Minicourses

- S. Boecherer (SB) (University of Mannheim):

     p-adic and complex L-functions on classical groups: admissible measures, special values I.

SB 1  - Representations of classical groups
The case GL(n), the symplectic case, representations of higher weights, examples

SB 2  - Structure of the Hecke algebra
Parameters of Satake, Euler products, examples

SB 3  - Complex and p-adic modular forms of GL(2) and Sp(2n)
Theta series, Eisenstein-Siegel series, Eisenstein-Klingen series

SB 3.5 - p-adic Siegel Modular Forms

SB 4  - Theta Operators, Maass operators
Integral representation of L-functions, main identity, doubling method

- G. Comte (GC) (University Savoie Mont Blanc):

     Real motivic Milnor fiber of semi-algebraic formulas, o-minimality

GC 1  - Tame geometry
Semialgebric sets, o-minimal structures, cell decompositions

GC 2  - Additive Invariants,
Euler-Poincaré characteristic, virtual Betti polynomial, Grothendieck rings

GC 3  - Singularities
Set-theoretic Milnor fibres of complex polynomial functions, monodromy
Resolution of singularities, Denef-Loeser motivic Milnor fibre

GC 4  - Real motivic Milnor fiber of formulas
Grothendieck ring of semialgebraic formulas, real motivic Milnor fibre,
realizations

- D. Essouabri (DE) (University Jean Monnet St-Etienne):

     Dirichlet series and zeta functions (in one and in several variables) and applications

DE 1  – Some tools from analytic number theory and complex geometry:
Integral representation formulas and residues in one and several variables,
Resolution of singularities, Tauberian theorems, etc.

DE 2  – Dirichlet series associated to polynomial of several variables:
Analytical properties (meromorphic continuation, moderate growth, localization of singularities,etc.);
Special values, periods, etc.; Application to the Arithmetic of number fields,
applications to some counting problems of integer points satisfying arithmetical or geometrical constraints

DE 3  – Fractal zeta functions and applications:
Fractal zeta functions, application to the geometry of discrete fractals

DE 4  – Height zeta functions:
Introduction to the Manin Conjecture on the rational points of algebraic varieties,
Definition of the height zeta function and link with the Manin Conjecture,
study of some examples

  DE 5 - Means values of Multivariable Arithmetic Functions and Applications

- Multivariable Tauberian theorems,

-Asymptotic properties of densities of multivariable arithmetic functions

-Application to Manin's conjecture on toric varieties

- A. Pantchichkine (AP) (University Grenoble Alpes) :

    p-adic and complex L-functions on classical groups : admissible measures, special values II.

AP 1  - Classical groups
The case GL(n), the symplectic case, the unitary case. Modular forms and automophic forms,
examples

AP 2  - Hermitian modular forms
Complex L-functions on classical groups, Hecke algebras, Methode of Rankin-Selberg

AP 3  - Distributions, measures, congruences of Kummer

     p-adic zeta function of Kubota-Leopoldt, Iwasawa algebra

AP 4  - p-adic L-functions on classical groups
Admissible measures, special values

- M. Raibaut (MR) (University Savoie Mont Blanc):

     Igusa Zeta functions, Motivic Zeta functions, Monodromy Conjecture

MR 1  - Integration in a valued field
p-adic integration, spaces of arcs and motivic integration

MR 2  - Igusa Zeta functions
Igusa p-adic and motivic Zeta functions, Poincaré series of an algebraic variety
Theorems of rationality and proof using the method of resolution of singularities

MR 3  - Rationality of zeta functions and theory des models
Model theory of valued fields and cell  decomposition, Proof of the theorem of rationality
of zeta functions, Theorems of uniformity in residual characteristic p

MR 4  - Monodromy Conjecture
Igusa Zeta functions and topological zeta functions of a polynomial, Zeta functions of the monodromy
of a polynomial, Monodromy Conjecture

- J.-L. Verger-Gaugry (VG) (CNRS, Université Savoie Mont Blanc):

     Limit Conjectures in number theory, Lehmer's Conjecture,
Conjecture of Schinzel-Zassenhaus, dynamical zeta function of the beta-shift

VG 1  – Conjecture of Lehmer, Conjecture of Schinzel-Zassenhaus

     Minoration of the Mahler measure, minoration of heights, problem of Lehmer. Analogues and
generalizations, limit problems. Perron numbers, Pisot numbers, Salem numbers. Boyd's Conjectures on
Salem numbers , Theorems of Boyd-Lawton and Doche, Mahler measures of multivariate polynomials,

     cohomological interpretations : Deninger, Boyd, Rodriguez-Villegas...

VG 2  – Dynamical zeta function of the beta-shift

     Beta-transformation, Perron-Frobenius operators, transfer operators, generalized Fredholm
determinants, kneading determinants of Milnor and Thurston, Parry Upper functions, some results from
ergodic theory

VG 3  – Rényi-Parry dynamical system, lacunarity, lenticularity

     Conditions of Parry, dynamics of Perron numbers, in real algebraic number basis,
Geometry and identification of the zeros of the Parry Upper functions, Solomyak's fractal,
questions of rationality, dichotomy of Carlson-Polya

VG 4  – Asymptotic expansions of the Mahler  measures

     Limit equidistribution of conjuguates and beta-conjugates (Bilu, Favre Rivera-Letelier), theory of d'Erdös-Turan,
asymptotic expansions and polylogarithms : Poincaré, Condon. Dobrowolski type inequalities and
minorations, examples. Methods of resolution.