Instructor Contact and General Information

Instructor: Luís Finotti
Office: Ayres Hall 251
Phone: 974-1321 (don't leave messages! -- me if I don't answer!)
Office Hours: MW 11:10-12:10 or by appointment.
Textbook: J. Silverman and J. Tate, "Rational Points on Elliptic Curves",
2nd Edition, 2015, Springer.
Prerequisite: one year of undergraduate algebra.
Class Meeting Time: MWF 12:20am to 1:10pm at Ayres 113. (Section 001.)
Exams: No exams.
Grade: Average of HWs, lowest score dropped.

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Course Description and Information

Course Content

In this semester we will discuss elliptic curves It will be pretty basic, with not much background required as prerequisites, but an undergraduate course in algebra will be assumed.

We will mostly follow the textbook, and will discuss topics such as the group law, group structure (Mordell-Weil's Theorem), elliptic curves over finite fields, applications to factorization and cryptography and integral points. I will like have a brief discussion about the Birch and Swinnerton-Dyer Conjecure.


Chapters and Topics

The plan is to cover, at least, Chapter 1 to 5 of the textbook. If time allows, we will proceed to Chapter 6, which would finish the book. I do intend to take some "detours" from the book on occasions, though, to cover topics I find interesting and relevant.


Additional Bibliography

There many good books on the topic, but most are more advanced than our textbook. Here are some of these:



Homework Policy

Homework will be regularly posted at the section Homework of this page. No paper copy of the HW assignments will be distributed in class. It is your responsibility to check this page often!

The HWs will "due" on Wednesdays. Each HW will have problems from the previous week (Monday, Wednesday and Friday lectures). I will not be very strict on dates, but I do not want you to turn in all your homework on the last day! I will also suggest a few problems, but you have the freedom to choose the ones you intend to do. As long as you turn in two to three (three if you choose really easy ones, please) a week, you should be fine. (Assuming you are getting most of the problems right.)

Of course, you get out what you put in, so students interested in algebra and number theory should at least try to do the harder problems, even if you do not turn them in. You are always welcome to come discuss problems with me.

In any event, no matter what you turn in, you should take a look at all exercises, as you will find interesting results and applications there. Sometimes we might need some results that were exercises from previous sections.

It's unlikely I will post solutions, but if you want to see some solution, you can come see me.

Besides HW, I would urge you to review the material often! I really don't want students to get lost in class, by forgetting theorems and definitions. Most of all, I want you to be able to follow the lectures! Of course, one way to make yourself keep up is to work on problems often, so try to not let your HWs be late often.

Also, you should try to come to my office hours if you are having difficulties with the course. I will do my best to help you. Please try to come during my scheduled office hours, but feel free to make an appointment if that would be impossible.

I will keep all your HW scores at Canvas.


Piazza (Discussion Board)

We will use Piazza for online discussions. The advantage of Piazza (over other discussion boards) is that it allows us (or simply me) to use math symbols efficiently and with good looking results (unlike Canvas).

To enter math, you can use LaTeX code. (See the section on LaTeX below.) The only difference is that you must surround the math code with double dollar signs ($$) instead of single ones ($). Even if you don't take advantage of this, I can use making it easier for you to read the answers.

You can access Piazza through the link on "Syllabus" page of Canvas or directly here: There is also links at the "Navigation" section on the top of this page and on the Links section.

To keep things organized, I've set up a few different folders/labels for our discussions:

I urge you to use Piazza often for discussions! (This is specially true for Feedback!) If you are ever thinking of sending me an e-mail, think first if it could be posted there. That way my answer might help others that have the same questions as you and will be always available to all. (Of course, if it is something personal (such as your grades), you should e-mail me instead.)

Note that you can post anonymously. (Just be careful to check the proper box!) But please don't post anonymously if you don't feel compelled to, as it would help me to know you, individually, much better.

Students can (and should!) reply to and comment on posts on Piazza. Discussion is encouraged here!

Also, please don't forget to choose the appropriate folder(s) (you can choose more than one, like a label) for your question. And make sure to choose between Question, Note or Poll.

When replying/commenting/contributing to a discussion, please do so in the appropriate place. If it is an answer to the question, use the Answer area. (Note: The answer area for students can be edited by other students. The idea is to be a collaborative answer. Only one answer will be presented for students and one from the instructor. So, if you want to contribute to answer already posted, just edited it.) You can also post a Follow Up discussion instead of (or besides) an answer. There can be multiple follow ups, but don't start a new one if it is the same discussion.

Also, you can send Private Messages in Piazza. So, if you have a math question not appropriate for the whole class (e.g., your question gives away the solution of a HW problem), you can send me a private message instead of an e-mail. That way my reply can have the math symbols nicely formatted.

Important: Make sure you set your "Notifications Settings" on Piazza to receive notifications for all posts: Click on the gear on the top right of the Piazza site, the choose "Account/Email Setting", then "Edit Email Notifications" and then check "Automatically follow every question and note". Preferably, also set "Real Time" for both new and updates to questions and notes. I will consider a post in Piazza official communication in this course, I will assume all have read every single post there!

You should receive an invitation to join our class in Piazza via your "" e-mail address before classes start. If you don't, you can sign up here: If you've register with a different e-mail (e.g., you do not need to register again, but you can consolidate your different e-mails (like and in Piazza, so that it knows it is the same person. (Only if you want to! It is recommended but not required as long as you have access to our course there!) Just click on the gear icon on the top right of Piazza, beside your name, and select "Account/Email Settings". Then, in "Other Emails" add the new ones.


Communications and E-Mail Policy

You are required to set up notifications for Piazza (as explained above) and for Canvas to be sent to you immediately. For Canvas, check this page and/or this video on how to set your notifications. Set notifications for Announcements to "right away"! (Basically: click on the the profile button on left, under UT's "T", then click "Notifications". Click on the check mark ("notify me right away")for Announcements.)

Moreover, I may send e-mails with important information directly to you. I will use the e-mail given to me by the registrar and set up automatically in Canvas. (If that is not your preferred address, please make sure to forward your university e-mail to it!)

All three (notifications from Piazza, notifications from Canvas and e-mails) are official communications for this course and it's your responsibility to check them often!



Please, post all comments and suggestions regarding the course using Piazza. Usually these should be posted as Notes and put in the Feedback folder/label (and add other labels if relevant). These can be posted anonymously (or not), just make sure to check the appropriate option. Others students and myself will be able to respond and comment. If you prefer to keep the conversation private (between us), you can send me an e-mail, but then, of course, it won't be anonymous.


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Legal Issues


All students should be familiar and maintain their Academic Integrity: from Hilltopics, pg. 46:

Academic Integrity

The university expects that all academic work will provide an honest reflection of the knowledge and abilities of both students and faculty. Cheating, plagiarism, fabrication of data, providing unauthorized help, and other acts of academic dishonesty are abhorrent to the purposes for which the university exists. In support of its commitment to academic integrity, the university has adopted an Honor Statement.

All students should follow the Honor Statement: from Hilltopics, pg. 16:

Honor Statement

"An essential feature of The University of Tennessee is a commitment to maintaining an atmosphere of intellectual integrity and academic honesty. As a student of the University, I pledge that I will neither knowingly give nor receive any inappropriate assistance in academic work, thus affirming my own personal commitment to honor and integrity."

You should also be familiar with the Classroom Behavior Expectations.

We are in a honor system in this course!



Students with disabilities that need special accommodations should contact the Office of Disability Services and bring me the appropriate letter/forms.


Sexual Harassment and Discrimination

For Sexual Harassment, Sexual Assault and Discrimination information, please visit the Office of Equity and Diversity.


Campus Syllabus

Please, see also the Campus Syllabus.


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This (LaTeX) is not necessary to our course! The only benefit for us is to help you post messages in Piazza with math symbols and type your homework. But, if you intend to write a thesis (or some math) at some point, might as well learn it now. But feel free to skip the rest of this section.

LaTeX is the most used software to produce mathematics texts. It is quite powerful and the final result is, when properly used, outstanding! Virtually all professional math text you will ever see is done with LaTeX, or one of its variants.

LaTeX is available for all platforms and freely available.

The problem is that it has a steep learning curve at first, but after the first difficulties are overcome, it is not bad at all.

One of the first difficulties one encounters is that it is not WYSIWYG ("what you see is what you get"). It resembles a programming language: you first type some code and then this code is processed to produce a nice document (a non-editable PDF file, for example). Thus, one has to learn how to "code" in LaTeX, but this brings many benefits.

I recommend that anyone with any serious interest in producing math texts to learn it! On the other hand, I don't expect all of you to do so. But note that there are processors that can make it "easier" to create LaTeX documents, by making it "point-and-click" and (somewhat) WYSIWYG.

Here are some that you can use online (no need to install anything and files are available online, but need to register):

If you want to install LaTeX in your computer (so that you don't need an Internet connection), check here.

A few resources:


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HW1 - Due on 02/15:

Chapter 1: Choose 2 or 3. Suggestions: 1.14 (there is a typo in the second edition; the equation should be $u^3+v^3=u+v+1$), 1.18 (skip (d)), 1.19, 1.20. (You don't have to do those! Just see what you find interesting.)


HW2 - Due on 02/29:

Chapter 2: Choose 2 or 3. Suggestions: I think everyone should do a few of the parts of 2.12. Also, 2.2(b), 2.4, 2.10, 2.11, 2.13(c) and (d).


HW3 - Due on 04/12:

Chapter 3: Choose 2 or 3. Suggestions: I think everyone should do a few of the parts of 3.9. Also, 3.5, 3.7(a) and (b), 3.8. It seems that 3.10 and after are for singular curves, which we skipped in class, so you might want to skip it too.


If it is already Friday afternoon and there still is a "More to come" after the HW assignment due on the coming week, write me an e-mail at, and I'll update it and let you know.

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