By Zannier, Umberto (ed.)

Contributions via a number of authors reminiscent of Michael G. Cowling.- Joseph A. Wolf.- Gisbert Wustholz and David Mumford

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3. [π] is a discrete summand of ζ . A representation class [π] ∈ N is L2 or square integrable or relative discrete series if its coefﬁcients f u,v (n) = f π:u,v (n) := u, π(n)v satisfy | f u,v | ∈ L 2 (N/Z ), in other words if its coefﬁcients are square integrable modulo Z . 2(3) justiﬁes the term “relative discrete series”. We say that N has square integrable representations if at least one class [π] ∈ N is square integrable. These representations satisfy an analog of the Schur orthogonality relations: 26 Joseph A.

As a nice example on can deduce the transcendence of values of the Beta-function B(a, b) = (a) (b) (a + b) at rational arguments a, b, such that none of the numbers a, b, a + b are integral. In fact these Beta-values are periods of differentials of the second kind on curves whose Jacobians are abelian varieties of CM-type and so Schneider’s results apply. In the abelian case the ﬁrst result without the hypothesis of complex multiplication in the very special case of a product of two elliptic curves was obtained by A.

As we have already indicated all the three conjectures are completely out of reach and of similar nature concerning depth, complexity and difﬁculty as the famous millennium problems. If the ground ﬁeld is a function ﬁeld instead of the ﬁeld of algebraic numbers then Conjecture 5 has been proved by Ax [2] in the case of a torus and in the case of elliptic curves there is some work of Brownawell and Kubota [4]. If we restrict the integration map to the linear part L(M1 ) of P(M1 ) generated linearly over Q by the 1-motives then one can try to give an answer for a weak version of the 1-motif conjecture.