which is easy to prove by multiplying both sides by 1−a. Using this identity, and denoting (q + q2) by a, we can rewrite the generating function for the Fibonacci numbers
as
Next, writing 1 − q − q2 as a product of linear factors, we find that
Using the above identity again, with , we find that the coefficient in front of qn in our generating function (which is Fn) is equal to
We therefore obtain a closed formula for the nth Fibonacci number, which is independent of the preceding ones.
Note that the number , which appears in this formula, is the so-called golden ratio. It follows from the above formula that the ratio Fn/Fn−1 tends to the golden ratio as n becomes larger. For more on the golden ratio and the Fibonacci numbers, see Mario Livio, The Golden Ratio, Broadway, 2003.
11. I follow the presentation of this result given in Richard Taylor, Modular arithmetic: driven by inherent beauty and human curiosity, The Letter of the Institute for Advanced Study, Summer 2012, pp. 6–8. I thank Ken Ribet for useful comments. According to André Weil’s book Dirichlet Series and Automorphic Forms, Springer-Verlag, 1971, the cubic equation we are discussing in this chapter was introduced by John Tate, following Robert Fricke.
12. This group is one of the so-called “congruence subgroups” of the group called , which consists of 2 × 2 matrices with integer coefficients and with the determinant 1, that is, the arrays of integers
such that ad − bc = 1. The multiplication of matrices is given by the standard formula
Now, any complex number q inside the unit disc may be written as for some complex number τ whose imaginary part is positive: , where y > 0 (see endnote 12 to Chapter 15). The number q is uniquely determined by τ, and vice versa. Hence, we can describe the action of the group on q by describing the corresponding action on τ. The latter is given by the following formula:
The group (more precisely, its quotient by the two-element subgroup that consists of the identity matrix I and the matrix −I) is the group of symmetries of the disc endowed with a particular non-Euclidean metric, which is called the Poincaré disc model. Our function is a modular form of “weight 2,” which means that it is invariant under the above action of a congruence subgroup of on the disc, if we correct this action by multiplying the function by the factor (cτ + d)2.
See, for example, Henri Darmon, A proof of the full Shimura–Taniyama–Weil conjecture is announced, Notices of the American Mathematical Society, vol. 46, December 1999, pp. 1397–1401. Available online at http://www.ams.org/notices/199911/comm-darmon.pdf
13. This picture was created by Lars Madsen and is published with his permission. I thank Ian Agol for pointing it out to me and a useful discussion.
14. See, for example, Neal Koblitz, Elliptic curve cryptosystems, Mathematics of Computation, vol. 49, 1987, pp. 203–209;
I. Blake, G. Seroussi, and N. Smart, Elliptic Curves in Cryptography, Cambridge University Press, 1999.
15. In general, this is true for all but finitely many primes p. There is also an additional pair of invariants attached to the cubic equation (the so-called conductor) and to the modular form (the so-called level), and these invariants are also preserved under this correspondence. For example, in the case of the cubic equation we have considered, they are both equal to 11. I also note that every modular forms that appears here has zero constant term, the coefficient b1 in front of q is equal to 1, and all other coefficients bn with n > 1 are determined by the bp corresponding to the primes p.
16. Namely, if a,b,c solve the Fermat equation an + bn = cn, where n is an odd prime number, then consider, following Yves Hellegouarch and Gerhard Frey, the cubic equation
Ken Ribet proved (following a suggestion of Frey and some partial results obtained by Jean-Pierre Serre) that this equation cannot satisfy the Shimura–Taniyama–Weil conjecture. Together with the case n = 4 (which was in fact proved by Fermat himself), this implies Fermat’s Last Theorem. Indeed, any integer n > 2 may be written as a product n = mk, where m is either 4 or an odd prime. Therefore the absence of solutions to the Fermat equation for such m implies their absence for all n > 2.
17. Goro Shimura, Yutaka Taniyama and his time. Very personal recollections, Bulletin of London Mathematical Society, vol. 21, 1989, p. 193.
18. Ibid., p. 190.
19. See footnote 1 on pp. 1302–1303 in the following article on the rich history of the conjecture: Serge Lang, Some history of the Shimura–Taniyama conjecture, Notices of the American Mathematical Society, vol. 42, 1995, pp. 1301–1307. Available online at http://www.ams.org/notices/199511/forum.pdf
Chapter 9. Rosetta Stone
1. The Economist, August 20, 1998, p. 70.
2. The pictures of Riemann surfaces in this book were created with Mathematica® software, using the code kindly provided by Stan Wagon. For more detail, see his book: Stan Wagon, Mathematica® in Action: Problem Solving Through Visualization and Computation, Springer-Verlag, 2010.
3. This is not a precise definition, but it gives the right intuition for real numbers. To get a precise definition, we should think of every real number as the limit of a converging sequence of rational numbers (also known as a Cauchy sequence); for example, the truncations of the infinite decimal expansion of yield such a sequence.
4. In order to do this, mark one point on the circle and put the circle on the line, so that this marked point touches point 0 on the line. Then roll the circle to the right until the marked point on the circle again touches the line (this will happen after the circle makes one full turn). This point of contact between the circle and the line will be the point corresponding to π.
5. The geometry of complex numbers (and other numerical systems) is beautifully explained in Barry Mazur, Imagining Numbers, Picador, 2004.
6. More precisely, we get the surface of the donut without one point. This extra point corresponds to the “infinite solution,” when both x and y tend to infinity.
7. To get a Riemann surface of genus g, we should put a polynomial in x of degree 2g + 1 on the right-hand side of the equation.
8. This link between algebra and geometry was a profound insight of René Descartes, first described in La Géométrie, an appendix to his book Discours de la Méthode, published in 1637. Here is what E.T. Bell wrote about Descartes’ method: “Now comes the real power of his method. We start with equations of any desired or suggested degree of complexity and interpret their algebraic and analytic properties geometrically.... Henceforth algebra and analysis are to be our pilots to the uncharted seas of ‘space’ and its ‘geometry.’ ” (E.T. Bell, Men of Mathematics, Touchstone, 1986, p. 54). Note however that Descartes’ method applies to solutions of equations in real numbers, whereas in this chapter we are interested in solutions in finite fields and in complex numbers.
9. For example, we learned in Chapter 8 that the cubic equation y2 + y = x3 − x2 has four solutions modulo 5. So, naively, the corresponding curve over the finite field of 5 elements has four points. But in fact there is a lot more structure because we can also consider solutions with values in various extensions of the finite field of 5 elements; for example, the field obtained by adjoining the solutions of the equation x2 = 2, which we discuss in endnote 8 to Chapter 14. These extended fields have 5n elements for n = 2,3,4,..., and so we get a hierarchy of solutions with values in these finite fields.
The curves corresponding to the cubic equations are called “elliptic curves.”
10. The Bhagavad-Gita, Krishna’s Counsel in Time of War, translated by Barbara Stoler Miller, Bantam Classic, 1986.
It is interesting to note that Weil spent two years in India in the early 1930s and, by his own admission, was influenced by the Hindu religion.
11. See, for example, Noel Sheth, Hindu Avatāra and Christian Incarnation: A comparison, Philosophy East and West, vol. 52, No. 1, pp. 98–125.
12. André Weil, Collected Papers, vol. I, Springer-Verlag, 1979, p. 251 (my translation).
13. Ibid., p. 253. Th
e idea is that given a curve over a finite field, we consider the so-called rational functions on it. These functions are ratios of two polynomials. (Note that such a function has a “pole” – that is, its value is undefined – at each point of the curve where the polynomial appearing in the denominator has a zero.) It turns out that the set of all rational functions on a given curve is analogous in its properties to the set of rational numbers, or a more general number field, like the ones we discussed in Chapter 8.
To explain this more precisely, let’s consider rational functions on Riemann surfaces; the analogy will still be valid. For example, consider the sphere. Using the stereographic projection, we can view the sphere as a union of a point and the complex plane (we can think of the extra point as one representing infinity). Denote by the coordinate on the complex plane. Then each polynomial P(t) with complex coefficients is a function on the plane. These polynomials are the analogues of the integers appearing in number theory. A rational function on the sphere is a ratio of two polynomials P(t)/Q(t) without common factors. These rational functions are the analogues of the rational numbers, which are ratios m/n of integers without common factors. Similarly, rational functions on a more general Riemann surface are analogous to elements of a more general number field.
The power of this analogy lies in the fact that for many results about number fields there will be similar results valid for the rational functions on curves over finite fields, and vice versa. Sometimes, it is easier to spot and/or to prove a particular statement for one of them. Then the analogy would tell us that a similar statement must be true for the other. This has been one of the vehicles used by Weil and other mathematicians to produce new results.
14. Ibid., p. 253. Here I use the translation by Martin H. Krieger in Notices of the American Mathematical Society, vol. 52, 2005, p. 340.
15. There were three Weil conjectures, which were proved by Bernard Dwork, Alexander Grothendieck, and Pierre Deligne.
16. There is a redundancy in this definition. To explain this, consider two paths on the plane shown on the picture below, one solid and one dotted. It is clear that one of them can be continuously deformed into another without breaking. It is reasonable and economical to declare two closed paths that may be deformed into one another in this way as equal. If we do this, we drastically reduce the number of elements in our group.
This rule is in fact similar to the rule we used in the definition of braid groups in Chapter 5. There, too, we declared equal two braids that could be deformed (or “tweaked”) into one another without cutting and sewing the threads.
So we define the fundamental group of our Riemann surface as the group whose elements are closed paths starting and ending at the point P, with the additional requirement that we identify the paths that can be continuously deformed into one another.
Note that if our Riemann surface is connected, which we tacitly assume to be the case throughout, then the choice of the reference point P is inessential: the fundamental groups assigned to different reference points P will be in one-to-one correspondence with each other (more precisely, they will be “isomorphic” to each other).
17. The identity element will be the “constant path.” It never leaves the marked point P. In fact, it is instructive to think of each closed path as a trajectory of a particle, starting and ending at the same point P. The constant path is the trajectory of the particle that just stays at the point P. It is clear that if we add any path to the constant path, in the sense described in the main text, we will get back the original path.
The inverse path to a given path will be the same path, but traversed in the opposite direction. To check that it is indeed the inverse, let us add a path and its inverse. We obtain a new path which traverses the same route twice, but in two opposite directions. We can continuously deform this new “double” path to the constant path. First, we tweak one of the two paths slightly. The resulting path may be contracted to a point, as shown on the pictures below.
18. Alternatively, as we discussed in endnote 10 to Chapter 5, the braid group Bn may be interpreted as the fundamental group of the space of monic polynomials of degree n with n distinct roots. We choose as the reference point P, the polynomial (x−1)(x−2)...(x−n) with the roots 1,2,...,n (these are the “nails” of the braid).
19. To see that the two paths commute with each other, let’s observe that the torus may be obtained by gluing the opposite sides of a square (polygon with 4 vertices). When we glue together two horizontal sides, a1 and a′1, we obtain a cylinder.
Gluing the circles at the opposite ends of the cylinder (which is what the other two vertical sides of the square, a2 and a′2, become after the first gluing), we obtain a torus. Now we see that the sides a1 and a2 become the two independent closed paths on the torus. Note that on the torus all four corners represent the same point, so these two paths become closed – they start and end at the same point P on the torus. Also, a1 = a′1 because we have glued them together, and likewise, a2 = a′2.
On the square, if we take the path a1 and then take the path a2, this will take us from one corner to its opposite. The resulting path is a1 + a2. But we can also go between these corners along a different path: first take a′2 and then a′1, which is the same as a1. The resulting path is a′2 + a′1. After gluing the opposite sides of the square, a′1 becomes a1, and a′2 becomes a2. So a′2 + a′1 = a2 + a1.
Now observe that both a1 + a2 and a2 + a1 can be deformed to the diagonal path, a straight line connecting the two opposite corners, as shown on the picture below (the dashed arrows show how to deform each of the two paths).
This means that the paths a1 + a2 and a2 + a1 give rise to the same element in the fundamental group of the torus. We have proved that
This implies that the fundamental group of the torus has a simple structure: we can express its elements as M·a1+N·a2, where a1 and a2 are two circles on the torus shown on the picture in the main body of the text, and M and N are integers. The addition in the fundamental group coincides with the usual addition of these expressions.
20. The easiest way to describe the fundamental group of the Riemann surface of a positive genus g (that is, with g holes) is again to realize that we can obtain it by gluing the opposite sides of a polygon – but now with 4g vertices. For example, let’s glue together the opposite sides of an octagon (the polygon with eight vertices). There are four pairs of opposite sides in this case, and we identify the sides in each pair. The result of this gluing is more difficult to imagine than in the case of the torus, but it is known that we obtain a Riemann surface of genus two (the surface of a Danish pastry).
This can be used to describe the fundamental group of a general Riemann surface similarly to the way we described the fundamental group of a torus. As in the case of a torus, we construct 2g elements in the fundamental group of the Riemann surface of genus g by taking the paths along 2g consecutive sides of the polygon. (Each of the remaining 2g sides will be identified with one of these.) Let us denote them by a1,a2,..., a2g. They will generate the fundamental group of our Riemann surface, in the sense that any element of this group may be obtained by adding these together, possibly several times. For example, for g = 2 we have the following element: a3 + 2a1 + 3a2 + a3. (But note that we cannot rewrite this as 2a3 + 2a1 + 3a2, because a3 does not commute with a2 and a1, so we cannot move the rightmost a3 to the left.)
In the same way as in the torus case, by expressing the path connecting two opposite corners of our polygon in two different ways, we obtain a relation between them, generalizing the commutativity relation in the torus case:
This actually turns out to be the only relation between these elements, so we obtain a concise description of the fundamental group: it is generated by a1, a2,..., a2g, subject to this relation.
21. To explain this more precisely, consider all rational functions on our Riemann surface, in the sense of endnote 13, above. They are analogous to the rational numbers. The relevant Galois group is defined as the gr
oup of symmetries of a number field obtained by adjoining solutions of polynomial equations such as x2 = 2 to the rational numbers. Likewise, we can adjoin solutions of polynomial equations to rational functions on a Riemann surface X. It turns out that when we do this, we obtain rational functions on another Riemann surface X′, which is a “covering” of X; that is, we have a map X′ → X with finite fibers. In this situation, the Galois group consists of those symmetries of X′ which leave all points of X unchanged. In other words, these symmetries act along the fibers of the map X′ → X.
Now observe that if we have a closed path on the Riemann surface X, starting and ending at a point P on X, we can take each point of X′ in the fiber over P and “follow” it along this path. When we come back, we will in general get a different point in the fiber over P, so we obtain a transformation of this fiber. This is the phenomenon of monodromy, which will be discussed in more detail in Chapter 15. This transformation of the fiber may be traced to an element of the Galois group. Thus, we obtain a link between the fundamental group and the Galois group.
Chapter 10. Being in the Loop
1. The word “special” refers to those orthogonal transformations that preserve orientation – these are precisely the rotations of the sphere. An example of an orthogonal transformation that does not preserve orientation (and hence does not belong to SO(3)) is a reflection with respect to one of the coordinate planes. The group SO(3) is closely related to the group SU(3), which we discussed in Chapter 2 in connection with quarks (the special unitary group of the 3-dimensional space). The group SU(3) is defined analogously to SO(3); we replace the real 3-dimensional space by the complex 3-dimensional space.
2. Yet another way to see that the circle is one-dimensional is to recall that it can be thought of as the set of real solutions of the equation x2 + y2 = 1, as we discussed in Chapter 9. So the circle is the set of points on the plane constrained by one equation. Hence, its dimension is the dimension of the plane, which is two, minus the number of equations, which is one.
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