no. 1
Positivity of quadratic base change L-functions
[Positivité des fonctions L du changement de base quadratique]
Bulletin de la Société Mathématique de France, Tome 129 (2001) no. 1, pp. 33-90.

On montre que certaines des fonctions L de Gl (2) obtenues par changement de base quadratique sont positives en leur centre de symétrie.

We show that certain quadratic base change L-functions for Gl (2) are non-negative at their center of symmetry.

DOI : 10.24033/bsmf.2386
Classification : 11F67, 11F70, 11F72, 22E55
Keywords: $L$-functions, positivity, quadratic base change
Mot clés : fonctions $L$, positivité, changement de base quadratique 
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     title = {Positivity of quadratic base change $L$-functions},
     journal = {Bulletin de la Soci\'et\'e Math\'ematique de France},
     pages = {33--90},
     publisher = {Soci\'et\'e math\'ematique de France},
     volume = {129},
     number = {1},
     year = {2001},
     doi = {10.24033/bsmf.2386},
     mrnumber = {1871978},
     zbl = {1069.11017},
     language = {en},
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Jacquet, Hervé; Nan, Chen. Positivity of quadratic base change $L$-functions. Bulletin de la Société Mathématique de France, Tome 129 (2001) no. 1, pp. 33-90. doi : 10.24033/bsmf.2386. http://archive.numdam.org/articles/10.24033/bsmf.2386/

[1] Asai (T.) - On certain Dirichlet series associated with Hilbert modular forms, Math. Ann., t.226 (1977), pp.81-94. | EuDML | MR | Zbl

[2] Arthur (J.) - A trace formula for reductive groups I: Terms associated to classes in G(ℚ), Duke Math. J., t.45 (1978), 911-952. | MR | Zbl

[3] Bernstein (J.) - P-invariant distributions of Gl (n) and the classification of unitary representations of Gl (n) (non Archimedean case), in Lie Group Representations II, College Park, Md., 1982/1983, Lecture Notes in Math. 1041, Springer-Verlag, 1984. | MR | Zbl

[4] Bump (D.), Friedberg (S.), Hoffstein (J.) - Eisenstein series on the metaplectic group and non-vanishing theorems for automorphic L-functions, Ann. of Math., t.131 (1990), pp.53-127. | MR | Zbl

[5] Baruch (E.M.), Mao (Z.) - Central critical value of automorphic L-functions, preprint | Zbl

[6] Casselman (W.), Hecht (H.), Miličic (D.) - Bruhat filtrations and Whittaker vectors for real groups, preprint. | Zbl

[7] Dixmier (J.), Malliavin (P.) - Factorizations de fonctions et vecteurs indéfiniment différentiables, Bull. Soc. Math. France, t.102 (1978), pp.305-330. | MR | Zbl

[8] Flicker (Y.) - Twisted tensors and Euler products, Bull. Soc. Math. France, t.116 (1988), pp.295-313. | EuDML | Numdam | MR | Zbl

[9] Flicker (Y.) - On distinguished representations, J. reine angew Math., t.418 (1991), pp.139-172. | EuDML | MR | Zbl

[10] Flicker (Y.) - A Fourier summation formula for the symmetric space Gl (n)/ Gl (n-1), Comp. Math., t.88 (1992), pp.39-117. | Numdam | MR | Zbl

[11] Flicker (Y.) - Orbital Integrals on symmetric spaces and spherical characters, J. Algebra, t.184 (1996), pp.705-754. | MR | Zbl

[12] Flicker (Y.), Zinoniev (D.) - On the poles of twisted L-functions, Proc. Japan Acad. Sci., t.71 (1995), pp.114-116. | MR | Zbl

[13] Friedberg (S.), Hoffstein (J.) - Non vanishing theorem for automorphic L-functions for Gl (2), Ann. of Math., t.142 (1995), pp.385-423. | Zbl

[14] Guo (J.) - On the positivity of the central critical value of automorphic L-functions for Gl (2), Duke Math. J., t.83 (1996), pp.157-189. | MR | Zbl

[15] Guo (J.) - On a generalization of a result of Waldspurger, Canad. J. Math., t.48 (1996), pp.105-142. | MR | Zbl

[16] Gross (B.) - Heights and the special values of L series, in Number Theory (Montreal, Que., 1985), 115-187, CMS Conf. Proc. 7, Americ. Math. Soc., Providence, RI, 1987. | MR | Zbl

[17] Gross (B.), Zagier (D.) - Heegner points and derivatives of automorphic L-functions for Gl (2), Invent. Math., t.84 (1986), pp.225-320. | MR | Zbl

[18] Hakim (J.) - Distinguished p-adic representations, Duke Math. J., t.62 (1991), pp.1-22. | MR | Zbl

[19] Harder (G.), Langlands (R.P.), Rapoport (M.) - Algebraische Zyklen auf Hilbert-Blumenthal-Flächen, J. reine angew. Math., t.366 (1986), pp.53-120. | MR | Zbl

[20] Jacquet (H.) - Sur un résultat de Waldspurger, Ann. Scient. École Norm. Sup., t.19 (1986), pp.185-229. | Numdam | MR | Zbl

[21] Jacquet (H.) - Sur un résultat de Waldspurger II, Comp. Math., t.63 (1987), pp.315-389. | Numdam | MR | Zbl

[22] Jacquet (H.), Lai (K.) - A relative trace formula, Comp. Math., t.54 (1995), pp.243-310. | Numdam | MR | Zbl

[23] Jacquet (H.), Lai (K.), Rallis (S.) A trace formula for symmetric spaces, Duke Math. J., t.70 (1993), pp.305-371. | MR | Zbl

[24] Jacquet (H.), Lapid (E.), Rogawski (J.) - Periods of automorphic forms, J. Amer. Math. Soc., t.12 (1999), pp.173-240. | MR | Zbl

[25] Jacquet (H.), Ye (Y.) - Une remarque sur le changement de base quadratique, C. R. Acad. Sci. Paris Sér. I Math., t.311 (1990), pp.671-676. | MR | Zbl

[26] Khonen (W.) - Fourier coefficients of modular forms of half integral weight, Math. Ann., t.271 (1985), pp.237-268. | MR | Zbl

[27] Kamal Khuri-Makdsi - On the Fourier coefficients of non holomorphic Hilbert modular forms of half integral weight, Duke Math. J., t.84 (1996), pp.399-452. | MR | Zbl

[28] Katok (S.), Sarnak (P.) - Heegner points, cycles, and Maass forms, Israel J. Math., t.84 (1993), pp.193-227. | MR | Zbl

[29] Kohnen (W.), Zagier (D.) - Values of L-series of modular forms at the center of the critical strip, Invent. Math., t.64 (1981), pp.175-198. | MR | Zbl

[30] Rohrlich (D.) - Non vanishing of L-functions for Gl (2), Invent. Math., t.97 (1989), pp.381-403. | Zbl

[31] Rader (C.), Rallis (S.) - Spherical characters on p-adic symmetric spaces, Amer. J. Math., t.118 (1996), pp.91-178. | MR | Zbl

[32] Shimura (S.) - On Fourier coefficients of Hilbert modular forms of half-integral weight, Duke Math. J., t.71 (1993), pp.501-557. | MR | Zbl

[33] Waldspurger (J.-L.) - Corespondances de Shimura, J. Math. Pures Appl., t.59 (1980), pp.1-132. | MR | Zbl

[34] Waldspurger (J.-L.) - Sur les coefficients de Fourier des formes modulaires de poids demi-entier, J. Math. Pures Appl., t.60 (1981), pp.375-484. | MR | Zbl

[35] Waldspurger (J.-L.) - Quelques propriétés arithmétiques de certaines formes automorphes sur Gl (2), Comp. Math., t.54 (1985), pp.121-171. | Numdam | MR | Zbl

[36] Waldspurger (J.-L.) - Sur les valeurs de fonctions L-automorphes en leur centre de symétrie, Comp. Math., t.54 (1985), pp.173-242. | Numdam | MR | Zbl

[37] Waldspurger (J.-L.) - Correspondances de Shimura et quaternions, Forum Math., t.3 (1991), pp.219-307. | MR | Zbl

[38] Wallach (N.) - Real Reductive Groups II, Pure and Applied Mathematics, Vol 132-II (1192), Academic Press. | Zbl

[39] Yang (T.) - Theta liftings and Hecke L-functions, J. reine angew. Math., t.485 (1997), pp.25-53. | MR | Zbl

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