CoCalc Public Fileshomework / 2016-02-22-homework-7 / 2016-02-22-homework-7.sagews
Author: William A. Stein

# Math 582: computational number theory

## 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


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## Problem 1. Think about a definition.

Let $f(z)$ be a weight 2 modular form for some congruence subgroup $G$.  Check *by hand* that
for all $\gamma\in G$ we have $f(\gamma(z))d(\gamma(z)) = f(z)dz$.  Here $\gamma$ acts by
linear fractional transformations.


## Problem 1. Think about a definition.

Let $f(z)$ be a weight 2 modular form for some congruence subgroup $G$. Check by hand that for all $\gamma\in G$ we have $f(\gamma(z))d(\gamma(z)) = f(z)dz$. Here $\gamma$ acts by linear fractional transformations.


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# Problem 2. Compute $\Delta$

Let $\Delta(q) = \sum a_n q^n$ be the weight $12$ cusp form $q\prod (1-q^n)^{24}$.

See http://wstein.org/books/modform/modform/level_one.html.

(a) Supersingular primes: Find all primes $p<10^6$ such that $p \mid a_p$.

(b) A big one?: There is one more known big prime $p>10^6$ with $p \mid a_p$.   Find it using the Internet.

(c) A congruence: Find a prime $\ell$ such that $a_p \equiv p^{11} + 1 \pmod{\ell}$ for all $p<1000$.


# Problem 2. Compute $\Delta$

Let $\Delta(q) = \sum a_n q^n$ be the weight $12$ cusp form $q\prod (1-q^n)^{24}$.

(a) Supersingular primes: Find all primes $p<10^6$ such that $p \mid a_p$.

(b) A big one?: There is one more known big prime $p>10^6$ with $p \mid a_p$. Find it using the Internet.

(c) A congruence: Find a prime $\ell$ such that $a_p \equiv p^{11} + 1 \pmod{\ell}$ for all $p<1000$.


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# Problem 3: Supersingular primes

Noam Elkies provided that if $E$ is an elliptic curve, then there are infinitely many primes $p$
such that $p \mid a_p(E)$.

Write down an elliptic curve of conductor 43.  How many primes $p<10^7$ are supersingular for $E$? (Hint: use E.aplist(...))
Plot a histogram showing the gaps between supersingular primes.  (Hint: use stats.TimeSeries([...]).plot_histogram())


# Problem 3: Supersingular primes

Noam Elkies provided that if $E$ is an elliptic curve, then there are infinitely many primes $p$ such that $p \mid a_p(E)$.

Write down an elliptic curve of conductor 43. How many primes $p<10^7$ are supersingular for $E$? (Hint: use E.aplist(...)) Plot a histogram showing the gaps between supersingular primes. (Hint: use stats.TimeSeries([...]).plot_histogram())


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## Problem 4: Modular forms

Compute the class groups of all of the number fields $K_f$ generated by the coefficients of the newforms of weight
$2$ on $\Gamma_0(N)$ for each $N<100$.  Make a conjecture.


## Problem 4: Modular forms

Compute the class groups of all of the number fields $K_f$ generated by the coefficients of the newforms of weight $2$ on $\Gamma_0(N)$ for each $N<100$. Make a conjecture.

# HINT:
N = Newforms(43,names='a')
N[1].hecke_eigenvalue_field().class_group()

Class group of order 1 of Number Field in a1 with defining polynomial x^2 - 2