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

The highest priority for L-functions is to reorganize the code so that all the information is stored in a database. This will make it much easier to implement the other planned changes. This work has been started for the L-functions of genus 2 curves.

• Reorganize the degree 3 and 4 navigation pages to allow browsing in terms of functional equation parameters in addition to browsing by underlying object.
• Add searching by first zero, coefficients, functional equation parameters, and underlying object.
• Show the sign of the Dirichlet L-functions as a color code on the browsing graph.

• Reorganize the home pages so that they look more like other home pages. In particular, have an "Invariants" section near the top.
• Add section for Special Values.
• Allow the user to choose arithmetic or analytic normalization (for arithmetic L-functions).
• 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.)
• Display more coefficients for L-functions with very sparse Dirichlet series, and condense the display for floating point coefficients.
• 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), twists, symmetric powers, other powers.
• Next/previous navigation for more L-functions (e.g. GL(2) Maass forms).

## Available data

• Add the L-functions of: Maass forms (some currently limited by the number of available coefficients), $sym^2$ of CM elliptic curves, $sym^2$ of other GL(2) objects, Siegel modular forms, Hilbert modular forms (Asai L-function), Artin representations.

# Elliptic modular forms

• Add a paragraph above each browsable table (Gamma_0(N) and Gamma_1(N)) describing how much data is currently available

• 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(?) commands
• produce zoom-able images of fundamental domains

## Available data

• Rewrite the whole backend for getting data for modular forms (spaces)
• Use (mostly) William's recently computed data
• Include Eisenstein series
• Add more data to the homepages of spaces of newforms (e.g., Hecke polynomial (factored?))

# Elliptic Curves

• Currently the search possibilities are rather limited, but if more search fields are introduced the search page will become very cluttered. At that point we should only have very common options on the main page, with a separate "advanced search" page. Examples include searching by [a1,a2,a3,a4,a6] or by [A,B]. Work on this is in progress.

• Add more to the Statistics page for elliptic curves over Q: currently this gives distributions of ranks, Shas and other things.
• 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,...

• Currently the database contains all elliptic curves defined over $\mathbb{Q}$ of conductor at most 360000.
In future it is planned to 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.
• Automatic construction of quadratic twists of each curve from its home page; if the twist is not in the database we can still create a home page for it, with limited data.
• 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

• Work has started to include elliptic curves over many number fields (real and imaginary quadratic and more). So far we have all curves over real quadratic fields whose corresponding Hilbert modular form is in the database; curves over the first 5 imaginary quadratic fields, which will be similarly linked to Bianchi newforms; and the Gunnells-Yasaki curves over the cubic field of discriminant -23.
• The above is currently only accessible from beta.lmfdb.org.
• Data available for each curve should be extended to include Mordell-Weil generators, which have in most cases not yet been computed.
• Automatic base change from the home page of every elliptic curve over $\Q$ to a number field, specified by its label. Whether or not the resulting curve is in the database, its home page can be displayed.

# Dirichlet characters

• improve speed
• add magma commands (and pari when characters are released)
• show a graph of characters above and below (parents and sons)
• show minimal polynomial and galois orbit

## Search

• group characters by Galois orbits
• search from values

## Available data

• add Dirichlet characters over number fields

# Genus 2 curves

• In the search by torsion order, it would be nice to know what possible torsion groups occur among the curves in the database.

• List automorphic friends that arise via base change.
• Compute Kummer surface.

# Hilbert modular forms

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

• Compute Eisenstein ideal (torsion).
• Simplify presentation of Hecke field (adjoin all elements to get an order, or mimic classical modular forms by working with the dual).
• 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

• Make browsing more intuitive (how do people actually want to browse / search Maass forms?)
• Is it possible to keep information about where in the browse list we are, when going to the page for a specific Maass form?
• Maass forms arising from quadratic fields.

• Make it possible to manipulate Fourier coefficients (i.e. test Hecke relations etc.)
• Get statics for Fourier coefficients, for example plots of value distribution (to see Sato-Tate) etc.
• Add plots of the Maass form in the fundamental domain.
• Add next / previous buttons (like on the Dirichlet character pages).

## Available Data

• All Maass form data (for GL(2)/Q) is currently recomputed using sage/psage/cython. These computations should continue until we have data for all levels, at least up to 100.
• Compute large sets of coefficients for all Maass forms in the database.

## Technical Issues

• The option to show "all" records does not work on the browse page.
• Fix the search so that it is also done with server-side dataTables so we can remove the limit of 200 records (currently the client crash if we return more records)
• Try to make "tabbed" dataTables to make it possible to flip between different levels when browsing.

## Possible new subcategories

• Maass forms on the Picard group (these already exist in the database, but the corresponding pages needs to be revised).
• Harmonic weak Maass forms.
• Maass forms of half-integral weight.

# Siegel Modular Forms

• Provide data for and make links to the L-function section.
• Improve the presentation of many and big Fourier coefficients: the user should be able to ask for a range, reduction modulo arbitrary ideals of the number field generated by the coefficients should be possible (currently one can only reduce modula rational primes).

## Database

• Compute generators and examples for the rings $\Gamma_0(2)$, $\Gamma_0(3)$, $\Gamma_0(4)$ (possibly with characters).
• Implement the data of Ibukiyama and Hayashida on half integral weight and vector valued modular forms.
• Let mongodb or sqlite keep the date.
• Normalise the Fourier coefficients so that they are integral and primitive.

# Artin representations

The search page will be expanded to include searches based on information depending on the Artin field (for instance the Galois group).

• A link from degree 1 Artin representations to the corresponding Dirichlet character needs to be added (and similarly for L-functions)
• 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.

## Available data

Very limited data is available right now (37000 representations). We will soon add more data, coming from computations of Tim Dokchitser.

# Global number fields

Currently have fields with small absolute discriminant in degrees $\leq 11$, and other complete sets of fields with degrees up to $15$. See completeness of global number field data for more details

• Data on narrow class numbers
• Include information on local algebras for ramified primes
• Pretty print names for abelian fields (e.g. cyclotomic fields, totally real subfield, biquadratic fields)

## Available Data

• Add fields with bigger degrees and discriminants, and with more diverse Galois groups.

# Local number fields

Currently have all degree $n$ fields over $\mathbb{Q}_p$ for $p<150$ and $1\leq n\leq11$.

• Allow searching by inputing a prime and a polynomial (return components of the local algebra)

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

## Available data

• Hope to add higher degree fields once data has been computed for wildly ramified fields of degree 12

# Galois groups

Currently the database contains all transitive subgroups of $S_n$ up to conjugation for $n\leq 23$.

• Better handling of character tables

# Future areas

Objects which will be added to the LMFDB include:

• hyperelliptic curves
• 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