Abstract:

Let $F$ be a number field and let $\Bbb{A}_F$ be its ring of adeles. Let $B$ be a quaternion algebra over $F$ and let $\nu:B\to F$ be the reduced norm. Consider the reductive monoid $M$ over $F$ whose points in an $F$-algebra $R$ are given by $$ M(R):=\big\{\big(\gamma_1,\gamma_2\big)\in\big(B\otimes_F R\big)^2:\nu\big(\gamma_1\big)=\nu\big(\gamma_2\big)\big\}. $$ Motivated by an influential conjecture of Braverman and Kazhdan we prove a summation formula analogous to the Poisson summation formula for certain spaces of functions on the monoid. As an application, we define new zeta integrals for the Rankin-Selberg $L$-function and prove their basic properties. We also use the formula to prove a nonabelian twisted trace formula, that is, a trace formula whose spectral side is given in terms of automorphic representations of the unit group of $M$ that are isomorphic (up to a twist by a character) to their conjugates under a simple nonabelian Galois group.

摘要：

假设$ F $为数字字段，并使$ \ Bbb {A} _F $为其环。假设$ B $是超过$ F $的四元数代数，并且使$ \ nu：B \ to F $是简化范数。考虑在$ F $上的归约类半体$ M $在$ F $-代数$ R $中的点由$$ M（R）给出：= \ big \ {\ big（\ gamma_1，\ gamma_2 \ big）\ in \ big（B \ otimes_F R \ big）^ 2：\ nu \ big（\ gamma_1 \ big）= \ nu \ big（\ gamma_2 \ big）\ big \}。$$受Braverman和Kazhdan的有影响力的猜想的驱使，我们证明了与等式上某些函数空间的泊松求和公式相似的求和公式。作为应用，我们为Rankin-Selberg $ L $函数定义新的zeta积分，并证明其基本性质。我们还使用该公式来证明一个非阿贝尔扭曲的跟踪公式，即