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Author: William A. Stein

Math 582: computational number theory

Homework 4 -- due by Friday Feb 5 at 9am

WARNING: I haven't exactly tried the problems below. I don't know how computationally difficult they are for sure.

fcaf67de-0ff6-4544-b9f2-d6d494060f6bi %md **Problem 1:** Let $\rho_{E,2}$ denote the mod 2 representation attached to $E$ for each of the following elliptic curves: 17a1, 32a1, 32a2, 37a1. For each, compute explicitly the matrix of $\rho_{E,2}(\text{Frob}_P)$, where $P$ is a choice of prime ideal over each of 3,5,7,11,13. Be sure to check that $X^2 - a_p(E)X + p$ is the charpoly of $\rho_{E,2}(\text{Frob}_P)$.

Problem 1: Let ρE,2\rho_{E,2} denote the mod 2 representation attached to EE for each of the following elliptic curves: 17a1, 32a1, 32a2, 37a1. For each, compute explicitly the matrix of ρE,2(FrobP)\rho_{E,2}(\text{Frob}_P), where PP is a choice of prime ideal over each of 3,5,7,11,13. Be sure to check that X2ap(E)X+pX^2 - a_p(E)X + p is the charpoly of ρE,2(FrobP)\rho_{E,2}(\text{Frob}_P).

fd00381e-10ad-454a-aefb-3bb40d7a131f d8b26975-ea3e-4b5d-8626-cb24fe336e60 81f969b4-b3ab-4008-a2ad-004aa6f0aa23 b7d0ad95-db24-4d1f-8eb5-af5f7ede042a 679c8d56-85fd-4034-bd07-3af1eefd70e3 6a1f1acc-7cb0-44a3-959b-7865ed4fea2a 56f2221a-e6a2-4dca-bd56-bae196a60aa2i %md **Problem 2:** Let $\rho_{E,4}$ denote the mod 4 representation attached to $E$ for the curve 32a1. This is the homomorphism $G_\QQ\to\text{GL}_2(\ZZ/4\ZZ)$ defined by the action of $G_\QQ$ on $E[4]$. Be sure to check that $X^2 - a_p(E)X + p$ is the charpoly of $\rho_{E,2}(\text{Frob}_P)$. Try to compute explicitly the matrix of $\rho_{E,4}(\text{Frob}_P)$, where $P$ is a choice of prime ideal over each of 3,5,7,11,13.

Problem 2: Let ρE,4\rho_{E,4} denote the mod 4 representation attached to EE for the curve 32a1. This is the homomorphism GQGL2(Z/4Z)G_\QQ\to\text{GL}_2(\ZZ/4\ZZ) defined by the action of GQG_\QQ on E[4]E[4]. Be sure to check that X2ap(E)X+pX^2 - a_p(E)X + p is the charpoly of ρE,2(FrobP)\rho_{E,2}(\text{Frob}_P).

Try to compute explicitly the matrix of ρE,4(FrobP)\rho_{E,4}(\text{Frob}_P), where PP is a choice of prime ideal over each of 3,5,7,11,13.

ee1e4474-8413-42d7-9b50-1d5482cb6db2 706970bc-3997-404d-a300-82d455cf4b81 6293159f-22be-4675-b607-5ff1160e2668 6a517ed1-19c5-43f6-8b06-a6359f979c5f c49d596e-916c-4cbe-a7e8-6ff12f6cf434 c7349679-e04f-4c1d-adc4-f279e791dbca 809ffae5-6a4e-4115-927e-9392867ca575 537fa1a1-325f-4988-a744-34fef7f59ff3