A Map of the LMFDB
Introduction and more
The research areas of the LMFDB are distinguished by the variety of connections among the various objects.
field of definition,
n-divison fields, etc
some varieties are modular
A number field is a finite dimensional extension
of the rational numbers.
The Dedekind zeta function is defined as a
product over the prime ideals of the number
An Artin representation is a finite dimensional
representation of the Galois group of a number
Associated to a number field is a Galois group:
the Galois group of its Galois closure.
The Rankin-Selberg convolution of a degree n
L-function and a degree m L-function is an
L-function of degree n*m.
A variety is 'modular' if its Hasse-Weil
L-function is the same as the standard
L-function of some modular form.
Shimura varieties are algebraic varieties
uniformized by the same linear algebraic
groups that give rise to automorphic forms.
A 'twist' is an algebraic variety defined
over the same field isomorphic to the
original one over some larger field.
Given a cuspidal automorphic representation
π of an adelic reductive group G and a finite
dimensional representation ρ of the Weil-
Deligne group of G, one can form the
automorphic L-function L(s,π,ρ).
A 'lift' is a map from the automorphic forms
on one group to the automorphic forms on
A 'converse theorem' is an assertion that
an L-function is the L-function of an
The local factors of the Hasse-Weil
L-function are determined by counting
points on the variety modulo prime powers.
The local factors of the Artin L-function are
constructed from the trace of Frobenius.
Algebraic varieties are geometric objects
defined by polynomial equations; a more
refined notion is the concept of a scheme.
Automorphic forms are L
certain double coset spaces of the adelic
points of a linear algebraic group.
An L-function is a Dirichlet series with a
functional equation and an Euler product.
Galois groups can be realized as permutations
acting on the roots of a polynomial.
The Langlands program predicts that Galois
representations are automorphic.
The étale cohomology with torsion coefficients
carries an action of the absolute Galois group.
The kernel is an n-division field.
This project is supported by
from the US National Science Foundation and the UK Engineering and Physical Sciences Research Council.
2017-10-23 09:15:37 -0400
· SageMath version 8.0