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## NOTES ON THE ARITHMETIC OF HILBERT MODULAR FORMS

Citations: | 7 - 2 self |

### Citations

356 | Continuous cohomology, discrete subgroups, and representations of reductive groups - Borel, Wallach - 2000 |

177 | Automorphic Forms and Representations - Bump - 1998 |

156 |
The special values of the zeta functions associated with Hilbert modular forms
- Shimura
- 1978
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Citation Context ...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... |

154 |
Valeurs de fonctions L et periodes d’integrales, in: Automorphic forms, representations, and L-functions
- Deligne
- 1979
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Citation Context ...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... |

153 | Fourier Analysis in Number Fields and Hecke’s Zeta-Functions, in: Algebraic Number Theory - Tate - 1967 |

137 | Automorphic Forms on
- Jacquet, Langlands
- 1970
(Show Context)
Citation Context ...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... |

130 | Algebraic number theory, Grundlehren der Mathematischen Wissenschaften [Fundamental - Neukirch - 1999 |

128 | Automorphic forms on adèle groups - Gelbart - 1975 |

87 |
Motifs et formes automorphes: applications du principe de fonctorialité, Automorphic forms, Shimura varieties
- Clozel
- 1990
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Citation Context ...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 ... |

78 | Automorphic forms and automorphic representations. In Automorphic forms, representations and $L$-functions (Proc. Sympos. Pure Math., Oregon State Univ - Borel, Jacquet - 1977 |

60 |
Cohomologie de sln et valeurs de fonctions zeta aux points entiers
- Borel
- 1977
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Citation Context ...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... |

58 |
Hilbert Modular Forms,
- Freitag
- 1990
(Show Context)
Citation Context ...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... |

53 |
Motives for Hilbert modular forms
- Blasius, Rogawski
- 1993
(Show Context)
Citation Context ... 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... |

45 | The local Langlands correspondence: the nonArchimedean case. Motives - Kudla - 1991 |

39 |
Hilbert Modular Surfaces. Ergebnisse der
- Geer
- 1988
(Show Context)
Citation Context ...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... |

39 | Basic number theory, Die Grundlehren der Mathematischen Wissenschaften, Band 144 - Weil - 1974 |

36 |
Local Langlands correspondence: the Archimedean case
- Knapp
- 1991
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Citation Context ...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... |

30 |
Lectures on L-functions, Converse Theorems, and Functoriality for GL(n),
- Cogdell
- 2004
(Show Context)
Citation Context ...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... |

25 | On liftings and cusp cohomology of arithmetic groups - Labesse, Schwermer - 1986 |

23 |
Quelques propriétés arithmétiques de certaines formes automorphes sur GL(2
- Waldspurger
- 1985
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Citation Context ...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... |

19 |
A refinement of the strong multiplicity one theorem for GL(2)’ [Appendix to R. Taylor ‘l-adic representations associated to modular forms over imaginary quadratic fields
- Ramakrishnan
- 1994
(Show Context)
Citation Context ...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... |

18 |
On automorphic forms on GL2 and Hecke operators
- Miyake
- 1971
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Citation Context ...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 ... |

17 |
Regularization theorems in Lie algebra cohomology.
- Borel
- 1983
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Citation Context ... 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... |

17 |
L-functions of 2 × 2 unitary groups and factorization of periods of Hilbert modular forms
- Harris
- 1993
(Show Context)
Citation Context ...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... |

16 | Non-Archimedean L-functions of Siegel and Hilbert Modular Forms - Panchishkin - 1991 |

15 | On the special values of certain Rankin-Selberg L-functions and applications to odd symmetric power L-functions of modular forms
- Raghuram
- 2010
(Show Context)
Citation Context ...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 )... |

15 | On certain period relations for cusp forms on GLn
- Raghuram, Shahidi
- 2008
(Show Context)
Citation Context ...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... |

15 | Some remarks on local newforms for GL(2 - Schmidt |

12 | Representation Theory and Automorphic - Gelfand, Graev, et al. - 1969 |

12 |
General aspects in the theory of modular symbols. Seminar on number theory
- Harder
- 1981
(Show Context)
Citation Context ...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... |

11 | Adjoint L-Values and Primes of Congruence for Hilbert Modular Forms - Ghate |

11 |
Relative modular symbols and Rankin-Selberg convolutions
- Kazhdan, Mazur, et al.
(Show Context)
Citation Context ...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 ... |

11 |
Cohomology of arithmetic groups, parabolic subgroups and the special values of L-functions of GLn
- Mahnkopf
- 2005
(Show Context)
Citation Context ...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... |

9 |
On the critical values of L-functions of GL(2) and GL(2
- Hida
- 1994
(Show Context)
Citation Context ...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 ... |

8 | Functoriality and special values of L-functions. Eisenstein series and applications - Raghuram, Shahidi - 2008 |

7 |
Holomorphic Hilbert modular forms. The Wadsworth & Brooks/Cole Mathematics Series
- Garrett
- 1990
(Show Context)
Citation Context ...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... |

4 |
On the Critical Values of L-Functions of GL(2) and GL(2)×GL(2
- Hida
- 1994
(Show Context)
Citation Context ...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 ... |

2 | in Algebraic Geometry I. Sheaves, cohomology of sheaves, and applications to Riemann surfaces. Second revised edition - Harder - 2011 |

2 | Algebraic number theory. Translated from the 1988 Russian edition. Reprint of the 1992 translation - Koch - 1997 |

1 |
On the fundamental periods of Hilbert modular forms
- Dou
- 1994
(Show Context)
Citation Context ..., 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... |