 So this talk will be about the vague conjectures for varieties of the finite fields and in particular I'll be talking about how Andre they came up with the vague conjectures. This is what he did in his paper, number of solutions of equations in finite fields. So this talk will be a sort of overview of his paper. First of all, we just recall that we have the zeta function of curve can be defined like this. So it's sum of n m q minus m s over m. Where m m is the number of solutions, the number of points on the curve over the finite field of order q to the m. So the zeta function kind of encodes the number of solutions to some equation over various finite fields. And what they did is he noted that this definition makes perfect sense not just for curves which he had been studying but for high dimensional varieties. And he calculated the zeta function explicitly for a firmer hyper surface. So this is a hyper surface defined by the equation a one x one r one plus a and x and the r n zero. So these firmer hyper surfaces are the almost the simplest non trivial examples of high dimensional varieties so I mean obviously right project space is a high dimensional variety but that turns out to be fairly easy and the firmer hyper surfaces are the about the simplest examples of non rational high dimensional varieties so they're a they're a traditional test case. And what they did is he calculated the number of solutions to this curve over various finite fields and what we're going to do is just do a slightly simplified special case of vase calculation which will give the idea. So in particular we're going to take q to be p to be prime. And we're going to take m to be one. So we're just looking at finite fields of prime order the case of prime power order is similar but does have a few extra complications. So. So how did they calculate the number of solutions, well, the number of solutions to this is to f one x one up to x n is common to zero mod p is given by one over p times the sum over all x one up to x n mod p. The sum over x mod p of zeta to the x times f of x one up to x and this zeta is not the zeta function but a root of unity so zeta p equals one zeta is complex so it might be something like e to the two pi i over p. That's not true. Well, and if zeta is a p through to one, then the sum over all x of zeta to the k x is equal to P, if K is connected to not mod P, and not, if K is not connected to not mod P. So you see this sum here. And is equal to P, if f x one up to x n equals not and not otherwise. So, so this sum here just counts P for each solution of this equation and so we divide by P and to get the number of solutions. And now we can rewrite this as follows, first of all, we can take all the terms to the x equal not which give us a term P to the n minus one. So we sum over all x one up to x n mod P and we sum over all x of not equal to zero mod P, zeta to the x times x one up to x n. So it should be one over P there. This is the sort of main term. It's the sort of expected average number of solutions to this curve. And then, and, and this term here is a sort of small error. It's not really an error but it's the difference between the, what we would guess would be roughly the number of terms and the exact number of terms. So what we need to do is to find a way of evaluating this sum here. And first of all, we can simplify it a bit by writing it as a product. What we want to do is to estimate sums like sum over x not equal zero sum over all x one up to x and zeta to the x times a one x one to the r one plus a n x n to the r n. Now I've swapped the sum over x and sum over all the x eyes. But this thing here, we can write it as a product. I equals one to n. We just sum over all the x i of zeta to the x a x i to the pi. So this reduces to evaluating sums of this form and we notice that x a is not equal to zero because we're summing over none zero x and a is none zero. So what we need to do is to estimate sums of the form sum over y from zeta to the a y to the r because and that's what each of these, these terms is. And this sum here is equal to sum over all z was zeta to the a z times. The number of solutions. Of white the R equals Z. Obviously. And the number of solutions is given as follows. First of all, it's P if Z equals zero. It's our P minus one, if Z is not zero, and Z is an R power. And it's equal to zero. If Z is not equal to zero and Z is not R power. And we're going to set this term here equal to D, because we're going to use it in the moment. And so to evaluate this, we now want to recall something about Dirichlet characters. So recall from number theory that a Dirichlet character is just a homomorphism from Z modulo P Z star the multiplicative group so this is all the P minus one to see star. And this is just a cyclic group of all the P minus one so there are P minus one Dirichlet characters mod P. And the Dirichlet characters form a base for the functions on Z modulo P Z star that they form an orthogonal base for a natural inner product. In particular, one function we're going to look at is just the function giving the number of solutions to why the R equals Z. And we find the number of solutions to why the R equals Z is just a sum of D equals P minus one are Dirichlet characters. In fact, it's just those of order dividing the number D. That's fairly easy to check. So from this we find that the sum over why of zeta to the a wide to the R which is the sum we wanted to evaluate is a sum of terms of the form. We have a z of chi of Z times zeta to the a Z where chi of Z is a Dirichlet character chi is a traditional letter for Dirichlet characters. So, what we've done is we've reduced the problem of working out the number of solutions of the equation over finite fields the problem of working out these expressions here. So, let's have a look at these. Well, so what we have is a sort of sum over Z of. This is chi of Z times zeta to the Z where you remember chi is a Dirichlet character. The Dirichlet character is just a homomorphism from Fp star to the nonzero complex numbers. And this is just a homomorphism from the additive group of Fp to the nonzero complex numbers. And these things are called Gauss sums. And what they really are they're really just a sort of variation of a gamma function over a finite field. So if you see this, let's write the formula for the gamma function let's see interval from zero to infinity of T to the s minus one times e to the minus T times dt. e to the minus T is just a homomorphism from the reals to the nonzero complex numbers, and the T to the s minus one is just a homomorphism from the nonzero real numbers to the complex numbers so that the Dirichlet character sort of is an analog of just a power of T over the reals and the power of a root of unity is the analog of e to the minus T. And pretty much any formula involved in the gamma function turns out to have an analog for these Gauss sums. In particular, there's a famous formula for the gamma function if this is gamma of s, so there's a gamma of s times gamma one minus s is equal to pi over sine pi s. And we're going to find the analog of this formula for Gauss sums, which will tell us the absolute value of the Gauss sum. So, what's the absolute value of the sum of chi z times zeta to the az. Well we just take this sum over z times chi of z zeta to the az and multiply it by its complex conjugate so let's have a sum over y zeta bar y zeta to the minus a y. So we're going to take a not equal to zero of course, we're also going to take chi, not the trivial character so there's a trivial character which is one everywhere, and we're going to assume it's not trivial. Furthermore, we can simplify it a bit, we can just set a equal one by changing zeta to a different root of unity if we want. So this is now equal to sum over all y and z of chi z y to the minus one, because we notice that chi of z chi of y bar is equal to the chi of z y minus one, and we have to multiply that by zeta to the z minus y. And now we change z to z times y, and we find this is a sum over y and z of chi of z times zeta to the z minus one times y. And incidentally if you've seen a proof that gamma of s times gamma one minus s is pi over sine pi s, the proof is formally very similar to this proof here. Well anyway we can now evaluate this bit here, because we're just summing this over or non zero y, and this is equal to minus one if z is not equal to one and equal to p minus one if z equals one. And from this we can easily evaluate this sum here, and it becomes p minus sum over z not equal to zero of chi of z, which is equal to p if chi is not the trivial character. So we need to use the fact that chi is not the trivial character to say that this sum here is zero. So this gives the absolute value of the Gaussian sum, the absolute value is just the square root of p. Now we can use this to estimate the number of solutions of our equations and so the number of solutions of a one x one to the r one plus plus a n x n r n is not is now p to the n minus one plus some sum of terms. So these is a is a product of n Gauss sums. So each of these terms has absolute value q to the n over two. Guess that should be p to the n over two. So this is equal to p to the n minus one plus an error where the error has absolute value less than or equal to some constant times p to the n over two and you can you can figure out an estimate for this constant it's something like product of the di minus one. In particular, suppose n is great for three and p is large, whatever large means it means it must be significantly bigger than this constant, then the number of solutions is at least one so there are none trivial solutions. The number of solutions is now greater than one in this obvious solution where all the exercise is zero. So they did this calculation, much more precisely and actually worked out. He did everything rather more explicitly than the rather vague survey we've done here, and in particular what he did was he looked at the following Fermat hyper surface. And what I'm doing here is I'm making all the exponents the same. And this is now homogenous a degree R. So it is a variety in n dimensional project space. And it has a zeta function log of zeta. So t is sum of n m times t to the n over n where n m is the number of solutions over the finite field of order q to the n. So they did the case where n is bigger than one which is slightly more complicated. And he found the following formula for the zeta function. The product of one minus alpha i t to the plus or minus one depending on whether the dimensions odd or even divide by one minus t one minus p t, all the way up to one minus p to the n t. And the, these exponents here are given by the product of Gauss sums. So the absolute value p to the n over two. So, this is a special case of the formula they conjectured for all varieties, where you can get to the zeta function was of the following form p one of t times p three of t and sometimes p to the minus one of t divided by p naught of t p to n of t. And so the special case of thermo hypersurfaces all these polynomials on the denominator have this particularly simple form and the numerator, only one of these polynomials is none zero. So, they then conjectured that not only did the zeta function of the following form but all the roots of the p is have absolute value p to the minus i over two. So this is this is the Riemann hypothesis part of the very conjectures. So what base calculation of this amounts to is proving the Riemann hypothesis for the special case of firma hypersurfaces. They also pointed out several other rather striking properties of this function, first of all, you can write the degree of these polynomials as appears to be the Betty number of the corresponding complex variety. So, for example, the degree of this factor here will be the middle Betty number of the firm hyper surface. And they also conjectured things like the function equation for this function that we mentioned in the first lecture. Okay, I think that's all for those paper.