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Math4805407 TOPICS IN MODERN MATH What are numbers Part I AAKirillov Dec 2007 Abstract The aim of this course is to show what meaning has the notion of number in modern mathematics tell about the problems arising in connection with different understanding of numbers and how these problems are being solved Of course I can explain only first steps of corresponding theories For those who want to know more I indicate the appropriate literature Preface The muzhiks near Vyatka lived badly But they did not know it and believed that they live well not worse than the others 7 AKrupin The live water When a school student rst meet mathematics she is told that it is a science which studies numbers and gures Later in a college she learns analytic geometry which express geometric notions using numbers So it seems that numbers is the only object of study in mathematics True if you open a modern mathematical journal and try to read any article it is very probable that you will see no numbers at all Instead au thors speak about sets functions operators groups manifolds categories etc Department of Mathematics The University of Pennsylvania Philadelphia PA 19104 6395 Eemail address kin39llovmathupennedu Nevertheless all these notions in one way or another are based on num bers and the nal result of any mathematical theory usually is expressed by a number So I think it is useful to discuss with math major students the ques tion posed in the title I want to show what meaning can have the term number in modern mathematics speak of some problems arising in this connection and of their solutions I hope this will help novices to orient themselves in the reach beautiful and complicated world of mathematics 1 ThechainNCZCQCRCCCHHCG This chain of subsequent extensions of the notion of number or at least rst 4 6 members of it you must already know The symbols occurring here are now the standard notations for the sets of natural integer rational real or complex numbers quaternions and octonions The latter are also known as octaves and Cayley numbers I want to discuss here the transition from one link of the chain to the next one and show that the ideas used in these transitions are working in other sometimes unexpected and beautiful theories 11 From N to Z and from Z to Q the Grothendieck group the Lie elds and derived categories We pull ourselves to the sky by the shoe laces JiPi Serre Local algebra and multiplicity theoryquot We can add natural numbers but not always to subtract them the in tegers can be multiplied but not always divide The wish to circumvent this inconvenience actually provoked the transition from natural numbers to integers and from integers to rationals Recall how these transitions are made If we want to subtract a natural number m from a natural number n then in case m 2 n the answer can not be a natural number Let us denote it temporally by n e m If we want that in the extended set of numbers the habitual rules were veri ed we have to identify 71 e m with all expressions of the form n k e m k k E N and also with expressions n 7 k e m 7 k 1 g k minm In other words the symbols n1 9 m1 and n2 9 m2 are identi ed if n1 m2 712 m1 Consider now all expressions of the form n e m nm E N We can not only add them componentwise but also subtract according to the rule n19miin29m2n1m2en2m1 1 Eg we have 0 e 0 7 m e n n e m One can check that equivalence classes form an additive group Do it yourself This exercise is for those who only start to study the group theory It is rater easy to establish that the group in questin is isomorphic to Z Indeed for m gt n all symbols of the form m k e n k are identi ed with the natural number m 7 n for m n they are identi ed with zero and for m lt n with the negative number m 7 n The procedure of constructing of multiplicative group Q consisting of non zero rational numbers from the semigroup ZO is completely analo gous Namely we consider the formal symbols m n where m n E and identify m1 n1 with m2 n2 if 7711712 mgnl It is clear that the equivalence class of the symbol m n can be identi ed with the rational number 1 More interesting example Consider the collection of all nite groups P In case when F1 is a normal subgroup in P and F2 is the quotient group FFl it is natural to say that P is divisible by F1 and the quotient is equal to P2 We denote by P the class of all nite groups isomorphic to P De ne a new group Q5 as follows By de nition Q5 is the abelian group generated by all symbols P with the relations Pl Pll lel 2 if F1 is a normal subgroup in P and F2 FFl One can check that any element of Q5 has the form 9n1 lP1l nklPkl 3 where P1 Pk are nite groups Theorem 1 The group Q5 is a free abelian group with countable set of gen erators As generators one can take the symbols P where P is a cyclic group Zp of a prime order p or a simple non abelian nite group So an element 9 6 Q5 can be uniquely written in the form 3 where P belong to the list of groups pointed out in the theorem Exercise 1 Show that in the group 5 the classes Z6 and 53 de ne the same element Z2 Z3 though Z6 and 53 are not isomorphic Exercise 2 Express in terms of generators the following elements of 5 a Zm n is not a prime 7 Sn where Sn is the group of permutations ofn objects c TnFq where TnFq is the group of all inuertible upper triangular matrices of order n with entries from a nite eld with q pk elements p is prime In all cases considered above we construct a group using the same method by introducing new elements negative numbers fractions formal linear combinations etc and by splitting a set into equivalence classes The most general form of this method until recently was the notion of so called Grothendieck group C of a small additive category C see 221 This group by de nition is commutative and generated by equivalence classes of objects of C with relations Al Bl Cl 4 where B is a subobject of A and C is the corresponding quotient object AB For instance the group 5 above is the Grothendieck group of the category of nite groups Exercise 3 Show that for the category A of all abelian groups with nite number of generators the group A is isomorphic to Z One of the most brilliant applications of this construction is the so called K theory where the initial material is the collection of vector bundles over a given smooth manifold The detailed exposition of this young but already very famous theory one can nd in 1 5 A further generalization is possible when the operation in question is non commutative Example 1 Let A denote an algebra with a unit ouer C generated by ele ments p and q with relations pqiqp1 5 1Unfortunately the notion of a category is not in the curriculum of any undergraduate course though in the modern mathematics it plays the role comparable with the notions of a set and a function The initial material about categories is given in 11 12 and more detailed information one can nd in 6 30 9 This algebra has a convenient realization as an algebra of di erential oper ators on the real line with polynomial coef cients The generators have the form at p UT di erentiation q m multiplication by z Exercise 4 Show that A has no zero divisors ie if ab 0 then a 0 or Let us call a right fraction the expression alf1 where a b E A and b 7 O Analogously a left fraction is an expression b la We say that a left fraction c ld is equivalent to a right fraction alf1 if ca db Two left respectfully right fractions we consider as equivalent if they are equivalent to the same right resp left fraction The following remarkable fact takes place Theorem 2 In any equivalence class there are both left and right fractions In any two classes there are left resp right fractions with a common denominator The theorem follows rather easily from the following important Lemma 1 For any two non zero elements ofA there exists a common right resp left multiple Proof of the Theorem Let A denote the subspace in A consisting of all elements which can be written as polynomials of degree n in generators p q Check yourself that dim AW 1n 2 Assume now that that a b are non zero elements of A They belong to Am for some m Consider the spaces2 a A and I A It is clear that both spaces are contained in AWN On the other hand for n big enough we have 2 dim AW n 1n 2 gt gm m 1n m 2 dim AW Therefore a A and I A have a common non zero element which is a desired common right multiple of a and b 2We denote by a AM the set of all elements of the form ax x E A n In general if A and B are some sets and as is an algebraic operation applicable to their elements then the symbol A as B denote the set of all a as b where a E A b E B Exercise 5 Find a common right multiple of p and q Using Theorem 2 we can de ne the quotient skew eld D for the algebra A Namely we cal take the equivalence classes of fractions as elements of D and de ne addition subtraction and division by the rules ac 1 i bc 1 a i bc71 ac 1 bc 1 abil 6 The multiplication by ab 1 can be de ned as the division by the inverse fraction ba l Of course we have to check the correctness of all these def initions ie independence of the result from the choice of representatives and also all the ordinary laws commutativity of addition distributivity of multiplication and associativity of both laws All this traditionally is left to the reader The skew eld D is a very interesting object Many of its properties are still not known We recommend 10 to the interested reader The construction of the skew eld D can be generalized First of all we can start with several pairs of generators pi qi 1 g i g n and relations 29in pm qiqj Mi pm mi 511 7 More essential and more interesting generalization we obtain by consid ering an associative algebra with generators m1 zn and relations of a special form V L mimj 7 mix Zcszk 1 i j g n 8 k1 where cf are some constants The left hand side of 8 is called a commutator of z and mi It is usually denoted by zh xi Exercise 6 Show that in any associative algebra the operation commuta torquot satis es the so called Jacobi identity 967 yl7 Zl 247 Zl7 ml 27 ml yl 0 9 It is worth to mention or recall here that a vector space with a bilinear skew symmetric operation satisfying 9 is called a Lie algebra This name is related to Lie groups which we shall discuss later Come back to our associative algebra We assume that the generators M are linearly independent Then from exercise 6 it follows that the constants cf satisfy the equation V L Zeacs 0 10 9 H for all i j k m In this case the linear span of 1 zn is a Lie algebra L and our associative algebra is denoted by UL and is called a universal enveloping algebra for L It is remarkable that the statements of the lemma and of the theorem 2 remain valid for the algebra UL Therefore for any Lie algebra L a skew eld DL is de ned as a quotient eld of UL The study of skew elds Dn and DL is one of most interesting parts of a new direction in mathematics called non commutative algebraic geometry see 11 In conclusion of this section we propose two simply formulated questions which are still unsolved 1 Does the Fermat equation Xk Yk zk 11 has a non trivial solution XYZ 7 const in the algebras A An UL It is known that in the polynomial algebra CLp q all solutions are trivial for k gt 2 In 10 a non trivial solution of 11 is found for k 3 Exercise 7 Find the general solution to 11 in the algebra Ch 2 Let P Q E A have the property PQ 7 QP 1 Then the map 4p p H P q H Q de nes an endomorphism of the algebra A into itself ls it true that 4p is actually an isomorphism In other words is 4p always invertible Or does A contain a proper subalgebra isomorphic to A The commutative analogue of this problem is also non solved This is so called Jacobian problem The exact formulation is Let P Q 6 Ch y satisfy Q E 34 55 1 12 3y 3y ls it true that the polynomial map 7 9072 H Hay 9072 has a polynomial inverse map In the same circle of ideas is the notion of a derived category which unexpectedly turned out to be a very effective method of solution of many di icult algebraic and geometric problems The idea of construction of a derived category is rather simple and recall simultaneously the construction of the Grothendick group and the construction of the quotient skew eld View to the lack of time place and competence I refer to the book 9 for the further information Answers and hints to the problems 1 In both groups there is a normal subgroup isomorphic to Z3 2 a Zn 2k akZpk if n Hkpgk is the decomposition of n in prime powers b lSZl Z2l7 Sal lzzl Z3l7 l54l 3lZ2l Z3l7 Snl lzzl Anl for n 2 5 Here A is a simple group of order consisting of all even permutations in Sn c If q pk p prime and q 7 1 Hkpgk is the decomposition in prime powers then TnFq mnn 7 1Zp 2k akZpk 3 Take Z as generator of 5 Use the fact that for any n we have ZnZ Zn 4 Introduce the notion of a leading term for elements of A so that the leading term of a E A A 1 is a E Cnba q Then check that M lal bl 539 172quot 27 qkpk pm W k0 7 One of the possibilities X A2 7 B2C Y 2ABC Z A2 B2C where A B C are arbitrary polynomials 12 From Q to R the idea of completion p adic numbers and adeles In a domain without center when center can be any random pointm AiKi Tolstoj Don Juanquot 121 p adic numbers The real numbers are obtained from rational ones by the procedure of com pletion This procedure can be applied to any metric space ie a set where a distance is de ned for any pair of points You can nd a rigorous de ni tion and basic theorems in any textbook in advanced calculus eg KG or Km Instead we ask a seditious question how natural is the ordinary de ni tion of a distance between rational numbers dltT1T2lT1 7 TZ is there an other way to describe the proximity between them It turns out that such a way exists Here is an example Let us choose a prime number p Any rational number r can be uniquely written in the form r pk where k E Z and g is an irreducible fraction whose numerator and denominator are relatively prime to p The quantity p k is called the p adic norm of r and is denoted by Her One can check that the distance 11201772 HM 7 WW 14 has many properties of an ordinary distance 13 For example it satis es the Triangle inequality 112017 T2 S 112017 T3 2 112027 T3 15 In the same time there are differences Eg with respect to the dis tance 14 all triangles are isosceles and moreover the equal sides are not shorter than the third side The metric spaces with this property are called ultrametric The Triangle inequality takes here a stronger form dpT1 T2 max dpT1 T3 dpT2 T3 Exercise 8 Show that in an ultrametric space the following simple criterion takes place A series 221 an converges i an tends to 0 when n gt 00 Another remarkable property of the p adic distance is that all integers form a bounded set of diameter 1 If we apply to this set the completion procedure we get a compact set 017 whose elements are called p adic inte gers They can be conveniently written in the form of in nitedigit numbers in a p adic numerical system Namely every a 6 017 can be uniquely written as aana2a1a0 Ogaigpil 17 It can be understood as a sum of the series 00 a Z akpk 18 k0 lndeed Hakpkll 19 so that terms of the series tend to 0 and the series is convergent By de nition a is the sum Exercise 9 Construct a bicontinuous bijection of O1 to the Cantor set It is especially simple forp 2 and slightly more complicated for generalp The p adic numbers form a ring we can add them subtract and multiply It is also convenient to write a mixed periodic p adic number AAAB in the form AB Exercise 10 Compute the following quantities in 05 a 4 01 b 7 01 c 1 x However the set Op unlike Z has no natural order Hence there are no positive and negative numbers Indeed the set N of natural numbers is dense in 017 for instance 71 limyH00 p 7 1 Nevertheless for p adic numbers we can de ne an analogue of the func tion signum which takes p different values We recall that the ordinary signum 1 if as gt 0 sgnm 0 if z O 71 if 95 lt 0 can be approximated on the segment 71 1 by the function z5 where e is a small rational number i with an odd denominator n to make sense for negative In the p adic situation the role of e i is played by p Theorem 3 t39sgn For any a 6 01 there exists a limit lim aquot for n a 00 19 It is denoted by sgnpa and has the properties a sgnpab sgnpa sgnpb b sgnpa depends only on the digit a0 of number a c sgnpa 0 if a0 0 and is a root of degreepi 1 from 1 if a0 7 0 11 Thus the p adic line has p 7 1 different directions For the proof of the theorem the following result is useful Lemma 2 If0 lt dpa b lt 1 then dpap 12 lt dpa 1 Unlike ordinary integers many p adic integers are invertible Namely if a 6 0p and sgna 7 0 then a 1 is also a p adic integer In particular all rational numbers with denominator relatively prime to p are p adic integers Exercise 11 Show that the number a of the form 17 is rational if and only if the sequence an is euentually periodic ie periodic starting with some place If we apply the completion procedure with respect to the distance 14 not to Z but to Q we get the set of all p adic numbers not necessary integers This set is denoted by Qp lts elements are conveniently written as mixed p adic fractions of the form a an a2a1a0a1 ak 20 In particular every p adic number has the form a p k where a 6 Op The rules of arithmetic operations on p adic numbers are very similar to the rules of operations on usual decimal fractions but with one additional principle all computations one must start with the last digit Here is an example of such computation in 05 123 123 123 101 101 101 2241331 30221124 3123 1230 123000 12300000 2100220423 Actually here some relations between rational numbers are written Can you tell which ones Here is a little more complicated example 47 4444 sgn2 lim 25quot 212 MK 12 Exercise 12 How knowing a p adic form of a rational number r to tell is it positive or negative 122 p adic analysis In the set Qp of p adic numbers all operations of analysis are de ned four arithmetic operations and the limit of a sequence So we can transfer to the p adic case almost all material of an advanced calculus which is studied in the undergraduate school Some theorem are true literally the other need some corrections and some are replaced by completely different or even opposite statements For example for the segment 0 1 the favorite object of real analysis a natural p adic analogue is the set Op of p adic integers This p adic segment as well as the real one is a ball Recall that in any metric space X a ball B or more precisely a ball Bra with a center a and a radius r is de ned as a subset of the form Bra m E X l dz a g r In our case X Qp B Op Here we have r 1 but unlike the usual ball the role of a center can be played by arbitrary point a 6 013 Moreover Op is a compact set hence has all the properties of the seg ment 0 1 which follow from its compactness Exercise 13 Prove that any continuous function on 0 with values in 0 can be wuifurualy n 39 t ber 39 with 53 39 t in 0 Exercise 14 Consider the map 5 Q1 gt R which sends a p adic number a of the form 20 to the real number 5a 2k akp k In other words the p adic form of 5a is obtained from the p adic form ofa by the reflection in the pointquot Show that s is continuous and maps Q1 onto R1 and 0 onto 0 1 You might think that s is bijective and bicontinuous but it is not so The reason is that the p adic form of a real number is not unique Note also that in the case p 2 the restriction of s to 02 is related to the so called Cantor ladder77 or in other terminology devil ladder To those who became interested in p adic analysis I recommend the following problems for the independent study What is a p adic analogue of 3May be AKTolst0j had in mind exactly this 13 a the signature of a quadratic form b exponential and logarithmic functions c the Fourier transform d P function and B function of Euler An additional material you can nd in 11 12 15 123 Adeles The main application of p adic analysis until now were in the number the ory In this language it is convenient to formulate different questions of divisibility and residues modulo an integer Last time however there were many attempts to use the p adic analysis in mathematical physics Some of these attempts based only on the belief that any mathematical construc tion must have a physical meaning and the simpler and more beautiful the construction is the more fundamental it meaning is Other attempts are essentially dd dbsurdum if the usual analysis is not enough let us try the p adic one Finally there is one more important fact for which one can look a physical explanation The point is that the usual real eld R can be united with all p adic elds Q17 in one beautiful object the ring A of adeles An adele a E A is by de nition a sequence a ama2a3a5 ap 21 where do0 E R LP 6 Qp and for almost all p ie for all but a nite number of them LP 6 Op The arithmetic operations and limits for adeles are de ned component wise The invertible adeles are called ideles4 The set of ideles is denoted by AX For an idele a we can de ne its norm by the formula Hall laool Hapllp 22 Z7 This in nite product make sense because almost all factors are equal to 1 Note now that the eld Q of rational numbers can be embedded into AX namely a rational number r can be considered as adele g r TTT where the rst r is considered as a real number the second one as a 2 adic number the third one as a 3 adic number etc The adeles of this form are called principal adeles It is clear that any principal adele is actually an idele Moreover the following is true 4Actually ideles appeared rst and provoked the invention of adeles as additive ideles 14 Exercise 15 Show that HIM1 for allrEQ 23 The map r gt gt I de nes an embedding of Q into A so from now on we shall not distinguish r and 1 Let now M be an algebraic manifold de ned over the eld of rational numbers le a system of algebraic equations with coef cients in Q Then we can consider the sets MK of solutions of this system over any Q algebra K For K R we get an ordinary real algebraic manifold and for K A it is an adelic manifold We come in this way to the so called adelic analysis It is remarkable that the adelic theorems relate together the real and p adic facts For ex ample many elementary and special higher transcendental functions have nice p adic and adelic analogues Here I consider only two examples of these analogues 124 Tamagawa numbers If M is a real algebraic manifold then to any differential form to of top degree on M we can associate a measure a lwl on M Assume that M is de ned over Q and in appropriate local coordinates to has rational coef cients When to is multiplied by a rational number r the measure lwl is multiplied by It turns out that the set MQp of all points of M over Qp also has a canonical measure ap Hpr and the the replacement of omega by r no leads to the multiplication of MI by Her Finally we can de ne an adelic manifold MA and a measure aA cor responding to the initial differential form to But now the replacement of omega by r w does not change the measure aA since the adelic norm of r is 1 Hence the integral HM w M MA lt24 depends only on M and w modulo multiplication on a rational number There is one case when such an equivalence class is naturally de ned suppose that M is an homogeneous manifold with respect to some algebraic group G acting rationally on M When a G invariant differential form of top degree if it exists is uniquely de ned up to constant factor The 15 simplest example is the homogeneous space M GAGQ In this case M7 w is called the Tamagawa number of the group G and is denoted by rG For many classes of groups this number is the product of the real volume of the manifold GRGz and p adic volumes of GOp for all primes p It is astonishing that rG is often a very simple number7 eg 1 Consider in details two particular cases where G is an additive group of the basic eld Q or the unit circle on plane 1 Additive group In this case GK K and the manifold M is AQ The differential form in question is w dm It de nes the ordinary Lebesgue measure a on R and the Haar measure up on Qp7 normalized by the condition aPOp 1 ie the measure of a unit ball is equal to 1 Exercise 16 Show that AQ is in a natural bijection with RZ X H Op p prime We see that in this case TG yawnZ x H uozpop 1 p prime 2 Circle group Here G is an algebraic manifold given by equation 2 y2 1 The group law is inspired by multiplication of complex numbers and is de ned by the formula 951 241W 242 1952 7 2412427 961142 902241 We choose the form to 17 7 as the top degree form on G with rational coef cients Consider the group G of rational points of G Let us suppose rst that z y 7 717 0 and rewrite the initial equation in the form 1 y217x1x7 or 135 ya The common value of last two fractions is a rational number which we denote by r Then we have a system of linear equation for z y yr1x 17ry with the solution 7 1 7 r2 7 2r m71r27 y71r239 When r run through the set Q of all rationals7 z y runs through G with the point 717 0 deleted 25 Actually we can make r run through the set Q Q U 00 and obtain the whole set of solutions including 71 0 In terms of parameter r the group multiplication law looks like 72 26 17 mm 7 1 7 2 Now let us try the same approach to study the group GQP Namely assume that z y 7 71 0 and denote by the common value of fractions 7 17 1y Then we obtain the system of linear equations y1x 17my with determinant 1 A2 For some p the expression 1 A2 never vanishes for E Qp Exercise 17 Prove that forp 2 and for a prime p of the form 4k 71 the equation 1A20 has no solutions in Qp So for this kind of primes we can parametrize the p adic circle by the points AEQPQPUOO 17V 2 1V y1V39 0U But ifp 4k1 the situation is different Recall that the non zero values ofthe function sgnm are roots of degree p71 from 71 When p 41H 1 two of these values are square root from 71 Denote by it the corresponding points in Qp The initial equation can be rewritten as Mim1 So in this case the group GQp is isomorphic to the multiplicative group Q of the eld Qp The corresponding parametrization looks like 1 7 1 7 7 28 z 2 7 y It turns out that the shape of the set GQP the p adic circle depends on the residue p mod 4 Theorem 4 The p adic circle GQP consists of a four disjoint balls of radius ifp 2 b p 71 disjoint balls of radius ifp E1 mod 4 c p 1 disjoint balls of radius ifp E 71 mod 4 So7 we have uolG02 UOlGQpE for p4k71 1 p 29 uolGopL for p4k1 P Finally7 observe that CZ consists of four points 117 07 07 11 Thus7 the set GRGz is a quarter of a unit circle and has the length Collecting all this together we obtain for the Tamagawa number of the group G the value 7139 p 7 1 q 1 we 7 H 7 7 30 2 p q I7 prime 1 prime F4k1 q4k71 Note7 that the both in nite products in this formula are actually divergent 7 the rst goes to 07 the second to 00 But we can make sense of the whole product7 rewriting it in the form of a conditionally convergent series For this end we use the equalities 71 71 19 229 7 qu kZO q kZO n n Zprzwkiw 2k 1 4 p prime q prime kZO kZO kZO p4k1 q4k71 The last equality is the famous result of Leibniz and also the Taylor series for arctan1 The nal result is rG 1 Exercise 18 The wellprepared readers can try to nd the Tamagawa num ber for some more complicated groups The most appropriate examples are the group 5037 R of all real orthogonal matrices of order three and the group SU27 C S U17 H of unit quaternions 18 125 p adic function The classical C function of Riemann is de ned as a sum of the series 8 271757 31 n21 which is convergent for 9N5 gt 1 We shall see in a moment that this function can be analytically extended to the whole complex plane C with the origin deleted Moreover the values of 5 at negative integer points have very interesting arithmetic properties For this we need some elementary facts from the complex analysis The notion of a holomorphic function and its analytic continuation the Cauchy residue theorem and some properties of elementary functions Consider the integral 29 dz I 77 32 s g 571 2 lt gt over the contour C starting at 00 going along real axis from below then along small circle surrounding clockwise the origin and going back along real axis from above to 00 Here the expression 29 is understood as esaog lzl 39argz and 0 argz 27139 on the contour C So the integrand is holomorphic in s and the integral converges for every 5 E C Therefore its value 5 is a holomorphic function on C We compute this integral in two different ways First assuming 9N5 gt 1 and contracting our contour C to the twice passed ray 0 00 we get 5 151752m0 3 d5 Emil s Further since 0 zsile mdz nislxs 0 by the substitution y nz this integral is reduced to the de nition of the F function we obtain 15 17 5W9rs s 72isinwse isrs s for ms gt 1 33 19 This equality shows that 5 can be analytically extended to C1 Second for 9 s lt 0 the value of the integrand on a big circle OR of radius R is 0R9 1 so the integral over CR tends to 0 when R a 00 If we complete the contour C by 03 we can compute the new integral IR5 by the Cauchy residue formula The integrand has poles at the points i2n7ri and we get the result 135 2m 2 27rin9 1 lnllt n0 When R a 00 we get in the limit 5 2m Z27rinsil n0 The sum over positive n gives Z2m2mn9 1 7 27055 r17 s n21 The sum over negative n gives 37ris Z2m72m51 7 707095 2 r17 s n21 So together we have Bwis 2 1 7 s 72min gamsemcu 7 s 34 15 7 270 7 5 Comparing 33 and 34 we get the famous Riemann functional equa tion for 5 27139 9 8 T 2cos Ps We mention some corollaries from this equation First of all replasing in 35 s by 17 s and comparing the two expressions we come to the Euler identity 1 e s 35gt 7r P P 1 7 s sin7rs Second for s 2k k E N we get 7 k 22k717r2k 210 7W 1 mCGiQk 36 20 On the other hand7 it is well known that The value 2 is a rational number To show it7 consider the sequence of functions 19490 n0 62min 27rink 39 37 This series is convergent absolutely for k 2 2 and conditionally for k 1 and z Z The sum is7 evidently a periodic function fz 1 It turns out that on the interval 07 1 this function coinsides with a certain polynomial of degree k namely J WE BWEVM 38 The polynomials Bk are called Bemoulli polynomials They can be uniquely determined by the properties a 3095 1 b kBk1x c folem 0 for k 1 The constant terms Bk0 are denoted by Bk and are called Bemoulli numbers It is easy to see that they are rational and that B2k1 0 except k O From 37 and 38 follows that gm 27ri2kf2k0 727ri2kB2k2kl and7 using 367 we get 17 2k ing2k 39 The following arithmetic properties of Bernoulli numbers were discovered by EKummer Theorem 5 Kummer congruences pri 1 is not a diuisor of k then a HBka S 1 b ifk E m mod p71pN then 1 ipkBkk E 17 pmBmm mod M B Let us de ne the p adic C function by the formula Cpk1pk 1 1 k 40 21 From the Kummer congruences it follows that pas is uniformly contin uous on every set Ma of the form a p71N a 1 2 p7 2 There fore it can be extended to the closure of Ma in Q1 which coincides with Op The culmination of these beautiful theory is the formula of Kubota Leopold which gives an integral presentation gpkO impact for 201 mod p71 where pa is a measure on 017 with support on 0 The details one can nd in 15 Besides number theoretic and probable physical applications the exis tence of elds Qp essentially enlarges the horizon and the intuition of math ematicians For any de nition theorem formula one can ask what is its p adic analogue Sometimes the very possibility of such question can help better understand the situation Here I stop the excursus into p adic analysis and address the interested readers to 15 7 25 see also some exercises in 11 12 Answers and hints to the problems 8 Follows from the inequality Hal 1 a2 aan maxi Haillp 9 Start from the case p 2 10 a 0 b 4 c 34 11 Recall the proof of the analogous property of decimal fractions 12 If r AB where the fragments A and B have the same length then r gt 0 iff A gt B in the usual sense 13 Express the indicator function of a given ball using the p adic signum 14 Follows from the de nition of convergence in Q17 15 Use the multiplicativity of the norm and check the statement for prime numbers and for 71 16 Let A0 denote the open5 subgroup in A consisting of those a for which a 6 017 for all p Show that AA0Q and A QZ 17 Show using the function sgnp and expp that for p 4k 7 1 the multiplicative group Q is isomorphic to Z x Z41 x Op 18 Answers TSO3 R 2 TSU2 C 1 5Actually the fact that AD is open in A de nes the topology on A 22 13 From Q to R the idea of order nonstandaed analysis Look at rst puddle and there you ll nd a sod which surpasses and blacks out all other sodsi Saltykov Shchedrin A story of town77 There is one more way to pass from rational numbers to reals This way does not use the notion of a distance but instead is based on the natural order in Q We de ne a real number as a section 0 in the set Q that is a partition of Q into two parts A and B so that every element of A is less than any element of B If the set A has maximal element an x or the set B has a minimal element 13mm we identify 0 with one of them Actually am and bmin can not exist simultaneously for a given 0 Indeed otherwise the number W does not belong neither A nor B lfA has no maximum and B has no minimum then the section 0 de nes a new number which does not belong to Q By de nition we consider 0 as bigger than any a E A but smaller than any I E B For the sections one can de ne all arithmetic operations and check that the set of all sections forms a eld This eld is by de nition the eld R of real numbers However there is here something to think about Can we go further and consider sections in R as elements of a still bigger eld For example can we introduce an in nitesimal number 6 which is positive but smaller than any fraction i Formally it contradicts to the well known theorem about least upper bound which claims that for any section 0 in R there exists either amax or 13mm If we analyse the proof of this theorem6 we nd out that it is based on the axiom of Archimedes which claims that for any positive real numbers M 8 there exist a natural number N such that NE gt M But if we agree to sacri ce the axiom of Archimedes we can indeed construct many non archimedean elds strictly containing R Till some time these elds were considered as funny examples and the analysis in these elds as a crazy theory without applications But in 1966 A Robinson and AP Bernstein using non standard analysis have solved a dif cult problem of functional analysis existence of a non trivial invariant subspace for a polynomially compact operator in Hilbert space 6Sometimes the existence of a least upper bound is taken as an axiom then the Achimedean property below becomes a theorem 23 This solution soon was translated to the ordinary mathematical language by PHalmos and a more general result was obtained by VlLomonosov But now nobody can say that the non standard analysis has no applications There are many popular introductions to the non standard analysis see 25 and the bibliography there Here I describe only one original approach to the construction of a non archimedean extension of a real eld This approach was invented by John Horton Conway a famous Princeton mathematician It requires as little of prerequisit that a ction book 14 and an article 13 in a journal Quant for school students were written based on this approach Conway himself calls his numbers surreal and we shall call them Conway numbers or C numbers First of all about notations ln arithmetics of C numbers only two digits are used T or up and l or down By de nition C number is any completely ordered word in the alphabet T L7 The cardinality of a word can be arbitrary but already countable words form a very big eld containing all real numbers and many non standard ones We shall see soon that the empty word plays the role of zero so we denote it by 0 There are two order relations on the set of C numbers First uses terms bigger and smaller denoted by gt and lt It is de ned lexicographically we compare two numbers a and 1 digit par digit If all digits coincide a b if the rst non equal digit in a is bigger than in b then a gt b As for digits we agree that Tgt 0 gtl For the second relation Coway uses the terms earlier or later and symbols lt and a By de nition a e b if a is an initial subword of b To de ne arithmetical operations we need the Theorem 6 Basic Lemma Let A and B be two sets of C numbers such thata lt b for anya E A b E B Then a There exist C numbers c which separate A and B ie a lt c lt b for all a E A b E B 17 Among all C numbers c separating A and B there exists a unique earliest number denoted by A B Conway de nes all the arithmetic operations following two principles succession and simplicity The rst principle means that an arithmetic op eration eg addition is de ned not at once for all C numbers at once but 7Recall that a completely ordered set is an ordered set in which any nonempty subset has a minimal element starting with earlier numbers According to the second principle the re sult must be the simplest possible ie the earliest number which does not contradicts to the results already known Example Let us nd the sum 0 0 Since 0 is the earliest number there are no results already known So we can choose the answer among all C numbers The earliest possibility is 0 Thus 0 0 O Certainly this example is curious but for a reader got accustomed to rigorous de nition of analysis it could seem a bit lightheaded Let us give the de nition of addition for a general case For this we introduce some notations We call upper slice of a C number z the set of all C numbers which are bigger and earlier than m Let us denote this set by Analogously we de ne a lower slice of x as the set x of all C numbers which are less and earlier than m Eg if as TLTT then ml T 90l m H 0 And if z T then ml 2 ml 0 Now we de ne the sum of two C numbers by the formula xyltzlyumylgtltlyUmylgtl lt41gt The formula 41 de nes z y under condition that we already know the sums of all earlier summands succession principle and makes it in a simplest way simplicity principle see the Basic Lemma Exercise 19 Let T and l denote C numbers written by n symbols T or Prove the equalities a rquot raw b l rem T if m gt n Tmln 0 if mn W m if m lt n 42 From the exercise we see that the set of C numbers contains a subgroup isomorphic to the group Z Exercise 20 Prove that H TLT Hint Til le Til 0 T TlTTl So the C number plays the role of one half of the number T Further you can check that TTT plays the role of one quarter of T TTTT is one eight of T etc After these simple example one can guess that all nite C numbers form a group isomorphic to the group of all dyadic fractions Moreover the writing of a dyadic fraction r 2 as a C number is nothing but record of searching this number in the following sense We start from 0 E R it is convenient to imaging the real line disposed vertically so that numbers increase from below upwards and are moving to our number r by steps of size 1 Each step is marked in the record by the symbol T or T depending on the direction of a move We continue this way until we reach r if it is integer or overstep it In the last case we keep going in the direction of r but each next step is twice shorter than the previous one As before every step is marked in the record by T or T depending on the direction of a move Eg for a number r 21376 the record of our search is described by the writing down r 3 7 7 i 7 g T i and leads to the C number TTTTTTT The same method can be applied to all real numbers and produce their writing down as in nite C numbers Exercise 21 Show that rational but not dyadic rational numbers correspond to eventually periodic C numbers ie periodic stating with some place For writing down periodic C numbers it is convenient to use the symbol z to denote a period For example the expression T KT denotes the C number TTTTTTT corresponding to a real number All C numbers occuring until now were just fancy written real numbers This new way of writing down though rather transparent is much less convenient than the ordinary decimal or dyadic system The advatage of it can be seen when we pass to non standard numbers which are written down as easy as the standard real numbers A A Consider for example the C numbers w T and 6 T Exercise 22 Prove that w gt n and 0 lt 6 lt 71 for any n E N So to is in nitely big and 6 is in nitely small number Moreover after we de ne a product of two positive C numbers by the formula M m 39yUylgt m ygtultzyigtTT lt43 26 we can check that to 6 T Unfotunately7 this check is rather long and tedious7 since we have to know the products of all earlier numbers But it is very instructive to those who wants to feel free with C numbers It is time now to recall that in the de nition of C numbers the notion of a complete order is used The remakable fact is that all nite sets with given cardinality admit essentially unique complete order Namely7 any two such sets are isomorphic objects in the category COS of completely ordered sets Still more remarkable fact is that the countable completely ordered sets form in nite and even uncountable equivalence classes You can nd more details in textbooks in set theory see also some exercises in 117 12 Observe that all C numbers occuring until now belong to the class of N Here is an example of other type the number T T It could seem that it is the same word as w 1 but these words are ordered differently the rst has a maximal element7 while the rst has not Exercise 23 Prove the equalities 0 T w n by LUZ Here I nish my introduction to non standard analysis The interested readers can make experiments with C numbers or try to read more serious articles see7 eg Answers and hints 1 Straightforward check by induction 3 Compare the deduction of the formula for the some of the in nite decreasing geometric progression 5 a Use induction b Use the formula 43 14 From R to C H and 0 Clifford algebras Dirac equations and the projective plane over the eld F2 Most of ignorant people understand by the occultism the tableturning It is not so Arkadij Averchenkoi Occult scienceslli 141 Complex numbers The role of complex numbers in mathematics is really outstanding First it is a simplest and the only one known to students example of algebraically closed eld It means that any polynomial with complex coef cients has a complex root hence decomposes into linear factors8 Second the complex analysis ie the the theory of complex valued functions of one or several complex variables is a natural way to study analytic functions of real variables Many purely real77 facts about analytic functions can be understood only by studying their extensions into complex domain For example why the Taylor series for sin z and cos x are coverging everywhere on R while the Taylor series of 21 and arctan z are coverging only for lt 1 Or why the antiderivative of 1 7 2 can be found explicitly for 04 but can not for 04 etc Finally the transition from real to complex numbers admits generaliza tions one of which is the theory of Cll ord algebras We assume that complex numbers are known well enough and will speak here about further generalizations 142 Quaternions The inventor of quaternions the famous irish mathematician Sir William Rowan Hamilton has spent many years in attempt to nd multiplication law for 3 vectors which would generalize the multiplication of complex num bers 2 vectors We know now that it is impossible Only when Hamilton dared to pass to 4 vectors he found the solution You can read about this in 25 These new numbers have been named quatemz39ons They form a 4 dimensional vector space H over reals with one real unit 1 and three imag inary units 1 j k satisfying the relations 12j2k2ijk71 44 The legend is that these very relations were carved by Hamilton into the side of the Broom Bridge on the Royal Canal in Dublin on October 16 of 1843 8Unfortunately the algebraic closures of other elds are more complicated For exam ple the algebraic closure of Qp is not complete with respect to natural extension of p adic norm The corresponding completion CF is described in 15 This eld is more and more used in modern number theory but its role is still incomparable with the role of complex numbers The algebraic closure of a nite eld le has rather simple structure it is a union of nite elds qu q pquot see the description in chapter 2 However this eld has no natural topology except discrete one But the algebraic structure of H is better re ected by another system of equations which is almost equivqlent to 44 12j2k271 ijjijkkjkiik0 45 Exercise 24 Show that 45 implies that ijk i1 So the algebras gen erated by i jk satisfying 45 and i jk satisfying 44 are isomorphic and the isomorphism is given by iHi jHj kHik The system 44 in its turn is equivalent to ai bj ck2 4a b2 02 for all a b c e R 46 Hamilton himself wrote a quaternion in the form q 0 114 sz m3k and call mo 6 R the scalar part of q and X 1 mg mg 6 R3 the vector part of q The product of two quaternions is de ned as follows The scalar parts are multiplied as ordinary real numbers The product of a scalar and a vector is also the ordinary product of a real vector by a real number As for the product of two vectors it has the form xyxyxxy 47 Here the rst summand is the so called scalar or dot product X y 90151 95252 95333 and the second summand is a vector product m1 m2 m3 X X y det 241 242 v3 2 j k These two operations are now used far beyond the quaternion theory The scalar or dot product is an essential part in the de nition of Euclidean and Hilbert spaces see chapter 2 The vector product is the rst example of a Lie algebra commutator see the de nition in section 1 It is convenient to realize the elements q E H by 2 x 2 complex matrices of a special form 900 i951 902 i963 72 img 0 7i1 q0X01izj3klt gtlt This correspondence observed all arithmetical operations Note also that the sub eld C C H is realized by matrices of the form 29 143 Clifford algebras The identity 46 suggests the general de nition ofa Cll ord algebra ClV Q related to a vector space V over a eld K and a quadratic form Q on V9 By de nition ClV Q is the algebra generated over K by the unit 1 and the space V with de ning relations 82 628 1 for any 8 e V 48 Exercise 25 Show that the algebras C and H are the cll ord algebras re spectively for K V R 7x2 and K R V R2 Qm y 7352 7 272 Exercise 26 Let K C V C Q is any non degenerate quadrath form on V all such form are equivalent Show that Mat28C for 712k 49 Mat28Ce Mat28C for n2k1 cw c2 2 The Clifford algebras over R are more diverse I am giving here the table of algebras CW Claw QZW where QZW is the quadratic form of the type Qpqz1 mpq m z127x1217 7zgq qp 0 1 2 3 4 5 6 7 0 18 218 182 C2 1112 2112 1114 C8 1 C 182 2182 184 C4 1114 2114 1118 2 11 C2 184 2184 188 C8 1118 2118 3 211 1112 C4 188 2188 1816 C16 1116 4 1112 2112 1114 C8 1816 21816 1832 C32 5 C4 1114 2114 1118 C16 1832 21832 1864 6 188 C8 1118 2118 1116 C32 1864 21864 7 2188 1816 C16 1116 21116 1132 C64 18128 8 1816 21816 1832 C32 1132 21132 1164 C128 In this table we use the short notation for the algebra MatnK and 2Kn for the algebra MatnK 63 MatnK 9Recall that a quadratic form is a map Q V gt K given by the formula 32 2 Where B V X gt K is a symmetric bilinear map Actually B can be restored from Q by the formula Bv1 v2 Qv1 02 QWi 30 The proof of these relations and also the role of Clifford algebras in topology are given in 5 see also 11 and exercises in chapter 1 Until now we considered only the Clifford algebras related to non degenerate quadratic forms The opposite case of a zero quadratic form is also of inter est This algebra is called erterior or Gmssmann algebra We shall speak about it in chapter 2 One of Clifford algebras was used after three quarters of a century since its discovery by the great british physicist PAM Dirac in the quantum electrodynamics The idea of Dirac was very simple but rather crazy He wanted to replace the wave equation10 Df8t278 8 8 f0 50 by some equivalent equation ofthe rst order in time For this Dirac assumed that the operator 50 is a square of some operator of the rst order D V081 l V181 l V2821 l V382239 51 Of course this equality is impossible if coe icients y are the ordinary num bers Too bad for ordinary numbers77 7 said Dirac and de ned a new sort of numbers which he needed Namely suppose that 2 7 2 7 2 7 2 7 7 70 7 7V1 7 7V2 7 7V3 71 and WW 77 7 0 for z 7g 3 52 Then 51 will be satis ed Of course our reader recognizes in 52 the de nition of a real Clifford algebra C13 Mat2llll So the new Dirac numbers are just 2 x 2 quaternionic matrices or 4 x 4 complex matrices of special kind Note that the algebra generated by m k O 1 2 3 is isomorphic to the Clifford algebra 031 Mat4R Therefore the 4 x 4 matrices 39yk can be chosen pure imaginary The famous Dirac equation which describes the elementary particles of Fermi type electrons muons neutrinos has the form k 50 mat mam way wage m0 53 where m is a real number the mass of the particle and 9 is a real 4 vector so called Majomna spinor 10The symbol 2 or tells that the equation in question is the de nition of the part to which the colon is directed In our case it is the de nition of the symbol D 31 144 Octonions 0r Cayley numbers Among real Clifford algebras exactly three are elds or skew elds R C and H The F robenius theorem claims that there are no other associative nite dimensional division algebras over R However if we drop the associa tivity restriction a curious exemple of a division algebra can be constructed It is an algebra O of so called Cayley numbers or octom39ons As a real vector space it is spanned by the ordinary unit 1 and by seven imaginary units 5k 1 g k g 7 with de ning relations 5 71 1 k g 7 55 iekw 54 where the choice of signs and the function k are de ned by the table 2 3 4 5 6 7 3 72 5 74 7 76 70 1 6 77 74 5 71 7O 7 6 75 74 77 70 1 2 3 77 6 71 70 3 72 4 75 72 73 70 1 5 4 73 2 71 70 ammswwwo l l ll 4701 on l a If on the intersection of i th row and j th column we see the number ik it means that 5139 5739 iek This table has remarkable properties which allow to recover it essentially uniquely 1 In each row and in each column all digits from 0 to 7 occur 2 In each row and in each column there are 4 signs and 4 signs 7 3 In each fragment of the type the number of and 7 are ia ib ib ia odd ie there are 3 and 1 7 or 1 and 3 7 We dicuss below the geometric interpretation of the table and now give another realization of 0 An element X E O is written as a pair of quater nions q r the addition is de ned componentwise and multiplication is given by the formula C117 1quot1 0127 F2 011 012 7 1 2 r17 1quot24111 F1412 55 Here the bar denotes the quaternionic conjugation mo X mo 7X 32 The distribution of signs and indices in 54 is related with a beautiful geometric con guration the projective plane over the eld R of two ele ments We recall that the projective plane over a eld K is a collection lP ZK of all 1 dimensional subspaces lines in a 3 dimensional vector space over K Usually a point in lP ZK is given by 3 homogeneous coordinates m0 1 m2 These coordinates can not vanish simultaneously and are de ned up to common factor In our case the only invertible element of K is 1 so the homogeneous coordinates are de ned uniquely So lPZGFg is identi ed with lForigin and consists of seven points A projective line on lPZGFZ is a subset de ned by a linear equation 010950 11901 12902 0 The coef cients a0 a1 a2 in this equation can be considered as homo geneous coordinates of a point a in the dual projective plane lPZlF2 So we have 7 lines and 7 points in lP ZGFg It is easy to understand that every line contains 3 points and every point belongs to 3 lines The multiplica tion table in O is related to the geometry of lPZGFZ in the following sense we can enumerate the points on the projective plane in such a way that e ej ieK iff the points pi pi and pk belong to the same line As for the sign in 54 it can be de ned by the orientation of lines Here by orientation we understand the cyclic order on a line ie a numeration of the points up to cyclis permutation The sign rule have the form e ej ek if the points pi pj pk de ne the orientation of the line in question In conclusion I propose to readers the following subject to think about The projective plane contains projective subspaces of smaller dimensions lines and points Let us consider the subalgebra of 0 generated by units corresponding to a given projective subspace a Show that this subalgebra is isomorphic to C for points and to H for lines b Which algebras if any correspond to projective spaces of bigger dimensions Exercise 27 How many points lines planes etc are in the n dimensional space over the nite eld qu with q pl ele rnentsf2 Hint Introduce the notation 7 qniquilill39l 1 if 1 q Answer there are 7 k dimensional subspaces q 33 References EEE EEEE E Atiyah M Lectures on K theory Berezin FA Method of second quantization Bratelli O7 Robinson DW Operator algebras and quantum statistic mechanics vol 46 No 11 19997 1999 1208 Cartier P Husemoller D Fibre bundles Springer Verlag7 1994 Grothendieck A Gelfand lM7 GraeV Ml7 Pyatetskij Shapiro ll Representation theory and automorphic functions Generalized functions7 Vl7 77Nauka 7 Moscow7 1966 Gelfand lM7 Kirillov AA The structure of the eld related to a split semisimple Lie algebra7 Funct Analysis and App vol 37 No 1719697 7 26 Gelfand Sl7 Manin Yul Methods of homological algebra 1 In troduction to cohomology theory and deriued categories7 77Nauka 7 Moscow7 1988 Dixmier J On the Weyl algebras Kirillov AA Elements of representation theory7 77Nauka 7 Moscow7 19727 1978 English transl Springer7 1976 Kirillov AA7 GVishiani AD Theorems and problems in Functional analysis 77Nauka 7 Moscow7 19797 1988 English transl Springer7 197 Kirillov AA7 Klumova lN7 Sossinski AB Surreal numbers Kvant No 7 1979 Knuth7 DE Surreal Numbers7 Addison Wesley Publishing Com pany Reading7 Massachusetts 1974 Koblitz N p adic numbers p adic analysis and dzeta function Leites DA Introduction to supermanifolds Soviet Math Uspekhi7 vol 35 19807 3 57 34 l17l l18l l H E l20l l21l l D 8 ml l24l Lesmoir Gordon NRood W and Edney R Fractal Geometry7 Icon Books7 UK7 Totem books7 USA7 2000 Mac Lane S Categories for working mathematician7 Graduate Texts in Math7 Vol57 Springer7 New York7 1971 Mandelbrot BThe fractal geometry of Nature7 Freeman7 San Fran cisco7 1982 Manin Yul Gauge elds and complex geometry 77Nauka 7 Moscow7 1984 English transl Reshetikhin NYu7 Tahtadjan LA7 FaddeeV LD Quantization of Lie groups and Lie algebras7 Algebra and Analysis vol 7 No 7 19897 178206 Encyclopaedia of Mathematical Sciences7 VOl l V7 77VlNlTY 7 Moscow7 1977 20077 English translation Springer7 Vladimirov VS7 Volovich lV Introduction to superanalysis Teor i Math Physics7 vol 597 No 17 vol 607 No 2 19847 pages 3 277 169 198 Weil A Basic number theory7 Wikipedia httpenwikipediaorgwikiPortalMathematics 35
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