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Week 1

Monday, June 20

• Introductions: http://tinyurl.com/3m6fs8f

• Tutorial about how to compute things about elliptic curves: http://nt.sagenb.org/home/pub/45/

• Cremona's online tables: http://www.warwick.ac.uk/~masgaj/ftp/data/

• Cremona's book (see in particular Table 1): http://www.warwick.ac.uk/~masgaj/book/fulltext/

• Assignment for today: Compute as much as you can about the curves listed here.

• Results of assignment:

  • William: See http://nt.sagenb.org/home/pub/46/ and in particular the pdf link at the bottom.

Tuesday, June 21

• 10am-12pm: Introduction to Sage in Communications B27. Video

• Presentation of assignments from June 20 (table up to norm conductor 124):

  • Rado Kirov: http://nt.sagenb.org/home/pub/47/

  • Ben Leveque: http://nt.sagenb.org/home/pub/50/

  • Andrew Ohana: http://nt.sagenb.org/home/pub/51/

  • Ashwath Rabindranath: http://nt.sagenb.org/home/pub/53/

  • Paul Sharaba: http://nt.sagenb.org/home/pub/56/

  • William Stein: http://nt.sagenb.org/home/pub/46/

• Demos of things you've done using Sage (e.g., interacts, etc.):

  • Ben Leveque's Farey Graph: http://nt.sagenb.org/home/pub/52/

• Get started with the Wiki a little: http://wiki.sagemath.org/reu/2011

• Andrew Ohana and Aly Deines: canonical stuff ohana1.pdf

• Mentioned: dumps, loads, save, load

• nosqlite: http://code.google.com/p/nosqlite/ along with a worksheet that illustrates it.

• Antwerp IV -- old tables of elliptic curves: http://modular.math.washington.edu/Tables/antwerp/

• Read Ribet's awesome paper 'Galois representations and modular forms', in particular Sections 3 and 5.

• Extensive data over Q(sqrt(5)) so far is here as plain text files. This can be used instead of http://db.modform.org, and is easier to parse.

Wednesday, June 22

• Rado's talk at 10am in Communications B27:

  • Demo: http://sagenb.org/javascript/graph_editor2/index.html

  • Best HTML5 canvas tutorial: https://developer.mozilla.org/en/canvas_tutorial

  • Graph editor source code: https://bitbucket.org/radokirov/js-graph-editor

• 11:30am -- meet in the Sage lab

• Aly Deines -- Global minimal models at 3:10pm. http://nt.sagenb.org/home/pub/62/

• Get started with Mercurial a little: http://code.google.com/p/uw-nt-reu2011/

• William's idea for determining whether there is a p-isogeny: http://nt.sagenb.org/home/pub/61/

• Elkies tables that have isogeny examples: http://www.math.harvard.edu/~elkies/nature.html

• William: Added initial template version of our goal paper to the repository.

• Here modifiedellipticcurvedatafile.txt is an initial table that adds |T|, signs, ord(disc), ord_(j), Tamagawa numbers, and Kodaira symbols to the data at http://wstein.org/Tables/hmf/sqrt5/finding_weierstrass_equations/matched.txt

• Here is the paper from the 2010 REU: On the Irreducibility of Galois Representations Associated to Elliptic Curves

• Ashwath: Here is some very preliminary code that confirms William's idea to check if a certain curve has odd prime isogenies using the reducibility of the characteristic polynomial of Frobenius at primes of good reduction: http://nt.sagenb.org/home/pub/63

• The isogeny source code in Sage by Dan Shumow: http://nt.sagenb.org/src/schemes/elliptic_curves/ell_curve_isogeny.py

• Documentation of Shumow's isogeny code: http://sagemath.org/doc/reference/sage/schemes/elliptic_curves/ell_curve_isogeny.html

• Computing isogeny degrees and filling out the norm conductor 31 isogeny class: http://nt.sagenb.org/home/pub/65/

• Project idea: For each aplist in http://wstein.org/Tables/hmf/sqrt5/ellcurve_aplists.txt find all possible isogeny degrees. This would be extremely interesting!!

• Hilbert modular forms: http://code.google.com/p/purplesage/source/browse/#hg%2Fpsage%2Fmodform%2Fhilbert%2Fsqrt5%253Fstate%253Dclosed

Thursday, June 23

• Official start meeting at 12pm at Pa$eo in Fremont. 4225 Fremont Ave N.

• Work afterwards in Sage lab starting 2pm.

• For each aplist in http://wstein.org/Tables/hmf/sqrt5/ellcurve_aplists.txt find all possible isogeny degrees. Isogeny Degrees The worksheet is published here: http://nt.sagenb.org/home/pub/70

• Ashwath, Ben, Andrew: We have re-written the prime isogeny code from yesterday, making it significantly faster: http://nt.sagenb.org/home/pub/69/

• Note: Initial curve data file above (modifiedellipticcurvedatafile.txt) is updated to have 0s where the ord(j) list is empty so it is easier to parse

• Paul has written the code for nosqlite to create a database for the elliptic curves. It is currently uploaded to the sage clusters, in the folder databases, labeled small_nosqlite.

• Andrew: I made a short write-up on the ordering scheme that I was working on. It very briefly mentions the background used with specialized definitions. http://wstein.org/home/ohanar/sqrt5/element_ordering/element_ordering.pdf

Friday, June 24

• 10am in the Sage lab for status reports and planning.

• Here's a table of isogenies. They were computed using just the ap values, and still include some really really big isogenies: IsogenyTable4.txt -- it corresponds to the curves in this table: matched.txt. This version does not have spaces in the isogenies column.

• Computed some isogeny graphs for specific curves using division polynomials. Task: Find an algorithm to do the same.

• Explicit isogeny finding worksheet: http://nt.sagenb.org/home/pub/71/

• Here's a table of isogenies that includes the a-invariants for easy reference. IsogenyTable7.txt

• Worksheet which uses William and Ashwath's code to check which isogeny degrees appearing in the table above are actually valid. http://nt.sagenb.org/home/pub/72/

Week 2

Monday, June 27

• Code for finding isogeny classes: http://nt.sagenb.org/home/pub/76/ Notice that it has a rating of 4.0!

• 2:30 pm - Talk by Ashwath on computing prime isogenies of Elliptic Curves.

• Progress on graphing isogeny classes: http://nt.sagenb.org/home/pub/79/

Tuesday, June 28

Wednesday, June 29

• 2:10pm in Sieg Hall -- talk by Ralph Greenberg on Galois Representation

• Created updated text file (Galois paired table) Paired_Table.txt

• Elkies algorithm: see here

• Here's our isogeny class/adjacency matrix data for N up to 3205! isogeny_partial_output.txt

Thursday, June 30

Friday, July 1

Week 3

Monday, July 4

Tuesday, July 5

Wednesday, July 6

Thursday, July 7

Friday, July 8

Week 4 (William in Budapest this week)

Monday, July 11

Tuesday, July 12

Wednesday, July 13

Thursday, July 14

Friday, July 15

Week 5

Monday, July 18

Tuesday, July 19

Wednesday, July 20

Thursday, July 21

Friday, July 22

Week 6

Monday, July 25

Tuesday, July 26

Wednesday, July 27

Thursday, July 28

Friday, July 29

Week 7

Monday, August 1

Tuesday, August 2

Wednesday, August 3

Thursday, August 4

Friday, August 5

Week 8

Monday, August 8

Tuesday, August 9

Wednesday, August 10

Thursday, August 11

Friday, August 12