The Poisson summation formula has had vast applications to number theory. Many recent applications have been obtained by a more general averaging of the equation in the Poisson formula as follows: Let \(k\) be a number field with adele ring \(A\) and \(f\), a Schwartz-Bruhat function on the adelic points \(V(A)\) of a finite dimensional vector space \(V\) defined over \(k\) with \(\widehat{f}\) as the Fourier transform of \(f\) and \(\widehat{V}\) as the vector space dual to \(V\). Let \(\overline{\Lambda}\) be a rational representation of a rational reductive algebraic group \(G\) on \(V\), and let \(\widehat{\overline{\Lambda}}\) be its contragradient. The averaged equation is the equality \[ \begin{split} \int_{G_v(A) G(k)\setminus G(A)} \phi(g) \sum_{\gamma\in v(k)} f(\overline{\Lambda} (g^{-1}) \gamma) dg\\ =\int_{G_v(A) G(k)\setminus G(A)} \phi(g)|\operatorname {det}\overline{\Lambda}(g)|\sum_{\widehat{\gamma}\in \widehat{V}(k)} \widehat{f} (\widehat{\overline{\Lambda}} (g^{-1}) \widehat{\gamma}) dg \end{split} \] valid when both integrals converge with \(G_v\) the kernel of the representation \(\overline{\Lambda}\), and \(\phi\) a slowly increasing function on the left quotient \(G_v(A) G(k)\setminus G(A)\). For general \(G\), \(\overline{\Lambda}\) and \(\phi\) the integrals in the above equation do not converge and a suitable regularization of the integrals is required. It was first done by Shintani who used a classical formulation of the problem. In an alternative form of a regularization first introduced by Arthur in the context of the Selberg trace formula, Levy has produced a regularized identity in the case that the group \(G\) is semisimple of rank at most 2 and \(\phi\) is the constant function 1.
In this paper a regularized identity is produced for \(G= SL(2)\) and \(V\) as the space of binary quartic forms. This is done using methods similar to those of Jacquet-Langlands and Labesse-Langlands. A zeta function has been obtained from the identity and its analytic continuation and poles have also been discussed.
For the entire collection see [Zbl 0919.00046].