Rutgers Experimental Math Seminar Talk
November 14, 2013 -- 5-5:48pm
Title: "The 290-Theorem and Representing Numbers by Quadratic Forms"

Abstract: This talk will describe several finiteness theorems for quadratic
forms, and progress on the question: "Which positive definite
integer-valued quadratic forms represent all positive integers?". The
answer to this question depends on settling the related question
"Which integers are represented by a given quadratic form?" for
finitely many forms. The answer to this question can involve both
arithmetic and analytic techniques, though only recently with the
use of computers has the analytic approach become practical.

We will describe the theory of quadratic forms as it relates to
answering these questions, its connections with the theory of modular
forms, and give an idea of how one can obtain explicit bounds to
describe which numbers are represented by a given quadratic form.
Comments/Corrections: (Here [minute:second] describes the time stamp referenced)

[5:58] Legendre's result on the sum of 4 squares was proved around 1770 (not in the 1800s), though Jacobi's result on the sum of 4 squares was proved around 1834.

[12:14] "representing something in this set" should be "representing everything in this set'.

[16:57] The escalator construction first appears in William Schneeberger's 1997 Princeton PhD thesis, and was later used by Bhargava to prove the finiteness theorems described in the talk. The escalator form approach is due to some combination of Conway and Schneeberger.

[17:20] As an example of the "something else" one needs to know, one needs to get lucky that no ternary forms could appear as leaves of the escalator tree. If ternary forms do appear, then the analytic theory presented becomes ineffective without assuming there are no Siegel zeros (or the stronger statement of GRH).

[20:55] The form in the genus of x^2 + 55y^2 representing 5 is 5x^2 + 11y^2.

[21:52] There are only finitely many classes of positive-definite quadratic forms with class number one. In the indefinite case this not true.

[22:27] The notion of spinor genus allows one to inderstand what numbers are represented by integer-valued indefinite quadratic forms in n >= 3 variables, however the given example of numbers represented by x^2 - 2y^2 cannot be easily understood in this way. This comment appears in the lower left corner of the transparency.

[23:59] The real local densities are not exactly given by the surface area of the level set ellipsoid, but it is something very similar to this.

[26:28] The stated quadratic form only has four variables, instead of the requisite five for the example, so I should also have said something like "+ 13 v^2" at the end.

[32:23] An Eisenstein series (in our sense of being in the subspace of modular forms orthogonal to all cusp forms) is non-zero at at least one of the cusps, though perhaps not at all of them at once.

[33:09] Actually the Eisenstein series grow more like m^{k-1}, but only when they are non-zero.

[35:30] Again, the lower bound only applies to the non-zero coefficients a_E(m).

[35:55] The number r_Q(m) is always non-negative, not always positive. (The whole point to for us is to understand exactly when it is positive, and when it is zero!)

[36:06] This comment is premature (i.e. doesn't follow from the previous theorem as stated), though it is stated appropriately in the next transparency at [36:16].

[37:10] I should have said what appears on the transparency, where one divides by p^{n-1} with n being the number of variables of the quadratic form (e.g. n=4).

[37:55] Again, this effective lower bound is valid only when a_E(m) is not zero, which follows from the condition stated there that m is locally represented.

[39:48] TO DO: Check this reference about the Ramanujan bound!

[41:18] Here "quadratic form" means positive-definite integer-valued quadratic form in at least four variables.

[43:21] My answer here that the the numerical methods always have bounds that worked is accurate, though with the caveat that for safety/comfort we consistently added one to the numerical bounds, and the exact answers always were within this "+1" tolerance that we used for further checking. Without this tolerance the numerical answers were not always sufficient, so it's very important to be aware of possible roundoff error!

[46:22] There is a subtlety here that in (3) one may need to check a few more primes than those not meeting the bound individually in (1) when assembling the eligible squarefree numbers in (2) which strictly speaking I have glossed over, but it is very minor.

[46:34] I meant that one doesn't want to give this to a graduate student not using a computer. I did actually give this to my graduate student as an exercise to check her understanding! =)

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