An exceptional arithmetic group and its Eisenstein series

An Exceptional Arithmetic Group and Its Eisenstein Series

Weyl Group Multiple Dirichlet Series, Eisenstein Series and Crystal Bases

If F is a local field containing the group μn of n-th roots of unity, and if G is a split semisimple simply connected algebraic group, then Matsumoto [27] defined an n-fold covering group of G(F ), that is, a central extension of G(F ) by μn. Similarly if F is a global field with adele ring AF containing μn there is a cover G̃(AF ) of G(AF ) that splits over G(F ). The construction is built on i...

Eisenstein Series*

group GC defined over Q whose connected component G 0 Q has no rational character. It is also necessary to suppose that the centralizer of a maximal Q split torus of G0C meets every component of GC. The reduction theory of Borel applies, with trivial modifications, to G; it will be convenient to assume that Γ has a fundamental set with only one cusp. Fix a minimal parabolic subgroup P 0 C defin...

The Second Term of an Eisenstein Series

The leading term of Eisenstein series and L-functions are well-known to be extremely important and to have very important arithmetic implications. For example, one may consider the analytic class number formula, The Siegel-Weil formula, the Birch and Swinnerton-Dyer conjecture, Stark’s conjectures, and Beilinson’s and Bloch-Kato conjectures. These are well-documented and are central in current ...

EISENSTEIN SERIES TWISTED BY MODULAR SYMBOLS FOR THE GROUP SLn

We define Eisenstein series twisted by modular symbols for the group SLn, generalizing a construction of the first author [12, 13]. We show that, in the case of series attached to the minimal parabolic subgroup, our series converges for all points in a suitable cone. We conclude with examples for SL2 and SL3.