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...χ) = L(s + (k0 − 1)/2, f, χ), where the left hand side is the standard L-function defined as in Jacquet and Langlands [22], and the right hand side is the defined via a Dirichlet series as in Shimura =-=[37]-=-. (2) (Algebraicity) (a) if k1 ≡ · · · ≡ kn ≡ 0 (mod 2) then Π(f) is algebraic; (b) if k1 ≡ · · · ≡ kn ≡ 1 (mod 2) then Π(f) ⊗ | | 1/2 is algebraic; (c) if ki ̸≡ kj (mod 2) for some i and j then no tw...

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...Suppose µ = (µ1, . . . , µn) where µj = (aj, bj) and aj ≥ bj. Then s = 1 1 + m ∈ 2 2 + Z is critical for L(s, Π) ⇐⇒ −aj ≤ m ≤ −bj, ∀j. Proof. Recall the definition (as stated, for example, in Deligne =-=[9]-=-) for a point to be critical. If we are working with an L-function for GLn then the so-called motivic normalization says that critical points are in the set n−1 2 + Z. In our situation, we would say s...

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...aracter χ of F ×\A × F we have an equality of (completed) Lfunctions: L(s, Π(f) ⊗ χ) = L(s + (k0 − 1)/2, f, χ), where the left hand side is the standard L-function defined as in Jacquet and Langlands =-=[22]-=-, and the right hand side is the defined via a Dirichlet series as in Shimura [37]. (2) (Algebraicity) (a) if k1 ≡ · · · ≡ kn ≡ 0 (mod 2) then Π(f) is algebraic; (b) if k1 ≡ · · · ≡ kn ≡ 1 (mod 2) the...

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...t F be a totally real field of degree n, and let AF be the adèle ring. Let G = ResF/Q(GL2). Given a regular algebraic cuspidal automorphic representation Π of G(AQ) = GL2(AF ), one knows (from Clozel =-=[7]-=-) that there is a pure dominant integral weight µ such that Π has a nontrivial contribution to the cohomology of some locally symmetric space of G with coefficients coming from the dual of the finite ...

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...riods by playing off these rational structures against each other. (Another word for these ‘periods’ might be ‘regulators’, as the definition our periods is very close in spirit to Borel’s regulators =-=[2]-=-.) The rest of 3.2 is a very brief summary of Raghuram-Shahidi [34]. As a matter of definition/notation, given a C-vector space V , and given a subfield E ⊂ C, by an E-structure on V we mean an E-subs...

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...akes no such restrictions on F . Besides, we could not find anywhere the answer to the question: is the dictionary Aut(C)-equivariant? Some of the standard books on Hilbert modular forms like Freitag =-=[11]-=-, Garrett [12] or van der Geer [39] do not have what we want; although Garrett’s book Date: October 26, 2010. 1991 Mathematics Subject Classification. 11F67; 11F41, 11F70, 11F75, 22E55. 12 A. RAGHURA...

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... write this article; we write down such a dictionary, give enough details to make the presentation self-contained, and also analyze its arithmetic properties. (We refer the reader to Blasius-Rogawski =-=[1]-=- and Harris [20] for some intimately related arithmetic issues about Hilbert modular forms.) We now describe the theorems proved in this paper in greater detail, toward which we need some notation. Le...

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...sides, we could not find anywhere the answer to the question: is the dictionary Aut(C)-equivariant? Some of the standard books on Hilbert modular forms like Freitag [11], Garrett [12] or van der Geer =-=[39]-=- do not have what we want; although Garrett’s book Date: October 26, 2010. 1991 Mathematics Subject Classification. 11F67; 11F41, 11F70, 11F75, 22E55. 12 A. RAGHURAM AND NAOMI TANABE has a definitive...

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...C ∗ appearing in the Langlands parameter of Π∞. We now proceed to describe these exponents, for which we need some preliminaries about the local Langlands correspondence; we refer the reader to Knapp =-=[24]-=-.8 A. RAGHURAM AND NAOMI TANABE 3.1.2. The Weil group of R. Let WR be the Weil group of R. Recall that as a set it is defined as WR = C∗ ∪ jC∗ . The group structure is induced from that of C∗ and the...

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...crete series representation Dkj−1 of lowest weight kj. To prove the first part of this theorem, let us recall some important theorems regarding automorphic representations. (See, for example, Cogdell =-=[8]-=-.) Theorem 4.8 (Multiplicity One Theorem). The representation ρ ˜ω 0 decomposes as the direct sum of irreducible representations, each of which appear with multiplicity one. Theorem 4.9 (Tensor Produc...

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...1 ≡ · · · ≡ kn (mod 2) with kj ≥ 2 for all j. Then, for any σ ∈ Aut(C) we have: σ (Π(f) ⊗ | | k0/2 ) = Π(f σ ) ⊗ | | k0/2 , where the action of σ on representations is as in Clozel [7] or Waldspurger =-=[41]-=-, and on Hilbert modular forms is as in Shimura [37]. (5) (Rationality field) Let Q(f) be the field generated by the Fourier coefficients of f, and let Q(Π(f)) be the subfield of complex numbers fixed...

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...nsion of F , and in general this would not be the case for a given f. On a related note, one can make an interesting observation based on a refined strong multiplicity one theorem due to Ramakrishnan =-=[35]-=-: suppose, f and g are primitive forms, and suppose fν = gν for all ν except, say, ν = ν0. This means that c(p, f) = c(p, g) for all prime ideals p whose class in the narrow class group is not represe...

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...ed, we have f = g. Furthermore, if a newform f is normalized and a common eigenfunction for Tp for all p not dividing n, then it is an eigenfunction for all Tm and its eigenvalue is N(m)c(m, f). (See =-=[29]-=- and [37].) Suppose f = (f1, . . . , fh) is a primitive form. One may ask whether f is determined by any one of its components fν. In general this is not true. For example, take χ to be a non-trivial ...

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... H • (g∞, K 0 ∞; C ∞ (G(Q)\G(A)) Kf ⊗ E v µ). The inclusion C ∞ cusp(G(Q)\G(A)) ↩→ C ∞ (G(Q)\G(A)) of the space of smooth cusp forms in the space of all smooth functions induces, via results of Borel =-=[3]-=-, an injection in cohomology; this defines cuspidal cohomology: H • (S G , E v µ) � H • (g∞, K 0 ∞; C ∞ (G(Q)\G(A)) ⊗ E v µ) H• cusp(SG , E v � � µ) � H • (g∞, K 0 ∞; C ∞ cusp(G(Q)\G(A)) ⊗ E v µ) Usin...

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...icle; we write down such a dictionary, give enough details to make the presentation self-contained, and also analyze its arithmetic properties. (We refer the reader to Blasius-Rogawski [1] and Harris =-=[20]-=- for some intimately related arithmetic issues about Hilbert modular forms.) We now describe the theorems proved in this paper in greater detail, toward which we need some notation. Let F be a totally...

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...on special values of L-functions that have been proved under the assumption that a quantity analogous to 〈[Π∞] ++ 〉 is nonzero. See, for example, Kazhdan-MazurSchmidt [23], Mahnkopf [28], or Raghuram =-=[32]-=-. 3.3.10. Concluding part of the proof of Theorem 1.2. We can now finish the proof as follows. Using Proposition 3.24 in the main identity of Theorem 3.23 we have ∫ c (3.25) 4n L(1/2, Π) . d∞ vol(Rf )...

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...ed in the union of these papers: Harder [18], Hida [21]; see also Dou [10]. What is different from these papers is an organizational principle based on the period relations proved in Raghuram-Shahidi =-=[34]-=- while working in the context of regular algebraic cuspidal automorphic representations. The point of view taken in this article is that one need only prove an algebraicity theorem for the most intere...

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...her proof of the above theorem, which is rather different from Shimura’s proof. However, before proceeding any further, let us mention that our proof is contained in the union of these papers: Harder =-=[18]-=-, Hida [21]; see also Dou [10]. What is different from these papers is an organizational principle based on the period relations proved in Raghuram-Shahidi [34] while working in the context of regular...

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...re are many conditional theorems on special values of L-functions that have been proved under the assumption that a quantity analogous to 〈[Π∞] ++ 〉 is nonzero. See, for example, Kazhdan-MazurSchmidt =-=[23]-=-, Mahnkopf [28], or Raghuram [32]. 3.3.10. Concluding part of the proof of Theorem 1.2. We can now finish the proof as follows. Using Proposition 3.24 in the main identity of Theorem 3.23 we have ∫ c ...

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...ditional theorems on special values of L-functions that have been proved under the assumption that a quantity analogous to 〈[Π∞] ++ 〉 is nonzero. See, for example, Kazhdan-MazurSchmidt [23], Mahnkopf =-=[28]-=-, or Raghuram [32]. 3.3.10. Concluding part of the proof of Theorem 1.2. We can now finish the proof as follows. Using Proposition 3.24 in the main identity of Theorem 3.23 we have ∫ c (3.25) 4n L(1/2...

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...f the above theorem, which is rather different from Shimura’s proof. However, before proceeding any further, let us mention that our proof is contained in the union of these papers: Harder [18], Hida =-=[21]-=-; see also Dou [10]. What is different from these papers is an organizational principle based on the period relations proved in Raghuram-Shahidi [34] while working in the context of regular algebraic ...

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...estrictions on F . Besides, we could not find anywhere the answer to the question: is the dictionary Aut(C)-equivariant? Some of the standard books on Hilbert modular forms like Freitag [11], Garrett =-=[12]-=- or van der Geer [39] do not have what we want; although Garrett’s book Date: October 26, 2010. 1991 Mathematics Subject Classification. 11F67; 11F41, 11F70, 11F75, 22E55. 12 A. RAGHURAM AND NAOMI TA...

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...f the above theorem, which is rather different from Shimura’s proof. However, before proceeding any further, let us mention that our proof is contained in the union of these papers: Harder [18], Hida =-=[21]-=-; see also Dou [10]. What is different from these papers is an organizational principle based on the period relations proved in Raghuram-Shahidi [34] while working in the context of regular algebraic ...

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..., which is rather different from Shimura’s proof. However, before proceeding any further, let us mention that our proof is contained in the union of these papers: Harder [18], Hida [21]; see also Dou =-=[10]-=-. What is different from these papers is an organizational principle based on the period relations proved in Raghuram-Shahidi [34] while working in the context of regular algebraic cuspidal automorphi...