People: Elkies, Hart, Tornaría, Watkins
June 2008 Workshop Goals and beyond
\star weight 3/2 form for congruent number curve to 10^{10} this week. DONE. To 10^{12} by project end. Needs large RAM computers and algorithmic additions. Will post-process data for larger ranks and statistics.
\star weight 5/2 form for 8k4 to 10^8 this week. DONE (maybe 10^9), at least for odd twists (even twists are now done [correctly!] in theory, though working code has not yet been produced). To 10^{10} by project end.
\star weight 5/2 form for various symmetric cubes of CM elliptic curves. Did 49A to 10^8 (I think). Progress Page
\ast odd parity: Heegner points mod p for 32A. Too busy this week. Mark has 10^9 via Heegner points over C, so the goal must be 10^{10} at least. [The asterisk is intentional, as it indicates undone work].
Fallout (ideas for eventual papers, etc.)
\bullet Bill/Gonzalo: Enumerating points in ellipsoids. Implementation details with cache-friendliness, etc. How to multiply huge polynomials that don't fit into RAM. Perhaps a theoretical description of how to reduce enumeration to essentially linear time for any lattice (albeit with a large constant). How it works specifically in dimension 3. Give specific examples that were done (congruent number curve, 8k4, Sym^3(\psi_{49a}), maybe others).
\bullet Noam/Gonzalo: How to compute weight 5/2 lifts for weight 4 forms using ternary quadratic forms. Specific example of Sym^3(\psi_{49a}). Can action of Sym^3 on Brandt matrices be described directly? Goal might be to get the weight 5/2 lift for Sym^3 for "all" CM curves (though only one class of quadratic twists for j=0,1728). See also this paper. Can include 8k4 as another example, and perhaps compare splittings for the weight 5/2 form(s).
\bullet Noam/Mark: Post-processing of data for even twists of the congruent number curve, in order to determine the rank 4 twists. Noam has already run a fast "descent by 2-isogeny" test to prune the number of candidates. See here and here for more details. Mark has written a faster version of Denis Simon's 2-descent, that computes the local 2-Selmer group. This took 42 seconds to do all the remaning curves (in one congruence class) up to 10^6, and so should take only a few weeks to finish the lot.
Current Status
The weight 5/2 lifts (both genera) for Sym^3(\psi_{49a}) have been found (in the appropriate space of modular forms), and similarly with the lifts of weight 7/2 and weight 9/2 for the fifth and seventh symmetric powers. Only one of these has yet been written as a linear combination of ternary theta series.
Similarly, for \psi_{27a} we have listed six relevant lifts. Some of these seem to be \eta-products/quotients, which might not be too surprising, given the extra automorphisms of X_0(27).
For \psi_{32a} we have listed 12 relevant lifts (twists by evens are considered separately), and they are all \eta-quotients. See here.
The first results (the symmetric cube) for \psi_{121B} have additionally been computed. The elliptic curve 121B has odd parity here, so extra care is needed when measuring any theoretical results.
Information for the symmetric cube of the newform of level 8 and weight 4