On the Central Critical Value of the Triple Product L–Function

Abstract

Starting from the work of Garrett and of Piatetskii-Shapiro and Rallis on integral representations of the triple product L-function associated to three elliptic cusp forms the critical values of these L-functions have been studied in recent years from different points of view. From the classical point of view there are the works of Garrett [9], Satoh [22], Orloff [21], from an adelic point of view the problem has been treated by Garrett and Harris [10], Harris and Kudla [12] and Gross and Kudla [11]. Of course the central critical value is of particular interest. Harris and Kudla used the Siegel-Weil theorem to show that the central critical value is a square up to certain factors (Petersson norms and factors arising at the bad and the archimedean primes); the delicate question of the computation of the factors for the bad primes was left open. In the special situation that all three cusp forms are newforms of weight 2 and for the group 0(N) with square free level N > 1, Gross and Kudla gave for the first time a completely explicit treatment of this L-function including Euler factors for the bad places; they proved the functional equation and showed that the central critical value is a square up to elementary factors (that are explicitly given).

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