Contents -- Preface -- Program of symposium -- The Birch and Swinnerton-Dyer conjecture for Q-curves and Oda's period relations Henri Darmon, Victor Rotger and Yu Zhao -- 1. Introduction -- 2. Background -- 2.1. The Birch and Swinnerton-Dyer conjecture in low analytic rank -- 2.2. Oda's period relations and ATR points -- 3. The Birch and Swinnerton-Dyer conjecture for Q-curves -- 3.1. Review of Q-curves -- 3.2. The main result -- 4. Heegner points on Shimura's elliptic curves -- 4.1. An explicit Heegner point construction -- 4.2. Heegner points and ATR cycles -- 4.3. Numerical examples -- 4.4. Proof of Proposition 4.1 -- References -- The supremum of Newton polygons of p-divisible groups with a given p-kernel type Shushi Harashita -- 1. Introduction -- 2. A catalogue of p-divisible groups with a given type -- 3. Preliminaries on F-zips -- 4. Lifting of F-zips -- 5. A reduction of the problem -- 6. Extensions by a minimal p-divisible group -- 7. Proof of Proposition 5.2 -- References -- Borcherds lifts on Sp2(Z) Bernhard Heim and Atsushi Murase -- 1. Introduction and the main results -- 1.1. Introduction -- 1.2. Siegel modular forms -- 1.3. The organization of the paper -- 1.4. Notation -- 2. Borcherds lifts -- 2.1. Jacobi forms -- 2.2. Humbert surfaces -- 2.3. Siegel modular forms with a nontrivial character -- 2.4. Borcherds lifts on -- 2.5. Examples of Borcherds lifts -- 3. Proof of the main results -- 3.1. The multiplicative symmetry -- 3.2. A characterization of powers of the modular discriminant -- 3.3. The multiplicative symmetry for Sym2(Mk( 1)) -- 3.4. Proofs of Theorem 1.1 and Theorem 1.2 (i) -- 4. The weight formula -- 4.1. Cohen numbers -- 4.2. The weight formula for Borcherds lifts -- Acknowledgement -- References -- The archimedean Whittaker functions on GL(3) Miki Hirano, Taku Ishii and Tadashi Miyazaki -- 1. Introduction.

2. Preliminaries -- 2.1. Notation -- 2.2. Basic objects -- 2.3. Whittaker functions on Gn -- 2.5. The contragradient Whittaker functions -- 2.6. The generalized principal series representations of Gn = GL(n -- R) -- 2.7. The principal series representations of Gn = GL(n -- C) -- 3. Whittaker functions on G3 = GL(3 -- R) -- 3.1. Irreducible representations of K3 = O(3) -- 3.2. Whittaker functions on G3 = GL(3 -- R) at the minimal K3-types -- 3.3. Whittaker functions on G3 = GL(3 -- R) at the multiplicity one K3-types -- 4. Whittaker functions on G3 = GL(3 -- C) -- 4.1. Irreducible representations of K3 = U(3) -- 4.2. Whittaker functions on G3 = GL(3 -- C) at the minimal K3-types -- 5. The archimedean local theory of the standard L-functions for GL(n1) GL(n2) (n1 > n2) -- 5.1. The local Langlands correspondence for GL(n) over R -- 5.2. The local Langlands correspondence for GL(n) over C -- 5.3. The archimedean zeta integrals for GL(n1) GL(n2) (n1 > n2) -- 6. Calculus of the archimedean zeta integrals -- 6.1. The archimedean zeta integrals for GL(3) GL(1) -- 6.2. The proof of Theorem 6.1 -- 6.3. The archimedean zeta integrals for GL(3) GL(2) -- References -- Arithmetic properties of p-adic elliptic logarithmic functions Noriko Hirata-Kohno -- 1. Introduction -- 2. Definition of p-adic elliptic logarithmic and polylogarithmic functions -- 3. Arithmetic properties of p-adic elliptic logarithmic functions and p-adic elliptic polylogarithmic functions -- References -- Spherical functions on U(2n)/(U(n) U(n)) and hermitian Siegel series Yumiko Hironaka -- 0. Introduction -- 1. Spherical function !T (x -- s) on XT and XT -- 2. Functional equations, possible zeros and poles -- 3. Explicit formulas -- 3.1. Set -- 4. Spherical Fourier transform on S(KnXT ) -- 5. An application to hermitian Siegel series -- References.

Fourier transforms of weighted orbital integrals on the real symplectic group of rank two Werner Hoffmann -- 0. Introduction -- 1. Parametrisation of Cartan subgroups -- 2. Parametrisation of representations -- 2.1. Discrete series -- 2.2. Limits of discrete series -- 3. Guide to the results -- 4. Global characters -- 5. Characters as Fourier transforms -- 6. Geometric descent -- 7. Abelian Fourier inversion -- 8. Differential equations I -- 9. Jump relations I -- 10. Differential equations II -- 11. Normalising factors -- 12. The asymptotic formula -- 13. Jump relations II -- 14. Spectral descent -- References -- An Atkin-Lehner type theorem on Siegel modular forms and primitive Fourier coe.cients T. Ibukiyama and H. Katsurada -- 1. Introduction -- 2. Main results -- 3. Proof -- 3.1. Action of Hecke operators -- 3.2. Some group theory -- 3.3. Level change -- 3.4. Proof of main results -- References -- Cohomology of Siegel modular varieties of genus 2 and corresponding automorphic forms Takayuki Oda -- 1. Introduction -- 2. The Matsushima isomorphism and related results -- 2.1. Cohomology of discrete subgroups -- 2.2. Shift to the relative Lie algebra cohomology groups -- 2.3. Matsushima isomorphism -- 2.4. "Classical" vanishing theorems -- 2.5. Enumeration and construction of unitary cohomological representations -- 2.6. Example, the classical case SL(2 -- R) -- References -- 3. Basic rudiments on Siegel modular varieties -- 3.1. The Lie group and the associated homogeneous space, -- 3.2. The discrete subgroups -- 4. Example: The Matsushima isomorphism for compact quotients of Sp(2 -- R) -- 4.1. Finite-dimensional representations of G -- 4.2. The representations of discrete series of Sp(2 -- R) -- 4.2.1. Discrete series, or square-integrable representations -- 4.2.2. Harish-Chandra's parametrization of discrete series for Sp(2,R).

4.2.3. The discrete series with the same infinitesimal characters -- 4.2.4. K-types of a discrete series -- 4.3. Cohomological representations -- 4.4. The list of cohomological representations with trivial infinitesimal characters -- 4.4.1. Discrete series -- 4.4.2. Non-tempered cohomological representations -- 4.5. The Matsushima isomorphism for compact quotients -- 4.6. Scholia for the Matsushima isomorphism for Sp(2 -- R) -- 4.6.1. The middle components -- 4.6.2. The Dirac-Schmid operator -- 4.7. Explicit description of the automorphic realization of the Schmid equation -- 4.7.1. The natural trivialization of the bundle 2 H2 1 H2 -- 4.7.2. The G-invariant Kahler metric and the associated metric form -- 4.7.3. The natural trivialization of 1 2 and 2 2 -- 4.7.4. Decomposition of the trivialization on2 2 1 2 -- 4.7.5. Automorphic realization of the Schmid operator -- References -- 5. Analytic aspect: Spherical functions -- 5.1. Fourier expansion -- 5.1.1. Whittaker functions -- 5.1.2. Application to automorphic L-functions -- 5.1.3. Siegel-Whittaker functions -- 5.1.4. Fourier-Jacobi functions -- 5.2. Spherical functions with respect to reductive subgroups -- 5.2.1. Matrix coefficients -- 5.2.2. The cases of other spherical subgroups -- 5.3. Intertwining operators between di.erent spherical models -- 5.4. Postscript for this section -- References -- 6. Cohomology groups of Siegel modular groups of genus two -- 6.1. Definition of variants of cohomology groups -- 6.2. Cohomology groups of degree i .= 3 -- 7. The second cohomology group of Siegel modular varieties -- 7.1. Modular divisors -- 7.2. Green function associated to modular divisors -- References -- Reducibility and discrete series in the case of classical p-adic groups -- an approach based on examples Marko Tadić -- Introduction -- 1. Historical observations.

2. Smooth representations and the unitary dual -- 3. The non-unitary dual and unitary dual -- 4. Square integrable representations -- 5. Parabolic subgroups -- 6. Parabolic induction, tempered representations and the Langlands classi.cation -- 7. Parabolic induction - basic facts -- 8. Jacquet modules -- 9. The geometric lemma -- 10. Some general consequences -- 11. The case of maximal parabolic subgroups -- 12. Hopf algebras in the case of general linear groups -- 13. Square integrable representations of general linear groups -- 14. Other classical groups -- 15. Reducibility - irreducibility -- 16. Square integrability criterion -- 17. Cuspidal reducibilities -- 18. Regular induced representations -- 19. Square integrable representations of Steinberg type -- 20. A reducibility criterion -- 21. Proving irreducibility -- 22. Some half-integral examples of irreducibility -- 23. Some integral examples of irreducibility (and reducibility) -- 24. A delicate case -- 25. Langlands parameters of irreducible subquotients -- 26. An interesting integral tempered irreducibility -- 27. On R-groups -- 28. Introductory remarks on invariants of square integrable representations -- 29. An important simple example of construction of square integrable representations -- 30. A little bit more complicated example of construction of square integrable representations -- 31. Partially de.ned function -- 32. Some examples of strongly positive representations -- 34. General strongly positive representations -- 35. The general step -- References -- A survey on Voronoı̈'s theorem Takao Watanabe -- Contents -- 1. Type one functions and Voronoı̈'s theorem -- 1.1. Type one functions and semikernels -- 1.2. Vorono 's theorem of m1/. -- 1.3. Geometric characterizations of perfect forms -- 1.4. Hermite like constants -- 2. Rankin's constant and Voronoı̈'s theorem.

2.1. Rankin's constant.