Periods of Hecke Characters

p-adic RankinL-series and rational points on CM elliptic curves

“…We will briefly recall some key definitions regarding algebraic Hecke characters, mainly to fix notation. The reader is referred to [Schappacher 1988, Chapter 0] for more details. Let K and E be number fields.…”

On critical values of L-functions of potentially automorphic motives

“…where · is the idèlic norm on A × K , and χ 0 is of finite order (see [27], Chapter 0). Let be a CM type for L/K , that is, ⊂ J L consists of a choice of one of the two possible extensions of each σ ∈ J K to L. Proposition 2.1.1.…”

“…As in [27], Chaper 0, any tuple of integers (n τ ) τ ∈J L with the property that n τ + nτ is independent of τ is the infinity type of an algebraic Hecke character χ of L. To arrive at the conditions of the proposition, choose an arbitrary w 0 , with the same parity as the a τ , and take the tuple (n τ ) τ ∈ as…”

“…When K = Q, such elements are classically given by Gauss sums (see [12], 6.4). There is no such simple relation with Gauss sums when K = Q (see [27], 3.4).…”

Mixed automorphic vector bundles on Shimura varieties

“…The category of real Hodge structures has a Tannakian structure, and the affine group scheme attached to the category and the forgetful functor is S = Resc/κG m (see [4], [28], [29]). If V has a real Hodge structure with decomposition F®C = 0 V™, then an element ^ G §(i) = C x acts on V VΆ as multiplication by z~p~z~q.…”