LfunctionsAndModularForms/ClassicalModularForms/070730.PM
Known Databases
Stein/Watkins: Modular forms http://sage.math.washington.edu/ecdb/
Cremona: Rational modular forms http://www.maths.nott.ac.uk/personal/jec/ftp/data/INDEX.html
Kohel: Quaternion Algebras http://echidna.maths.usyd.edu.au/~kohel/dbs/index.html
Basmaji's mysterious tables http://modular.fas.harvard.edu/scans/papers/basmaji/
- Skoruppa: weight 1 icosahedral
- G Tornaria: weight 3/2
- Buzzard: Weight-1 modular forms (william probably has a copy somewhere)
Conrey/Farmer: http://lfunctions.org
M. Rubinstein's L-function database and calculator: http://pmmac03.math.uwaterloo.ca/~mrubinst/L_function_public/L.html
- Stein's mailbox...
Jodi Quer et al. have a table of \mathbb{Q}-curves.
Other collections
- Cohen et al. (not electronic)
The Number Theory Web lists a bunch of tables (070730) http://www.numbertheory.org/ntw/N1.html#tables
- Iwasawa invariants of elliptic curves (Robert Pollack)
Table of Q-isomorphism classes of elliptic curves over Q, with at least one 2-torsion rational over Q and conductor 2mp, with prime p < 29 (Wilfrid Ivorra - pdf file)
- Database of Automorphic L-functions (Stephen Miller)
- The Cunningham project
- Modular Polynomials (Masanari Kida)
- Minimal-known positive and negative k for Mordell curves of given rank (Tom Womack)
- Data on Mordell's curve y2=x3+k (SIMATH)
- Noam Elkies' computational number theory page (eg congruent numbers tables)
- A Database for Number Fields (Jurgen Kluners and Gunter Malle)
- Tables of Fermat near-misses (Noam Elkies)
- Tables related to Marshall Hall's conjecture (Noam Elkies)
- Tables of vanishing Fermat quotients, R. Ernvall and T. Metsankyla, 1995
- High rank elliptic curves with prescribed torsion (Andrej Dujella)
- Tables of Dedekind zeta functions at negative integers (Eyal Goren)
- Tables accompanying Rational eigenvectors in spaces of ternary forms, Larry Lehman, Math. Comp. 66 (1997) 833-839
- Tables of zeros of the zeta function (Andrew Odlyzko)
- Field Arithmetic Preprints Archive, Ben Gurion University of the Negev
- John Cremona's Elliptic Curve Data
- Online version of John Cremona's elliptic curve tables
- Number fields of low degree : (John Jones, Arizona State University)
LiDia preprint server, J. Buchmann's group (English Version)
- TNT (Tools on Number Theory)
- Tables of rank 1 Drinfeld modules (David Hayes)
- Classical Modular Polynomials (Michael Rubinstein)
- Mathematical constants and computation (Xavier Gourdon and Pascal Sebah)
- Mathematical Constants (Steve Finch)
- Elliptic curves with unusual torsion (Tom Womack)
- Tables of imaginary quadratic fields with class number not exceeding 23
List of class numbers for real and imaginary quadratic fields for discriminants 1000 < |D| < 10000 (David Terhune)
- New Modular Hyperelliptic Curves (E. Gonzalez-Jimenez)
- Genus 2 curves with modular jacobians (E. Gonzalez-Jimenez)
Basic questions informing the design
- What do we want from the database?
- What motivates its creation?
- How will we use it?
- Content:
- Computed information
- Computational state?
- Links to related databases
- Algorithms
- Code/Snippets
- Conjectures
- Centers of activity
Some content ideas
- Trace values
All eigenforms that are \eta-products (Conrey/Farmer)
Zeros of L-functions associated to modular forms (M. Rubinstein)
Zeros of modular forms (e.g., \Delta)
Sometimes, we want a_n as well as a_p (e.g., when considering degree-4 L-functions, we need both a_p and a_{p^2}).
- Both an eigenform basis and a rational basis are of interest (e.g., to keep the size of the coefficients small).
- Petersson norm (modular symbols needed?)
- Special values of modular forms
- Identify those newforms that have rational coefficients
q-expansions of Eisenstein series
- Weight-1 forms - can't get these directly from modular symbols or quaternion arithmetic
Weight-1 forms - can get these indirectly from modular symbols (see Springer, LNM 1585); thesis of Edixhoven students; Buzzarrd
- Half integral weight forms - can get indirectly from modular symbols
- Congruences between modular forms, congruence graphs (w. labeled edges)
- Construct models of curves: a modular Abelian variety as the Jacobian of a curve. Steven Galbraith's tables.
- Signs, under Frike and Atkin-Lehner involutions
- Order of vanishing at the critical central value; parity of the order.
Twist relations: entry A is a twist of entry B (by character \chi).
- Values of modular forms at interesting points.
- special values at CM points
Applications
Equations of modular curves (X_1(N))
- Cryptography
- Study Modular Abelian varieties
- Images of Galois representations
Labeling can be derived from first principles:
- [ level ] [ base 26 ] k [ weight ] [ character ]
Within the database, how do we order the entries for a given (weight, level)? William likes to use the trace values of the a_p for p \geq 1. We get the dimension as the first ordering.
Eidixhoven has a program to compute coefficients of \Delta in polynomial time; this is of more theoretical interest, but does it relate to any of this work?
Do we (who will) address the complexity of the computations?
Wild and Crazy thoughts (Wish List)
Given a list (a_2, a_3, ...) of coefficients, find forms in the database that are congruent to these, modulo primes of a given residue character.
With a newform, record its reductions modulo a collection of primes.
Computing reductions modulo primes can be tricky, without Magma; in particular, there are problems when you have non-p-maximal orders, which occur only for a finite set of primes.
Record all BSD invariants. The regulator is the hard part, of course.
Shimura lifts (cross-referenced to \frac12-integral-weight modular forms).
Graded rings of a given level: M(N)=\oplus_kM_k(N)
Include equations for modular curves.
We need checks on all data: Consistency, accuracy, precision, validity.
For the Maass forms, a lot of the information is numeric, so accuracy is important.
Modular forms in terms of \theta-series.