Math 582: computational number theory
Homework 7 -- due by Monday, Feb 29 at 11am
Topic: computing with modular forms
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NOTE: For this assignment, make sure you know how to make for
loops in Python or you will be screwed. If not read the sage tutorial, especially http://doc.sagemath.org/html/en/tutorial/programming.html#loops-functions-control-statements-and-comparisons
Problem 1. Think about a definition.
Let be a weight 2 modular form for some congruence subgroup . Check by hand that for all we have . Here acts by linear fractional transformations.
Problem 2. Compute
Let be the weight cusp form .
See http://wstein.org/books/modform/modform/level_one.html.
(a) Supersingular primes: Find all primes such that .
(b) A big one?: There is one more known big prime with . Find it using the Internet.
(c) A congruence: Find a prime such that for all .
Problem 3: Supersingular primes
Noam Elkies provided that if is an elliptic curve, then there are infinitely many primes such that .
Write down an elliptic curve of conductor 43. How many primes are supersingular for ? (Hint: use E.aplist(...)
) Plot a histogram showing the gaps between supersingular primes. (Hint: use stats.TimeSeries([...]).plot_histogram()
)
Problem 4: Modular forms
Compute the class groups of all of the number fields generated by the coefficients of the newforms of weight on for each . Make a conjecture.