Ongoing work on the LMFDB will add more data, include more mathematical
objects, provide better searching and browsing, and give more detailed
information about each object in the database. The lists below
describe some of the planned enhancements. To make comments or
suggestions about the future of LMFDB, please see the
feedback page for information about
the mailing list and bug-report form.

# L-functions

## Navigation pages

- Add searching by first zero, coefficients, functional equation parameters, and underlying object.

## Home pages

- Smoothly handle the zeros and Z-function plot in the cases where not enough coefficients are available. (Will be automatic for L-functions coming from the database.)
- Recognize non-primitive L-functions and provide links to the primitive components. (sym^2 of CM curves, Dedekind zeta-function in the nonabelian case, etc)
- Links to more friends: dual L-function (if not self-dual), etc.

# Elliptic modular forms

## Home pages

- Give an option to show the dimensions of the whole space (including oldforms)
- Make it possible to show a basis for the whole space, not "only" representatives for the Galois orbits
- Show sage/magma/gp commands
- Include Eisenstein series
- Add links to the lifts (e.g., Saito-Kurokawa) of classical MFs
- Add Satake parameters at "bad" primes

# Elliptic Curves

## Additional pages

- Add page with list of elliptic curves over Q that have a point of canonical height at most some small'ish epsilon. (Noam Elkies has this data.)
- Add page with elliptic curves of smallest known conductor and Mordell-Weil group of the form F x Z^r for the 15 possible finite groups F and r = 0,1,2,...

## Additional data (over Q)

- Currently the database contains all elliptic curves defined over $\mathbb{Q}$ of conductor at most 379998. Add the much larger Stein-Watkins database, which includes curves with larger conductor. This would not be complete. Currently we do not have as much data about all the Stein-Watkins curves as for those in the Cremona database: for example we do not have generators for all the curve of rank 1. Also Mike Bennett's 2015 data, Randall Rathbun's data on congruent number curves for parameters up to 1000000.
- More data about each curve:
- Fisher has explicit models of genus one curves representing every element of order 3 and order 5 in Sha, and one could easily do similar for elements of order 2. These could be shown on a curve's home page.
- Base change: the user could select an extension field and see the page for the base-changed curve. This would be easy for cases where the base-change curve is already in the database.

## Elliptic curves over number fields

- Data available for each curve should be extended to include Mordell-Weil generators, which have in most cases not yet been computed.

# Dirichlet characters

## Home pages

- improve speed
- add magma commands
- show a graph of characters above and below (parents and sons)

## Search

- search from values

## Available data

- add Hecke characters

# Genus 2 curves

## Home pages

- List automorphic friends that arise via base change.
- Compute Kummer surface.
- List known rational points.
- List rank and generators of the Jacobian (when known).

## Available data

- Add curves with Jacobians isogenous to existing entries.

# Hilbert modular forms

## Navigation pages

- Spaces of Hilbert modular forms need web pages. Could list Hecke-Sturm bound data, list tables of dimensions, associated Shimura curve if relevant, etc.

## Home pages

- Compute Eisenstein ideal (torsion).
- Simplify presentation of Hecke field.
- For the CM forms, prove the CM and identify explicitly the Hecke character.
- For the base change forms, prove this rigorously (if possible) and identify explicitly the base change form.

## Possible new subcategories

- Higher weight, odd weight, half-integral weight, and characters.
- Maass forms.

# Maass forms

## Navigation pages

- Make browsing more intuitive (how do people actually want to browse / search Maass forms?)
- Maass forms arising from quadratic fields.

## Home pages

- Get statistics for Fourier coefficients, for example plots of value distribution (to see Sato-Tate) etc.
- Add plots of the Maass form in the fundamental domain.

## Available data

- Compute large sets of coefficients for all Maass forms in the database.

## Possible new subcategories

- Maass forms on the Picard group.
- Harmonic weak Maass forms.
- Maass forms of half-integral weight.

# Siegel Modular Forms

## Home Page

- Provide data for and make links to the L-function section.
- Improve the presentation of Fourier coefficients.

# Artin representations

## Home pages

- Non integral root numbers need to be calculated (for 1-dimensional, based on Gauss sums, for higher dimensional reps, based on analytic approximation)
- A link from degree 2 Artin representations to the corresponding modular form will be added.

# Global number fields

## Home pages

- Data on narrow class numbers
- Include information on local algebras for ramified primes
- search by root discriminant

# Local number fields

## Navigation

- Allow download of fields coming from a search

## Home pages

- Give components of interesting local resolvent algebras, such as twins for sextics and $D_4$ quartics

# Galois groups

## Home Pages

- Better handling of character tables

## Sato-Tate groups

## Home pages

- Add generators for all groups
- Add formula for the Haar measure

## Available data

- Sato-Tate groups for all Dirichlet characters in the LMFDB
- Sato-Tate groups for all Artin representations in the LMFDB
- Add links to Sato-Tate group on L-function pages
- Sato-Tate groups for abelian 3-folds (weight 1 degree 6)

# Future areas

Objects which will be added to the LMFDB in the future include:

- hyperelliptic curves of higher genus
- Galois representations
- GL(N) automorphic forms of cohomological type
- K3 surfaces and Calabi-Yau 3-folds
- Examples of every type of L-function of degree $\le 4$
- Automorphic representations
- Bianchi modular forms