Solving Cubic Equations

Benedict Gross and William Stein December 6, 2011

Algebraic equations Algebraic equations Algebraic equations Algebraic equations Differential equations

F′(t) = F(t), F(0) = C has the unique solution F(t) = Ce^{t}

Pythagorean triples

a^{2} + b^{2} = c^{2} has solutions (3, 4, 5), (5, 12, 13), ... There are solutions on a Babylonian tablet from 1800 BCE: [[make some revealing of this stuff]]

(3, 4, 5) (5, 12, 13) (7, 24, 25) (9, 40, 41) (11, 60, 61) (13, 84, 85) (15, 8, 17) (21, 20, 29) (33, 56, 65) (35, 12, 37) (39, 80, 89) (45, 28, 53) (55, 48, 73) (63, 16, 65) (65, 72, 97)

The general solution of a^{2} + b^{2} = c^{2}

a^{2} + b^{2} = c^{2} has general solution a = p^{2} − q^{2}, b = 2pq, c = p^{2} + q^{2}

x = \frac{1 − t^{2}}{1 + t^{2}} y = \frac{2t}{1 + t^{2}}

[[circle]] [[line on circle] [[then labels]] [[then formula for a,b,c]] [[make the labels be abstract]] [[t = p/q]]

Cubic curves

y^{2} − y = x^{3} − x

[[3 parts: first equation; then graph; then line through two solutions giving a third. Get rid of next slide]]

New solutions from old ones

[[nix this slide]] Can turn this into a group law on the set of solutions. Is it finite or infinite?

Large solutions

If the number of solutions is infinite, the solutions become very large. [[put up equation of cube to the right]]

(0, 0)
(1, 0)
(-1, -1)
(2, -3)
(1/4, -5/8)
(6, 14)
(-5/9, 8/27)
(21/25, -69/125)
(-20/49, -435/343)
(161/16, -2065/64)
(116/529, -3612/12167)
(1357/841, 28888/24389)
(-3741/3481, -43355/205379)
(18526/16641, -2616119/2146689)
(8385/98596, -28076979/30959144)
(480106/4225, 332513754/274625)
(-239785/2337841, 331948240/3574558889)
(12551561/13608721, -8280062505/50202571769)
(-59997896/67387681, -641260644409/553185473329)
(683916417/264517696, -18784454671297/4302115807744)
(1849037896/6941055969, -318128427505160/578280195945297)
(51678803961/12925188721, 10663732503571536/1469451780501769)
(-270896443865/384768368209, 66316334575107447/238670664494938073)

Even the simplest solution can be large

y^{2} + y = x^{3} − 5115523309x − 140826120488927

[[add the number of digits]] Numerator of x-coordinate of smallest solution [[this many digits]]:

Denominator:

rank(E) = the number of independent solutions of infinite order

▶ rank(E) = 0 means there are finitely many solutions. ▶ The curve E(a) with equation y(y + 1) = x(x − 1)(x + a)

has rank = 0, 1, 2, 3, 4 for a = 0, 1, 2, 4, 16. ▶ Can the rank be arbitrarily large?

▶ rank(E) = 0 means there are finitely many solutions. ▶ The curve E(a) with equation y(y + 1) = x(x − 1)(x + a)

has rank = 0, 1, 2, 3, 4 for a = 0, 1, 2, 4, 16. ▶ Can the rank be arbitrarily large?

▶ rank(E) = 0 means there are finitely many solutions. ▶ The curve E(a) with equation y(y + 1) = x(x − 1)(x + a)

has rank = 0, 1, 2, 3, 4 for a = 0, 1, 2, 4, 16. ▶ Can the rank be arbitrarily large?

Current record is rank(E) = 28

y^{2} + xy + y = x^{3} − x^{2} − 20067762415575526585033208209338542750930230312178956502x+

344816117950305564670329856903907203748559443593191803612660082962919394 48732243429

Bryan Birch and Peter Swinnerton-Dyer made a prediction for the rank, based on the average number of solutions at each prime number p.

A Prime is a number not divisible by any smaller number:

[[make each of the four appear separately]] 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, …

There are infinitely many primes. The largest explicit prime known is 2^{43112609} − 1 with 12,978,189 digits.

The Prime Number 2^{9689} − 1 =

478220278805461202952839298660005909741497172402236500851334510991837895094266297027892768
611270789458682472098152425631930658505267683408748083442943326479742589324762368833102163
320895484735480579994334130982598901374380618710958104314868081377832153049671560156328262
441404039814320762203627219040859079053720347525610556407157926386787524098557335652265610
854212857732105787905232886503535587361567936365588992571157442015383209175242284304691881
142740066213555930351685370397681268638575037622778794958058208183126172570100349820651232
987267723348951095346937568303703837399969677158578890563911552261340549570718452415821920
822376644205901459333065700972215396237685342377048613857808977562130116781129916640736174
660669780818675796691467124607371290420058840892318638773788767529288695379706698096740605
353012285353903696549022478492464900795489867850331465554647550450168618735486696437455261
412064078294962245202778896213860266593314768769632208950427879162465151931232783175655377
937719452467339581928148666857638401959072017941334958297031939388438881049454604034208753
656362833215207318161430072176937142623851754052084521466531330118355196259184955893849902
534878037671647707393063443684008446825593744345169031599934913766463896897261419901530490
654781905622717122494707073971630095377574344130792050186353223446654564569577433188504497
825014866346737213039209989485214519099823287877248665051301081676990289251871925006694721
570653621624869624056925686555429622155221156042777866254593699880107018616260147647429345
983018365127336346273267588306070141035925482914977433929717368076561095959991130918978823
835013163567266143596921823997719693387439540399662367558052821120713639637085805605116078
177098545257698803233381293927275210194462952749031383555198519709592888523641530178921867
514101454120309619127093436903952209828031766894206132557234964363840305648734929088422378
629288747223121903238528103409182430661894774072726552428489330447486145494207679904173944
716583828167141043583120679050191452732628737033997470720601688256282740427017032260672798
034347932642573009183981307771932245539476396060658821432660315614149074055769805516626304
444758375671151649018119344223685942415184379538933576543212994405485534515585927342456182
514681371472060628778102124092370802149229834963517952727030296297015692768651163505008040
728267425236264469571076976886613730278931360967438271901738550848466337347612084356798306
505955807293511063754424080735066708298723377976887493898358452309563899612061631863439196
711208646438464947096323007272920091258614726799976249670985276950353573392441620265772074
124868359220282898331114083392330243391779797699031142584361935093675448381119440881276338
808420445180491245438388418080094527562666805762895476338464130510775377324708249580453335
571748196502507081973046642282610569751056428979895118219288597635222905389894873761464213
9910911535864505818992696826225754111

What do we mean by a solution of the cubic equation at the prime number p? [[pause]]

There are finitely many solutions A(p)? [[pause]]

[[put equation and p = 23 and p = 71 below these.]] In fact, p + 1 − 2\sqrt{p} < A(p) < p + 1 + 2\sqrt{p}. [[pause]] It is common to write A(p) = p + 1 − a(p) [[pause]] and to define the L-function of E by the infinite product

L(E, s) = ∏_{p}(1 − a(p)p^{−s} + p^{1−2s} )^{−1} = ∑ a(n)n^{−s}

This only makes sense as a function when s > 3/2, where the product converges. [[pause]] [[squish plot]] If we formally set s = 1 in the product however, we get

∏_{p}(1 − a(p)p^{−1} + p^{−1})^{−1} = ∏_{p}p/A(p)

[[pause]] If A(p) is large on average compared with p, this will approach 0. The larger A(p) is on average, the faster it will tend to 0. [[pause]] For the curve Elkies found of rank (at least) 28:

p A(p)  \frac{p}{A(p)} ∏ \frac{p}{A_{p}} 23  33 0.697 0.697 29  40 0.725 0.505 31  40 0.775 0.392 37  49 0.755 0.296 41  52 0.788 0.233 43  56 0.768 0.179 47  60 0.783 0.140 53  63 0.841 0.118

The precise conjecture of Birch and Swinnerton-Dyer

[[pause between 1,2,3]]

1. The function L(E, s) has an analytic continuation to a neighborhood of s = 1.

2. The order of vanishing at s = 1 is equal to the rank of E.

3. This conjecture was refined by John Tate, to give the leading term in the Taylor expansion at s = 1 in terms of other arithmetic invariants of E

L(E, s) ∼ c(E).(s − 1)^{rank(E)} s → 1

Picture of Tate The analytic continuation was proved using the method of Andrew Wiles and Richard Taylor: the function

F(τ) = ∑ a(n)e^{2πinτ}

is a modular form.

[[pause]] Picture of Wiles and Taylor [[I should have a picture in email]] Combining work I did with Don Zagier with work of Victor Kolyvagin, one can show

If L(E, 1) ≠ 0 the rank is zero.

If L(E, 1) = 0 and L′(E, 1) ≠ 0 the rank is one.

[[pause]] Pictures of Zagier and Kolyvagin (and me, if you are giving the talk). [[get in email]] When the order at s = 1 is greater than one, the computer has been a great guide. [[pause]]

A short summary of the evidence for first rank 2 curve: ▶ order of vanishing is as predicted ▶ no ordinary prime up to 48,859 divides the order of group

appearing in leading term (excluding p = 16231). [[pause]] [[change to backside air picture]]

The average rank

Manjul Bhargava has recently made progress on the study of the average rank, for ALL cubic curves with rational coefficients. [[pause]]

Enumerating the curves

▶ Every such curve has an equation of the form y^{2} = x^{3} + Ax + B where A and B are integers. It is unique if no prime p satisfies p^{4} divides A and p^{6} divides B. [[pause]]

▶ Define the height H(E) as the maximum of the positive

integers A^{4} and B^{6}. [[pause]] ▶ For any positive real number X, there are only finitely many

curves with H(E) ≤ X. Call this number N(X). It grows at the same rate as (X)^{1/4}(X)^{1/6} = X^{5/12}.

▶ Define the average rank by the limit as X → ∞ of

\frac{1}{N(X)} ∑_{H(E)≤X}rank(E)

[[pause]] ▶ We suspect that this limit exists, and is equal to 1/2. In

fact, we think that on average half the curves have rank zero and half have rank one. [[pause]]

▶ Bhargava and Shankar have shown why there is an upper bound on the limit, and have obtained a specific upper bound which is less than 1.

Thank you...