By Zannier, Umberto (ed.)

Contributions through numerous authors equivalent to Michael G. Cowling.- Joseph A. Wolf.- Gisbert Wustholz and David Mumford

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**Example text**

The morphism α induces a homomorphism J(α) : J(X) → E between the Jacobians. We denote by κ : E → J(X) the kernel of J(α). The holomorphic 1form dx/y on the elliptic curve E pulls back to a holomorphic 1-form ω = J(α)∗ dx/y on J(X). Let ι : X → J(X) be the canonical embedding of X into the Jacobian. This leads to a commutative diagram ι E × J (X) X /E κ ι κ / J (X) X MMM MMM MαMM MMM J (α) M& E with E ×J(X) X a closed subscheme of X and of E . We choose a path γ with class [γ ] in the relative homology H1 (X, E ×J(X) X, Z) of X with respect to E ×J(X) X such that γ (0) is the base point of X.

23] M. E. WALTER, A duality between locally compact groups and certain Banach algebras, J. Funct. Anal. 17 (1974), 131–160. Classical analysis and nilpotent Lie groups Joseph A. Wolf Classical Fourier analysis has an exact counterpart in group theory and in some areas of geometry. Here I’ll describe how this goes for nilpotent Lie groups and for a class of Riemannian manifolds closely related to a nilpotent Lie group structure. There are also some inﬁnite dimensional analogs but I won’t go into that here.

The answer depends, as it turns out, on the differential form and on the path as can be easily seen from examples: the differential form could be exact. Then obviously the integral takes algebraic values. Non-trivial examples can be obtained in the following way. Let E be an elliptic curve deﬁned over a number ﬁeld and let ⊂ E × E be the graph of an endomorphism ϕ of E. Then the differential form ξ = pr1∗ ϕ ∗ dx/y− pr2∗ dx/y vanishes on so that the integral I (ξ, γ ) is zero when the path γ is contained in .