: orthogonal and symplectic groups). However, in the book he takes some things for granted: The ...
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### Fundamental lemma: why is the transfer factor a power of q

Let $k$ be a finite field of sufficiently large characteristic, $F = k((t))$ and $\mathfrak{o} = k[[t]]$. Let $G$ be a reductive algebraic group defined over $\mathfrak{o}$. Roughly stated, for sake ...
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### Analytic continuation of intertwining operator

I was trying to understand the paper "Forms of GL(2) from the analytic point of view", by Gelbart and Jacquet. On Page 226 in Remark (4.13) they mention that the kernel of the local intertwining ...
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### A source for integral operators in the context of Arthur-Selberg trace formula

Could you suggest a textbook for integral operators (in the context of Arthur-Selberg trace formula)?
1 vote
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### About Theorem $3.1.3$ in Kubota's book: Elementary theory of eisenstein series

My question is about the proof of Theorem $3.1.3$ given in kubota's book, which shows how the function $\varphi(s)$ appearing in the Fourier expansion of eisenstein series can be continued ...
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### Regularity assumption in the simple trace formula

In the simple trace formula of Deligne Kazhdan one assumes that the test function is supported at the elliptic regular elements at one place and is a supercusp form at another place. Why can't one ...
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### What is the logarithmic derivative of an (intertwining) operator?

The constant term of the Eisenstein series (for an adele group $GL_2$, say) contains an intertwining operator, often written as $M(s)$. In the form given in Gelbart-Jacquet's Corvallis paper, for ...
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### What is the Twisted Trace Formula?

I am studying the trace formula using "An Introduction to the Trace Formula" by James Arthur. I would like to understand the twisted trace formula, but unfortunately I never came across a good ...
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### Spectral synthesis for central functions on locally compact groups

There is a large literature on harmonic analysis on locally compact group, that I am just beginning to discover. However I have not seen so far anything that emphasizes the central functions on $G$. A ...
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### Trace Class Functions on locally compact groups

Let $G$ be a locally compact subgroup, $\mu$ a Haar-measure. For $f \in L^1(G)$, and for $\pi$ a unitary, topology irreducible, representation of $G$ on an Hilbert space $H_\pi$, it is customary to ...
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### Arthur-Clozel Prop 3.1 for Function Fields?

The subject says it all. I would like to know if Proposition 3.1 in Arthur-Clozel's book on the trace formula holds for local fields of positive characteristic. Thanks! EDIT: Here is Prop 3.1 of ...
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### Base change and Langlands' combinatorial exercise

Is it correct that Langlands' combinatorial exercise (as he terms it in his paper "Shimura varieties and the Selberg trace formula") is to establish base change identities between orbital integrals of ...
We have these huge tables of elliptic curves, which were generated by computing modular forms of weight $2$ and level $\Gamma_0(N)$ as N increased. For abelian surfaces over $\mathbb{Q}$ we have very ...