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Author: William A. Stein

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

146f3bf0-07ab-4fc0-a4f6-f2b622010c69i %md ## 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)f(z) be a weight 2 modular form for some congruence subgroup GG. Check by hand that for all γG\gamma\in G we have f(γ(z))d(γ(z))=f(z)dzf(\gamma(z))d(\gamma(z)) = f(z)dz. Here γ\gamma acts by linear fractional transformations.

3d6eaa53-4a1c-4ff6-9087-35a937ae9389 2aaeaed7-6e0b-44e0-bf63-6bd11a2025bb 5f5ac7df-2080-4631-a000-560ef723f470 28f0b5f8-2846-4a24-84fc-ff99e288f846 c0dbbe33-3994-4b16-9af7-a2235220c4cei %md # 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 Δ\Delta

Let Δ(q)=anqn\Delta(q) = \sum a_n q^n be the weight 1212 cusp form q(1qn)24q\prod (1-q^n)^{24}.

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

(a) Supersingular primes: Find all primes p<106p<10^6 such that papp \mid a_p.

(b) A big one?: There is one more known big prime p>106p>10^6 with papp \mid a_p. Find it using the Internet.

(c) A congruence: Find a prime \ell such that app11+1(mod)a_p \equiv p^{11} + 1 \pmod{\ell} for all p<1000p<1000.

26a92423-5528-4efc-96f3-021eebe19fcf 0e0ede5f-6caa-4e23-beed-9ef37e056c3e f141ad70-e6c7-4c54-9a7b-963af9f8ff8c f6a69c21-7405-4632-a190-4d22bc44a5e2 407f8762-9fe7-4ef1-9afd-e49585b4cf8a 1d78d6b6-9ee5-43cd-ba44-8c024c93665b d9d7445d-ce8b-4aae-9ea2-6e54fb03db9ci %md # 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 EE is an elliptic curve, then there are infinitely many primes pp such that pap(E)p \mid a_p(E).

Write down an elliptic curve of conductor 43. How many primes p<107p<10^7 are supersingular for EE? (Hint: use E.aplist(...)) Plot a histogram showing the gaps between supersingular primes. (Hint: use stats.TimeSeries([...]).plot_histogram())

809af2d0-7a43-4d2c-9edb-91f271eeb5dd 8096d9a7-4fed-4d7f-9f11-e9898dea5a02 8d747588-bdbe-4248-999d-cd75cf8d5ca9 ed7a4c05-dde3-48c0-863c-5a38380121fb 6a941b43-79fe-4519-92d9-b00e6a109d62i %md ## 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 KfK_f generated by the coefficients of the newforms of weight 22 on Γ0(N)\Gamma_0(N) for each N<100N<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