# Questions tagged [trace-formula]

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

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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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### 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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### 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 ...
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### How to read the paper of Arthur on trace formula on general reductive groups

My question is about the correct order to read the papers by Arthur on trace formula. Arthur's papers are perfectly well-written, but maybe a little too hard for me to go through easily. I would like ...
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### Truncation and weighted orbital integrals in hyperbolic term of trace formula for $\mathrm{GL}(2)$

$\DeclareMathOperator\GL{GL}\DeclareMathOperator\PGL{PGL}$I am looking at Gelbart--Jacquet's article in the first Corvallis volume (the article entitled Forms of $\GL(2)$ from an analytic point of ...
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### The kernel $K(x,y)$ as an integral over Eisenstein series for $\operatorname{GL}_2$

Let $G = \operatorname{GL}_2$, $f \in C_c^{\infty}(G(\mathbb A)/Z(\mathbb A))$, and $V = L^2( G(\mathbb Q)Z(\mathbb A)\backslash G(\mathbb A))$ (trivial central character). Then the operator $R(f)$ ...
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### Arthur's Simple Trace Formula

In Deligne–Kazhdan–Vigneras's "Représentations des groupes réductifs sur un corps local," they use the Simple Trace Formula to prove cases of the local Jacquet–Langlands correspondence ...
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Let $E/F$ be a cyclic extension of order $\ell$ (not assumed prime) of fields of characteristic $0$, and $\Sigma$ its Galois group; we denote by $\sigma$ a generator of $\Sigma$. We denote by $G(E), G(... 5 votes 1 answer 259 views ### Combinatorial Skeleton of a Riemannian manifold In Chung and Yau's paper: "A combinatorial trace formula" (MSN), they proved a combinatorial version of Selberg's trace formula for lattice graphs. I learned also in the setup that it makes sense to ... 6 votes 2 answers 382 views ### Reduction to Lie algebra version of fundamental lemma? Ngo famously proved the Langlands-Shelstad fundamental lemma for Lie algebras using the geometry of the Hitchin fibration. For the purposes of the trace formula, one actually needs the fundamental ... 1 vote 1 answer 316 views ### Small questions in studying Arthur 's book 'Introduction to the Trace formula' I am reading Arthur's book "Introductionto the trace formula". In reading the book, two small question has arised and so I would like to ask it. Let$G$be a connected reductive group over$\mathbb{...
A standard idea when dealing with the Arthur-Selberg trace formula (or a relative trace formula, for that matter) is to impose local conditions on the test function $f=\prod_vf_v$ to obtain a simple ...