The bridge between number theory and complex analysis

[Music] in 1916 the mathematician ramanujan looked at the following very strange function which he called delta of q q times 1 minus q times 1 minus q squared times 1 minus q cubed and on and on and then all of this raised to the 24th power for no particular reason he decided to multiply it out and when he did he got this strange sequence of numbers while most people wouldn't have looked twice at this list ramanujan noticed something very strange if you took this number and multiplied it by this number it equaled this number another observation if you wrote out some more coefficients and you took this number times this number it equaled this number in other words the coefficients of this function were multiplicative the nth coefficient times the nth coefficient equals the nmth coefficient whenever n and m were co-prime ramanujan himself didn't know how to prove this and he conjectured it in what is now a very famous 1916 paper what we now know is that ramanujan's identity is reflective of the fact that these are no ordinary numbers they actually contain some very deep meaning in this video we'll see what the true meaning of these numbers is and how they form a bridge between number theory and complex analysis we'll then see how that bridge leads to the proof of firma's last theorem in a bit more detail that's because the function that ramanujan discovered is not alone but actually belongs to a very huge family of functions that we now know are very deep the numbers that appear in their coefficients have an interesting hidden meaning these are functions that we call modular forms this video is an intro to what modular forms are and how they proved for ma's last theorem for the sake of concreteness i'll focus on this one here our story begins in the 1950s with mathematicians martin eichler and goro shimura they looked at the coefficients of this modular form and wanted to understand their true meaning so to speak the first key step was to visualize this function in the complex plane here the variable q is a complex number in the interior of the unit circle in the complex plane to find the true meaning of the coefficients of the modular form they took a curved arc in the circle and integrated the modular form over that curved arc that is they added up the values of this function over the arc that gives you a complex number let's plot that over here on the right now take another arc in the circle and integrate the function over that arc that gives you another complex number let's plot it on the right now do that for more and more arcs in the circle slowly you notice that a pattern starts forming on the right the values on the right form a lattice in the complex plane the next insight is that this lattice has a very deep connection to number theory to see it it might be helpful to look at one dimension lower suppose that you have a one-dimensional lattice just a bunch of evenly spaced dots on the number line how do you relate this lattice to something that lives in the world of number theory well you can do the following thing take a function that repeats in every portion of this lattice so what we've drawn is the sine function it's periodic then set the variable x to equal sine of t and set the variable y to equal cosine of t and notice that they satisfy the following equation x squared plus y squared equals one so we started off with a lattice and ended up with an equation which lives in the world of numbered theory what we'll do now is an analogous process in higher dimensions given this lattice we'll look at a higher dimensional analog of sine which repeats in every square of the lattice the function you're looking at right now is called the wire strass p function set x equal to p of z and y equal to p prime of c then x and y satisfy this rather complicated looking equation so just like before we started off with a lattice and we cooked up something which lives in the world of number theory an algebraic equation now this equation over here is given a fancy name it's called an elliptic curve let's summarize what we've done so far we started off with a modular form from it we cooked up a lattice and from that lattice we produce this equation over here an elliptic curve but we started off the video wanting to know what is the hidden meaning of the coefficients of this modular form the point that we're leading up to is that they are exactly reflected in the number theory of this elliptic curve but how is that so well in the world of number theory a standard question people ask when given an equation is the following how many solutions are there to this equation mod p where p is a prime number for example if p equals 5 then the pair 4 comma 1 is a solution mod p that is if you substitute x equals 4 and y equals 1 you'll see that the two sides are congruent mod 5. similarly one can check with the brute force search that there are four solutions to this equation mod 5 listed here we can do this for every prime number p so for every prime number p we can count the number of solutions to this equation mod p for p equals two there are five solutions for p equals three there are five solutions likewise you can fill this out for all primes p now this list of numbers doesn't look too interesting at the moment here comes the magic instead of considering the number of solutions mod p consider this slightly weirder thing one plus p minus the number of solutions mod p here are the updated numbers the point is these numbers exactly match the coefficients of our modular form for example the number here is negative 2 and the second coefficient of our modular form is negative two the number here is minus one and the third coefficient of the modular form is minus one likewise for all the other numbers [Music] we started off the video as saying that the coefficients of a modular form have a hidden meaning we can now see what this is these coefficients somehow count the number of solutions to an algebraic equation living in an opposite corner of math they have a strange predictive power in other words we have two worlds modular forms and elliptic curves what we've done is to a modular form we've attached to it an elliptic curve but it wasn't until many years later that people dared to suggest that you might be able to go back that maybe every elliptic curve came from a modular form now when you first hear this a reasonable reaction is that it seems very unlikely but in 1967 this feeling was formalized into a conjecture that became known as the taneyama shimura conjecture every elliptic curve over q is modular that is for every elliptic curve over q there is a modular form whose coefficients are essentially the number of solutions mod p to this elliptic curve now the trouble with this conjecture is that nobody knew how to begin attacking it john coates a very well-known number theorist said beautiful though this problem was it seemed impossible to actually prove and then 20 years later the stakes were raised even higher when frey ser and finally ribbit showed that the taneyama shimura conjecture implied firma's last theorem there was a joke at the time that this was reducing one impossible problem to another equally impossible problem but unbeknownst to the rest of the math world somebody had been working in secret to prove the tanayamashimura conjecture for seven years and in september 1994 andrew wiles announced that he could prove enough of the tanya mashumura conjecture that firma's last theorem would then follow specifically you prove the following statement every blank elliptic curve over q is modular the blank over here is an adjective semi-stable and it was known that this statement was enough to prove that formazla's theorem was true slowly slowly in the years after people started removing conditions on the theorem so the tanayamashimura conjecture was then proven for all curves that were semi-stable at two and three then it was proven for all curves satisfying an even weaker technical condition and then in 2001 the full conjecture was proven by bray conrad diamond and taylor and thus a theorem that first seemed impossible to attack just 40 years later was completely solved that's all for this video with that thanks for watching and i'll see you next video you

Channel: Aleph 0

Views: 198,932

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Id: zLEyIT_BCgk

Length: 9min 58sec (598 seconds)

Published: Thu Apr 14 2022