Oliver Lorscheid

Toroidal automorphic forms for function fields

The space of toroidal automorphic forms was introduced by Zagier in 1979. Let F be a global field. An automorphic form on GL(2) is toroidal if it has vanishing constant Fourier coefficients along all embedded non-split tori. The interest in this space stems from the fact (amongst others) that an Eisenstein series of weight s is toroidal if s is a non-trivial zero of the zeta function, and thus a connection with the Riemann hypothesis is established.In this paper, we concentrate on the function field case. We show the following results. The (n−1)-th derivative of a non-trivial Eisenstein series of weight s and Hecke character x is toroidal if and only if L(x, s+1/2) vanishes in s to order at least n (for the “only if” part we assume that the characteristic of F is odd). There are no non-trivial toroidal residues of Eisenstein series. The dimension of the space of derivatives of unramified Eisenstein series equals h(g−1)+1 if the characteristic is not 2; in characteristic 2, the dimension is bounded from below by this number. Here g is the genus and h is the class number of F. The space of toroidal automorphic forms is an admissible representation and every irreducible subquotient is tempered.

A blueprinted view on

We extend the big and

\mathbb F_1

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-geometry

-typical Witt vector functors from commutative rings to commutative semirings. In the case of the big Witt vectors, this is a repackaging of some standard facts about monomial and Schur positivity in the combinatorics of symmetric functions. In the

On the relation between hyperrings and fuzzy rings

p

Schubert decompositions for quiver Grassmannians of tree modules

-typical case, it uses positivity with respect to an apparently new basis of the

Toroidal Automorphic Forms, Waldspurger Periods and Double Dirichlet Series

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The geometry of blueprints: Part I: Algebraic background and scheme theory

-typical symmetric functions. We also give explicit descriptions of the big Witt vectors of the natural numbers and of the nonnegative reals, the second of which is a restatement of Edreis theorem on totally positive power series. Finally we give some negative results on the relationship between truncated Witt vectors and

Mapping F_1-land:An overview of geometries over the field with one element

k

Algebraic groups over the field with one element

-Schur positivity, and we give ten open questions.Lecture notes of a course on A. Smirnovs approach to the ABC-conjecture via F1-geometry and an attempt to relate this to Jim Borgers F1-geometry based on lambda-rings.In this paper we answer a question raised in [25], Sec. 4, by showing that the genus zero moduli operad {M_(0,n+1)} can be endowed with natural descent data that allow it to be considered as the lift to Spec Z of an operad over F_1. The relevant descent data are based on a notion of constructible sets and constructible functions over F_1, which describes suitable differences of torifications with a positivity condition on the class in the Grothendieck ring. More generally, we do the same for the operads {T_(d,n+1)} (whose components were) introduced in [5]. Finally, we describe a blueprint structure on {M_(0,n)} and we discuss from this perspective the genus zero boundary modular operad {M^0_(g,n+1)}.In this very short and sketchy chapter, we draw some pictures on the arithmetic theory of