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This Week's Finds in Mathematical Physics (Week 217)
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no,location=no, scrollbars=yes,resizable=yes,status=no,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>\nAlso available as http://math.ucr.edu/home/baez/week217.html\n\nMay 30, 2005\nThis Week\'s Finds in Mathematical Physics - Week 217\nJohn Baez\n\nLast week I described lots of different zeta functions, but didn\'t say much\nabout what they\'re good for. This week I\'d like to get started on fixing\nthat problem.\n\nPeople have made lots of big conjectures related to zeta functions.\nSo far they\'ve just proved just a few... but it\'s still a big deal.\n\nFor example, Andrew Wiles\' proof of Fermat\'s Last Theorem was just\na tiny spin-off of his work on something much bigger called the\nTaniyama-Shimura conjecture. Now, personally, I think Fermat\'s Last\nTheorem is a ridiculous thing. The last thing I\'d ever want to know is\nwhether this equation:\n\nx^n + y^n = z^n\n\nhas nontrivial integer solutions for n > 2. But the Taniyama-Shimura\nConjecture is really interesting! It\'s all about the connection between\ngeometry, complex analysis and arithmetic, and it ties together some big\nideas in an unexpected way. This is how it usually works in number theory:\ncute but goofy puzzles get solved as a side-effect of deep and interesting\nresults related to zeta functions and L-functions - sort of like how the\npowdered drink "Tang" was invented as a spinoff of going to the moon.\n\nFor a good popular book on Fermat\'s Last Theorem and the Taniyama-Shimura\nConjecture, try:\n\n1) Simon Singh, Fermat\'s Enigma: The Epic Quest to Solve the World\'s\nGreatest Mathematical Problem, Walker, New York, 1997.\n\nDespite the "world\'s greatest mathematical problem" baloney, this book does\na great job of telling the story without drowning the reader in math.\n\nBut you read This Week\'s Finds because you *want* to be drowned in math,\nand I wouldn\'t want to disappoint you. So, let me list a few of the big\nconjectures and theorems related to zeta functions.\n\nHere goes:\n\nA) The Riemann Hypothesis - the zeros of the Riemann zeta function in\nthe critical strip\n\n0 <= Re(s) <= 1\n\nactually lie on the line Re(s) = 1/2.\n\nFirst stated in 1859 by Bernhard Riemann; still open.\n\nThis implies a good estimate on the number of primes less than a given\nnumber, as described in "week216".\n\nB) The Generalized Riemann Hypothesis - the zeros of any Dirichlet L-function\nthat lie in the critical strip actually lie on the line Re(s) = 1/2.\n\nStill open, since the Riemann Hypothesis is a special case.\n\nA "Dirichlet L-function" is a function like this:\n\nL(chi,s) = sum_{n > 0} chi(n)/n^s\n\nwhere chi is any "Dirichlet character", meaning a periodic complex function\non the positive integers such that\n\nchi(nm) = chi(n) chi(m)\n\nIf we take chi = 1 we get back the Riemann zeta function.\n\nDirichlet used these L-functions to prove that there are infinitely many\nprimes equal to k mod n as long as k is relatively prime to n. The\nGeneralized Riemann Hypothesis would give a good estimate on the number\nof such primes less than a given number, just as the Riemann Hypothesis\ndoes for plain old primes.\n\nErich Hecke established the basic properties of Dirichlet L-functions\nin 1936, including a special symmetry called the "functional equation"\nwhich Riemann had already shown for his zeta function. So I bet Hecke\nmust have dreamt of the Generalized Riemann Hypothesis, even if he didn\'t\ndare state it.\n\nC) The Extended Riemann Hypothesis - for any number field, the zeros of its\nzeta function in the critical strip actually lie on the line Re(s) = 1/2.\n\nStill open, since the Riemann Hypothesis is a special case.\n\nI described the zeta functions of number fields in "week216".\nThese are usually called "Dedekind zeta functions". Hecke also\nproved a functional equation for these back in 1936.\n\nD) The Grand Riemann Hypothesis - for any automorphic L-function,\nthe zeros in the critical strip actually lie on the line Re(s) = 1/2.\n\nThis is still open too, since it includes A)-C) as special cases!\n\nI don\'t want to tell you what "automorphic L-functions" are yet.\nFor now, you can just think of them as grand generalizations of both\nDirichlet L-functions and zeta functions of number fields.\n\nE) The Weil Conjectures - The zeros of the zeta function of any smooth\nalgebraic variety over a finite field lie on the line Re(s) = 1/2.\nAlso: such zeta functions are quotients of polynomials, they satisfy a\nfunctional equation, and they can be computed in terms of the topology\nof the corresponding *complex* algebraic varieties.\n\nFirst stated in 1949 by Andre Weil; proof completed by Pierre Deligne\nin 1974 based on much work by Michael Artin, J.-L. Verdier, and especially\nAlexander Grothendieck. Grothendieck invented topos theory as part of\nthe attack on this problem!\n\nF) The Taniyama-Shimura Conjecture - every elliptic curve over the rational\nnumbers is a modular curve. Or, equivalently: every L-function of an\nelliptic curve is the L-function of a modular curve.\n\nThis was first conjectured in 1955 by Yukata Taniyama, who worked on it\nwith Goto Shimura until committing suicide in 1958. Around 1982 Gerhard\nFrey suggested that this conjecture would imply Fermat\'s Last Theorem; this\nwas proved in 1986 by Ken Ribet. In 1995 Andrew Wiles and Richard Taylor\nproved a big enough special case of the Taniyama-Shimura Conjecture to get\nFermat\'s Last Theorem. The full conjecture was shown in 1999 by Breuil,\nConrad, Diamond, and Taylor.\n\nI don\'t want to say what L-functions of curves are yet... but they are\na lot like Dirichlet L-functions.\n\nG) The Langlands Conjectures - Any automorphic representation pi of a\nconnected reductive group G, together with a finite-dimensional representation\nof its L-group, gives an automorphic L-function L(s,pi). Also: these\nL-functions all satisfy functional equations. Furthermore, they depend\nfunctorially on the group G, its L-group, and their representations.\n\nZounds! Don\'t worry if this sounds like complete gobbledygook! I only\nmention it to show how scary math can get. As Stephen Gelbart once wrote:\n\nThe conjectures of Langlands just alluded to amount (roughly)\nto the assertion that the other zeta-functions arising in\nnumber theory are but special realizations of these L(s,pi).\n\nHerein lies the agony as well as the ecstacy of Langlands\'\nprogram. To merely state the conjectures correctly requires\nmuch of the machinery of class field theory, the structure\ntheory of algebraic groups, the representation theory of real\nand p-adic groups, and (at least) the language of algebraic\ngeometry. In other words, though the promised rewards are\ngreat, the initiation process is forbidding.\n\nI hope someday I\'ll understand this stuff well enough to say something more\nhelpful! Lately I\'ve been catching little glimpses of what it\'s about....\n\nBut, right now I think it\'s best if I talk about the "functional equation"\nsatisfied by the Riemann zeta function, since this gives the quickest way\nto see some of the strange things that are going on.\n\nThe Riemann zeta function starts out life as a sum:\n\nzeta(s) = 1^{-s} + 2^{-s} + 3^{-s} + 4^{-s} + ....\n\nThis only converges for Re(s) > 1. It blows up as we approach s = 1,\nsince then we get the series\n\n1/1 + 1/2 + 1/3 + 1/4 + ....\n\nwhich diverges. However, in 1859 Riemann showed that we can analytically\ncontinue the zeta function to the whole complex plane except for this pole\nat s = 1.\n\nHe also showed that the zeta function has an unexpected symmetry:\nits value at any complex number s is closely related to its value at 1-s.\nIt\'s not true that zeta(s) = zeta(1-s), but something similar is true,\nwhere we multiply the zeta function by an extra fudge factor.\n\nTo be precise: if we form the function\n\npi^{-s/2} Gamma(s/2) zeta(s)\n\nthen this function is unchanged by the transformation\n\ns |-> 1 - s\n\nThis symmetry maps the line\n\nRe(s) = 1/2\n\nto itself, and the Riemann Hypothesis says all the zeta zeros in\nthe critical strip actually lie on this magic line.\n\nThis symmetry is called the "functional equation". It\'s the tiny tip of a\npeninsula of a vast and mysterious continent which mathematicians are still\nstruggling to explore. Riemann gave two proofs of this equation. You can\nfind a precise statement and a version of Riemann\'s second proof here:\n\n2) Daniel Bump, Zeta Function, lecture notes on "the functional\nequation" available at http://math.stanford.edu/~bump/zeta.html\nand http://www.maths.ex.ac.uk/~mwatkins/zeta/fnleqn.htm\n\nThis proof is a beautiful application of Fourier analysis. Everyone\nshould learn it, but let me try to sketch the essential idea here.\n\nI will deliberately be VERY rough, and use some simplified nonstandard\ndefinitions, since the precise details have a way of distracting your\neye just as the magician pulls the rabbit out of the hat.\n\nWe start with the function zeta(2s):\n\n1^{-s} + 4^{-s} + 9^{-s} + 25^{-s} + ....\n\nThen we apply a curious thing called the "Mellin transform", which turns\nthis function into\n\nz^{1} + z^{4} + z^{9} + z^{25} + ....\n\nWeird, huh? This is almost the "theta function"\n\ntheta(t) = sum_n exp(pi i n^2 t)\n\nwhere we sum over all integers n. Indeed, it\'s easy to see that\n\n(theta(t) - 1)/2 = z^{1} + z^{4} + z^{9} + z^{25} + ....\n\nwhen\n\nz = exp(pi i t)\n\nThe theta function transforms in a very simple way when we replace\nt by -1/t, as one can show using Fourier analysis.\n\nUnravelling the consequences, this implies that the zeta function\ntransforms in a simple way when we replace s by 1-s. You have to\ngo through the calculation to see precisely how this works... but\nthe basic idea is: a symmetry in the theta function yields a symmetry\nin the zeta function.\n\nHmm, I\'m not sure that explained anything! But I hope at least the\nmystery is more evident now. A bunch of weird tricks, and then *presto* -\nthe functional equation! To make progress on understanding the Riemann\nHypothesis and its descendants, we need to get what\'s going on here.\n\nI feel I *do* get the Mellin transform; I\'ll say more about that later.\nBut for now, note that the theta function transforms in a simple way, not\njust when we do this:\n\nt |-> -1/t\n\nbut also when we do this:\n\nt |-> t + 2\n\nIndeed, it doesn\'t change at all when we add 2 to t, since exp(2 pi i) = 1.\n\nNow, the maps\n\nt |-> -1/t\n\nand\n\nt |-> t + 1\n\ngenerate the group of all maps\n\nat + b\nt |-> --------\nct + d\n\nwhere a,b,c,d form a 2x2 matrix of integers with determinant 1.\nThese maps form a group called PSL(2,Z), or the "modular group".\n\nA function that transforms simply under this group and doesn\'t blow up\nin nasty ways is called a "modular form". In "week197" I gave the precise\ndefinition of what counts as transforming simply and not blowing up in\nnasty ways. I also explained how modular forms are related to elliptic\ncurves and string theory. So, please either reread "week197" or take my\nword for it: modular forms are cool!\n\nThe theta function is almost a modular form, but not quite. It doesn\'t\nblow up in nasty ways. However, it only transforms simply under a subgroup\nof PSL(2,Z), namely that generated by\n\nt |-> -1/t\n\nand\n\nt |-> t + 2\n\nSo, the theta function isn\'t a full-fledged modular form.\nBut since it comes close, we call it an "automorphic form".\n\nIndeed, for any discrete subgroup G of PSL(2,Z), functions that transform\nnicely under G and don\'t blow up in nasty ways are called "automorphic forms"\nfor G. They act a lot like modular forms, and people know vast amounts\nabout them. It\'s the power of automorphic forms that makes number theory\nwhat it is today!\n\nWe can summarize everything so far in this slogan:\n\nTHE FUNCTIONAL EQUATION FOR THE RIEMANN ZETA FUNCTION SAYS\n"THE THETA FUNCTION IS AN AUTOMORPHIC FORM"\n\nBefore you start printing out bumper stickers, I should explain....\n\nThe point of this slogan is this. We *thought* we were interested in\nthe Riemann zeta function for its own sake, or what it could tell us\nabout prime numbers. But with the wisdom of hindsight, the first thing we\nshould do is hit this function with the Mellin transform and repackage all\nits information into an automorphic form - the theta function.\n\nZeta is dead, long live theta!\n\nThe Riemann zeta function is just like all the fancier zeta functions and\nL-functions in this respect. The fact that they satisfy a "functional\nequation" is just another way of saying their Mellin transforms are\nautomorphic forms... and it\'s these automorphic forms that exhibit the\ndeeper aspects of what\'s going on.\n\nNow let me say a little bit about the Mellin transform.\n\nIgnoring various fudge factors, the Mellin transform is basically just\nthe linear map that sends any function of s like this:\n\nn^{-s}\n\nto this function of z:\n\nz^n\n\nIn other words, it basically just turns things upside down, replacing the\nbase by the exponent and vice versa. The minus sign is just a matter of\nconvention; don\'t worry about that too much.\n\nSo, the Mellin transform basically sends any function like this, called a\n"Dirichlet series":\n\na_1 1^{-s} + a_2 2^{-s} + a_3 3^{-s} + a_4 4^{-s} + ....\n\nto this function, called a "Taylor series":\n\na_1 z^1 + a_2 z^2 + a_3 z^3 + a_4 z^4 + ....\n\nNow, why would we want to do this?\n\nThe reason is that multiplying Taylor series is closely related to *addition*\nof natural numbers:\n\nz^n z^m = z^{n+m}\n\nwhile multiplying Dirichlet series is closely related to *multiplication*\nof natural numbers:\n\nn^{-s} m^{-s} = (nm)^{-s}\n\nThe Mellin transform (and its inverse) are how we switch between these two\npleasant setups!\n\nIndeed, it\'s all about algebra - at least at first. We can add natural\nnumbers and multiply them, so N becomes a monoid in two ways. A "monoid",\nrecall, is a set with a binary associative product and unit. So, we have\ntwo closely related monoids:\n\n(N,+,0)\n\nand\n\n(N,x,1)\n\nGiven a monoid, we can form something called its "monoid algebra" by taking\nformal complex linear combinations of monoid elements. We multiply these\nin the obvious way, using the product in our monoid.\n\nIf we take the monoid algebra of (N,+,0), we get the algebra of Taylor\nseries! If we take the monoid algebra of (N,x,1), we get the algebra of\nDirichlet series!\n\n(Actually, this is only true if we allow ourselves to use *infinite* linear\ncombinations of monoid elements in our monoid algebra. So, let\'s do that.\nIf we used finite linear combinations, as people often do, (N,+,0) would give\nus the algebra of polynomials, while (N,x,0) would give us the algebra of\n"Dirichlet polynomials".)\n\nOf course, algebraically we can combine these structures. (N,+,x,0,1) is\na rig, and by taking formal complex linear combinations of natural numbers\nwe get a "rig algebra" with two products: the usual product of Taylor series,\nand the usual product of Dirichlet series. They\'re compatible, too, since\none distributes over the other. They both distribute over addition.\n\nHowever, if we\'re trying to get an algebra of functions on the complex plane,\nwith pointwise multiplication as the product, we need to make up our mind:\neither Taylor series or Dirichlet series! We then need the Mellin transform\nto translate between the two.\n\nSo, what seems to be going on is that people take a puzzle, like\n\n"what is the sum of the squares of the divisors of n?"\n\nor\n\n"how many ideals of order n are there in this number field?"\n\nand they call the answer a_n.\n\nThen they encode this sequence as either a Dirichlet series:\n\na_1 1^{-s} + a_2 2^{-s} + a_3 3^{-s} + a_4 4^{-s} + ....\n\nor a Taylor series:\n\na_1 z^1 + a_2 z^2 + a_3 z^3 + a_4 z^4 + ....\n\nThe first format is nice because it gets along well with multiplication of\nnatural numbers. For example, in our puzzle about ideals, every ideal is\na product of prime ideals, and its norm is the product of the norms of those\nprime ideals... so our Dirichlet series will have an Euler product formula.\n\nThe second format is nice *if* our Taylor series is an automorphic form.\nThis will happen precisely when our Dirichlet series satisfies a functional\nequation.\n\n(For experts: I\'m ignoring some fudge factors involving the gamma function.)\n\nI still need to say more about *which* puzzles give automorphic forms,\nwhat it really means when they *do*. But, not this week! I\'m tired,\nand I bet you are too.\n\nFor now, let me just give some references. There\'s a vast amount of material\non all these subjects, and I\'ve already referred to lots of it. But right now\nI want to focus on stuff that\'s free online, especially stuff that\'s readable\nby anyone with a solid math background - not journal articles for experts, but\nnot fluff, either.\n\nHere\'s some information on the Riemann Hypothesis provided by the Clay\nMathematics Institute, which is offering a million dollars for its solution:\n\n3) Clay Mathematics Institute, Problems of the Millenium:\nthe Riemann Hypothesis, http://www.claymath.org/millennium/\n\nThe official problem description by Enrico Bombieri talks about evidence\nfor the Riemann Hypothesis, including the Weil Conjectures. The article by\nPeter Sarnak describes generalizations leading up to the Grand Riemann\nHypothesis. In particular, he gives a super-rapid introduction to\nautomorphic L-functions.\n\nHere\'s a nice webpage that sketches Wiles and Taylor\'s proof of Fermat\'s last\ntheorem:\n\n4) Charles Daney, The Mathematics of Fermat\'s Last Theorem,\nhttp://www.mbay.net/~cgd/flt/fltmain.htm\n\nI like the quick introductions to "Elliptic curves and elliptic functions",\n"Elliptic curves and modular functions", "Zeta and L-functions", and "Galois\nRepresentations" - they\'re neither too detailed nor too vague, at least for\nme.\n\nHere\'s a nice little intro to the Weil Conjectures:\n\n5) Runar Ile, Introduction to the Weil Conjectures,\nhttp://folk.uio.no/~ile/WeilA4.pdf\n\nJames Milne goes a lot deeper - his course notes on etale cohomology include\na proof of the Weil Conjectures:\n\n6) James Milne, Lectures on Etale Cohomology,\nhttp://www.jmilne.org/math/CourseNotes/math732.html\n\nwhile his course notes on elliptic curves sketch the proof of Fermat\'s Last\nTheorem:\n\n7) James Milne, Elliptic Curves,\nhttp://www.jmilne.org/math/CourseNotes/math679.html\n\nHere\'s a nice history of what I\'ve been calling the Taniyama-Shimura\nConjecture, which explains why some people call it the Taniyama-Shimura-Weil\nconjecture, or other things:\n\n8) Serge Lang, Some history of the Shimura-Taniyama Conjecture,\nAMS Notices 42 (November 1995), 1301-1307. Available at\nhttp://www.ams.org/notices/199511/forum.pdf\n\nHere\'s a quick introduction to the proof of this conjecture, whatever\nit\'s called:\n\n9) Henri Diamond, A proof of the full Shimura-Taniyama-Weil Conjecture\nis announced, AMS Notices 46 (December 1999), 1397-1401. Available\nat http://www.ams.org/notices/199911/comm-darmon.pdf\n\nI won\'t give any references to the Langlands Conjectures, since\nI hope to talk a lot more about those some other time.\n\nAnd, I hope to keep on understanding this stuff better and better!\n\nQuote of the week:\n\n"If I were to awaken after having slept for a thousand years, my\nfirst question would be: Has the Riemann hypothesis been proven?" -\nDavid Hilbert\n\n-----------------------------------------------------------------------\nPrevious issues of "This Week\'s Finds" and other expository articles on\nmathematics and physics, as well as some of my research papers, can be\nobtained at\n\nhttp://math.ucr.edu/home/baez/\n\nFor a table of contents of all the issues of This Week\'s Finds, try\n\nhttp://math.ucr.edu/home/baez/twf.html\n\nA simple jumping-off point to the old issues is available at\n\nhttp://math.ucr.edu/home/baez/twfshort.html\n\nIf you just want the latest issue, go to\n\nhttp://math.ucr.edu/home/baez/this.week.html\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>Also available as http://math.ucr.edu/home/baez/week217.html
May 30, 2005 This Week's Finds in Mathematical Physics - Week 217 John Baez Last week I described lots of different [itex]\zeta[/itex] functions, but didn't say much about what they're good for. This week I'd like to get started on fixing that problem. People have made lots of big conjectures related to [itex]\zeta[/itex] functions. So far they've just proved just a few... but it's still a big deal. For example, Andrew Wiles' proof of Fermat's Last Theorem was just a tiny spin-off of his work on something much bigger called the Taniyama-Shimura conjecture. Now, personally, I think Fermat's Last Theorem is a ridiculous thing. The last thing I'd ever want to know is whether this equation: [tex]x^n + y^n = z^n[/tex] has nontrivial integer solutions for n > 2. But the Taniyama-Shimura Conjecture is really interesting! It's all about the connection between geometry, complex analysis and arithmetic, and it ties together some big ideas in an unexpected way. This is how it usually works in number theory: cute but goofy puzzles get solved as a side-effect of deep and interesting results related to [itex]\zeta[/itex] functions and L-functions - sort of like how the powdered drink "Tang" was invented as a spinoff of going to the moon. For a good popular book on Fermat's Last Theorem and the Taniyama-Shimura Conjecture, try: 1) Simon Singh, Fermat's Enigma: The Epic Quest to Solve the World's Greatest Mathematical Problem, Walker, New York, 1997. Despite the "world's greatest mathematical problem" baloney, this book does a great job of telling the story without drowning the reader in math. But you read This Week's Finds because you *want* to be drowned in math, and I wouldn't want to disappoint you. So, let me list a few of the big conjectures and theorems related to [itex]\zeta[/itex] functions. Here goes: A) The Riemann Hypothesis - the zeros of the Riemann [itex]\zeta[/itex] function in the critical strip [tex]<= Re(s) <= 1[/tex] actually lie on the line Re(s) [itex]= 1/2[/itex]. First stated in 1859 by Bernhard Riemann; still open. This implies a good estimate on the number of primes less than a given number, as described in "week216". B) The Generalized Riemann Hypothesis - the zeros of any Dirichlet L-function that lie in the critical strip actually lie on the line Re(s) [itex]= 1/2[/itex]. Still open, since the Riemann Hypothesis is a special case. A "Dirichlet L-function" is a function like this: [tex]L(\chi,s) = sum_{n > 0} \chi(n)/n^s[/tex] where [itex]\chi[/itex] is any "Dirichlet character", meaning a periodic complex function on the positive integers such that [tex]\chi(nm) = \chi(n) \chi(m)[/tex] If we take [itex]\chi = 1 we[/itex] get back the Riemann [itex]\zeta[/itex] function. Dirichlet used these L-functions to prove that there are infinitely many primes equal to k mod n as long as k is relatively prime to n. The Generalized Riemann Hypothesis would give a good estimate on the number of such primes less than a given number, just as the Riemann Hypothesis does for plain old primes. Erich Hecke established the basic properties of Dirichlet L-functions in 1936, including a special symmetry called the "functional equation" which Riemann had already shown for his [itex]\zeta[/itex] function. So I bet Hecke must have dreamt of the Generalized Riemann Hypothesis, even if he didn't dare state it. C) The Extended Riemann Hypothesis - for any number field, the zeros of its [itex]\zeta[/itex] function in the critical strip actually lie on the line Re(s) [itex]= 1/2[/itex]. Still open, since the Riemann Hypothesis is a special case. I described the [itex]\zeta[/itex] functions of number fields in "week216". These are usually called "Dedekind [itex]\zeta[/itex] functions". Hecke also proved a functional equation for these back in 1936. D) The Grand Riemann Hypothesis - for any automorphic L-function, the zeros in the critical strip actually lie on the line Re(s) [itex]= 1/2[/itex]. This is still open too, since it includes [itex]A)-C)[/itex] as special cases! I don't want to tell you what "automorphic L-functions" are yet. For now, you can just think of them as grand generalizations of both Dirichlet L-functions and [itex]\zeta[/itex] functions of number fields. E) The Weil Conjectures - The zeros of the [itex]\zeta[/itex] function of any smooth algebraic variety over a finite field lie on the line Re(s) [itex]= 1/2[/itex]. Also: such [itex]\zeta[/itex] functions are quotients of polynomials, they satisfy a functional equation, and they can be computed in terms of the topology of the corresponding *complex* algebraic varieties. First stated in 1949 by Andre Weil; proof completed by Pierre Deligne in 1974 based on much work by Michael Artin, J.-L. Verdier, and especially Alexander Grothendieck. Grothendieck invented topos theory as part of the attack on this problem! F) The Taniyama-Shimura Conjecture - every elliptic curve over the rational numbers is a modular curve. Or, equivalently: every L-function of an elliptic curve is the L-function of a modular curve. This was first conjectured in 1955 by Yukata Taniyama, who worked on it with Goto Shimura until committing suicide in 1958. Around 1982 Gerhard Frey suggested that this conjecture would imply Fermat's Last Theorem; this was proved in 1986 by Ken Ribet. In 1995 Andrew Wiles and Richard Taylor proved a big enough special case of the Taniyama-Shimura Conjecture to get Fermat's Last Theorem. The full conjecture was shown in 1999 by Breuil, Conrad, Diamond, and Taylor. I don't want to say what L-functions of curves are yet... but they are a lot like Dirichlet L-functions. G) The Langlands Conjectures - Any automorphic representation \pi of a connected reductive group G, together with a finite-dimensional representation of its L-group, gives an automorphic L-function [itex]L(s,\pi)[/itex]. Also: these L-functions all satisfy functional equations. Furthermore, they depend functorially on the group G, its L-group, and their representations. Zounds! Don't worry if this sounds like complete gobbledygook! I only mention it to show how scary math can get. As Stephen Gelbart once wrote: The conjectures of Langlands just alluded to amount (roughly) to the assertion that the other [itex]\zeta-functions[/itex] arising in number theory are but special realizations of these [itex]L(s,\pi)[/itex]. Herein lies the agony as well as the ecstacy of Langlands' program. To merely state the conjectures correctly requires much of the machinery of class field theory, the structure theory of algebraic groups, the representation theory of real and p-adic groups, and (at least) the language of algebraic geometry. In other words, though the promised rewards are great, the initiation process is forbidding. I hope someday I'll understand this stuff well enough to say something more helpful! Lately I've been catching little glimpses of what it's about.... But, right now I think it's best if I talk about the "functional equation" satisfied by the Riemann [itex]\zeta[/itex] function, since this gives the quickest way to see some of the strange things that are going on. The Riemann [itex]\zeta[/itex] function starts out life as a sum: [tex]\zeta(s) = 1^{-s} + 2^{-s} + 3^{-s} + 4^{-s} + .[/itex]... This only converges for Re(s) > 1. It blows up as we approach [itex]s = 1,[/itex] since then we get the series [itex]1/1 + 1/2 + 1/3 + 1/4 + .[/itex]... which diverges. However, in 1859 Riemann showed that we can analytically continue the [itex]\zeta[/itex] function to the whole complex plane except for this pole at [itex]s = 1[/itex]. He also showed that the [itex]\zeta[/itex] function has an unexpected symmetry: its value at any complex number s is closely related to its value [itex]at 1-s[/itex]. It's not true that [itex]\zeta(s) = \zeta(1-s),[/itex] but something similar is true, where we multiply the [itex]\zeta[/itex] function by an extra fudge factor. To be precise: if we form the function [itex]\pi^{-s/2} \Gamma(s/2) \zeta(s)[/tex] then this function is unchanged by the transformation [tex]s |-> 1 - s[/tex] This symmetry maps the line Re(s) [itex]= 1/2[/itex] to itself, and the Riemann Hypothesis says all the [itex]\zeta[/itex] zeros in the critical strip actually lie on this magic line. This symmetry is called the "functional equation". It's the tiny tip of a peninsula of a vast and mysterious continent which mathematicians are still struggling to explore. Riemann gave two proofs of this equation. You can find a precise statement and a version of Riemann's second proof here: 2) Daniel Bump, [itex]\Zeta[/itex] Function, lecture notes on "the functional equation" available at http://math.stanford.edu/~bump/\zeta.html and http://www.maths.ex.ac.uk/~mwatkins/\zeta/fnleqn.htm This proof is a beautiful application of Fourier analysis. Everyone should learn it, but let me try to sketch the essential idea here. I will deliberately be VERY rough, and use some simplified nonstandard definitions, since the precise details have a way of distracting your eye just as the magician pulls the rabbit out of the hat. We start with the function [itex]\zeta(2s):1^{-s} + 4^{-s} + 9^{-s} + 25^{-s} + .[/itex]... Then we apply a curious thing called the "Mellin transform", which turns this function into [tex]z^{1} + z^{4} + z^{9} + z^{25} + .[/itex]... Weird, huh? This is almost the "[itex]\theta[/itex] function" [itex]\theta(t) = sum_n \exp(\pi i n^2 t)[/tex] where we sum over all integers n. Indeed, it's easy to see that [tex](\theta(t) - 1)/2 = z^{1} + z^{4} + z^{9} + z^{25} + .[/itex]... when [itex]z = \exp(\pi i t)[/tex] The [itex]\theta[/itex] function transforms in a very simple way when we replace t by [itex]-1/t,[/itex] as one can show using Fourier analysis. Unravelling the consequences, this implies that the [itex]\zeta[/itex] function transforms in a simple way when we replace s by 1-s. You have to go through the calculation to see precisely how this works... but the basic idea is: a symmetry in the [itex]\theta[/itex] function yields a symmetry in the [itex]\zeta[/itex] function. Hmm, I'm not sure that explained anything! But I hope at least the mystery is more evident now. A bunch of weird tricks, and then *presto* - the functional equation! To make progress on understanding the Riemann Hypothesis and its descendants, we need to get what's going on here. I feel [itex]I *do*[/itex] get the Mellin transform; I'll say more about that later. But for now, note that the [itex]\theta[/itex] function transforms in a simple way, not just when we do this: [tex]t |-> -1/t[/tex] but also when we do this: [tex]t |-> t + 2[/tex] Indeed, it doesn't change at all when we add 2 to t, since [itex]\exp(2 \pi i) = 1[/itex]. Now, the maps [tex]t |-> -1/t[/tex] and [tex]t |-> t + 1[/tex] generate the group of all maps [tex]at + bt |->[/itex] -------- [itex]ct + d[/tex] where a,b,c,d form a 2x2 matrix of integers with determinant 1. These maps form a group called PSL(2,Z), or the "modular group". A function that transforms simply under this group and doesn't blow up in nasty ways is called a "modular form". In "week197" I gave the precise definition of what counts as transforming simply and not blowing up in nasty ways. I also explained how modular forms are related to elliptic curves and string theory. So, please either reread "week197" or take my word for it: modular forms are cool! The [itex]\theta[/itex] function is almost a modular form, but not quite. It doesn't blow up in nasty ways. However, it only transforms simply under a subgroup of PSL(2,Z), namely that generated by [tex]t |-> -1/t[/tex] and [tex]t |-> t + 2[/tex] So, the [itex]\theta[/itex] function isn't a full-fledged modular form. But since it comes close, we call it an "automorphic form". Indeed, for any discrete subgroup G of PSL(2,Z), functions that transform nicely under G and don't blow up in nasty ways are called "automorphic forms" for G. They act a lot like modular forms, and people know vast amounts about them. It's the power of automorphic forms that makes number theory what it is today! We can summarize everything so far in this slogan: THE FUNCTIONAL EQUATION FOR THE RIEMANN [itex]\ZETA[/itex] FUNCTION SAYS "THE [itex]\THETA[/itex] FUNCTION IS AN AUTOMORPHIC FORM" Before you start printing out bumper stickers, I should explain.... The point of this slogan is this. We *thought* we were interested in the Riemann [itex]\zeta[/itex] function for its own sake, or what it could tell us about prime numbers. But with the wisdom of hindsight, the first thing we should do is hit this function with the Mellin transform and repackage all its information into an automorphic form - the [itex]\theta[/itex] function. [tex]\Zeta[/itex] is dead, long live [itex]\theta![/tex] The Riemann [itex]\zeta[/itex] function is just like all the fancier [itex]\zeta[/itex] functions and L-functions in this respect. The fact that they satisfy a "functional equation" is just another way of saying their Mellin transforms are automorphic forms... and it's these automorphic forms that exhibit the deeper aspects of what's going on. Now let me say a little bit about the Mellin transform. Ignoring various fudge factors, the Mellin transform is basically just the linear map that sends any function of s like this: [tex]n^{-s}[/tex] to this function of z: [tex]z^n[/tex] In other words, it basically just turns things upside down, replacing the base by the exponent and vice versa. The minus sign is just a matter of convention; don't worry about that too much. So, the Mellin transform basically sends any function like this, called a "Dirichlet series": [tex]a_1 1^{-s} + a_2 2^{-s} + a_3 3^{-s} + a_4 4^{-s} + .[/itex]... to this function, called a "Taylor series": [itex]a_1 z^1 + a_2 z^2 + a_3 z^3 + a_4 z^4 + .[/itex]... Now, why would we want to do this? The reason is that multiplying Taylor series is closely related to *addition* of natural numbers: [itex]z^n z^m = z^{n+m}[/tex] while multiplying Dirichlet series is closely related to *multiplication* of natural numbers: [tex]n^{-s} m^{-s} = (nm)^{-s}[/tex] The Mellin transform (and its inverse) are how we switch between these two pleasant setups! Indeed, it's all about algebra [itex]- at[/itex] least at first. We can add natural numbers and multiply them, so N becomes a monoid in two ways. A "monoid", recall, is a set with a binary associative product and unit. So, we have two closely related monoids: [tex](N,+,0)[/tex] and (N,x,1) Given a monoid, we can form something called its "monoid algebra" by taking formal complex linear combinations of monoid elements. We multiply these in the obvious way, using the product in our monoid. If we take the monoid algebra of [itex](N,+,0), we[/itex] get the algebra of Taylor series! If we take the monoid algebra of (N,x,1), we get the algebra of Dirichlet series! (Actually, this is only true if we allow ourselves to use *infinite* linear combinations of monoid elements in our monoid algebra. So, let's do that. If we used finite linear combinations, as people often do, [itex](N,+,0)[/itex] would give us the algebra of polynomials, while (N,x,0) would give us the algebra of "Dirichlet polynomials".) Of course, algebraically we can combine these structures. [itex](N,+,x,0,1)[/itex] is a rig, and by taking formal complex linear combinations of natural numbers we get a "rig algebra" with two products: the usual product of Taylor series, and the usual product of Dirichlet series. They're compatible, too, since one distributes over the other. They both distribute over addition. However, if we're trying to get an algebra of functions on the complex plane, with pointwise multiplication as the product, we need to make up our mind: either Taylor series or Dirichlet series! We then need the Mellin transform to translate between the two. So, what seems to be going on is that people take a puzzle, like "what is the sum of the squares of the divisors of n?" or "how many ideals of order n are there in this number field?" and they call the answer [itex]a_n[/itex]. Then they encode this sequence as either a Dirichlet series: [itex]a_1 1^{-s} + a_2 2^{-s} + a_3 3^{-s} + a_4 4^{-s} + .[/itex]... or a Taylor series: [itex]a_1 z^1 + a_2 z^2 + a_3 z^3 + a_4 z^4 + .[/itex]... The first format is nice because it gets along well with multiplication of natural numbers. For example, in our puzzle about ideals, every ideal is a product of prime ideals, and its norm is the product of the norms of those prime ideals... so our Dirichlet series will have an Euler product formula. The second format is nice *if* our Taylor series is an automorphic form. This will happen precisely when our Dirichlet series satisfies a functional equation. (For experts: I'm ignoring some fudge factors involving the [itex]\gamma[/itex] function.) I still need to say more about *which* puzzles give automorphic forms, what it really means when they *do*. But, not this week! I'm tired, and I bet you are too. For now, let me just give some references. There's a vast amount of material on all these subjects, and I've already referred to lots of it. But right now I want to focus on stuff that's free online, especially stuff that's readable by anyone with a solid math background - not journal articles for experts, but not fluff, either. Here's some information on the Riemann Hypothesis provided by the Clay Mathematics Institute, which is offering a million dollars for its solution: 3) Clay Mathematics Institute, Problems of the Millenium: the Riemann Hypothesis, http://www.claymath.org/millennium/ The official problem description by Enrico Bombieri talks about evidence for the Riemann Hypothesis, including the Weil Conjectures. The article by Peter Sarnak describes generalizations leading up to the Grand Riemann Hypothesis. In particular, he gives a super-rapid introduction to automorphic L-functions. Here's a nice webpage that sketches Wiles and Taylor's proof of Fermat's last theorem: 4) Charles Daney, The Mathematics of Fermat's Last Theorem, http://www.mbay.net/~cgd/flt/fltmain.htm I like the quick introductions to "Elliptic curves and elliptic functions", "Elliptic curves and modular functions", "[itex]\Zeta[/itex] and L-functions", and "Galois Representations" - they're neither too detailed nor too vague, at least for me. Here's a nice little intro to the Weil Conjectures: 5) Runar Ile, Introduction to the Weil Conjectures, http://folk.uio.no/~ile/WeilA4.pdf James Milne goes a lot deeper - his course notes on etale cohomology include a proof of the Weil Conjectures: 6) James Milne, Lectures on Etale Cohomology, http://www.jmilne.org/math/CourseNotes/math732.html while his course notes on elliptic curves sketch the proof of Fermat's Last Theorem: 7) James Milne, Elliptic Curves, http://www.jmilne.org/math/CourseNotes/math679.html Here's a nice history of what I've been calling the Taniyama-Shimura Conjecture, which explains why some people call it the Taniyama-Shimura-Weil conjecture, or other things: 8) Serge Lang, Some history of the Shimura-Taniyama Conjecture, AMS Notices 42 (November 1995), [itex]1301-1307[/itex]. Available at http://www.ams.org/notices/199511/forum.pdf Here's a quick introduction to the proof of this conjecture, whatever it's called: 9) Henri Diamond, A proof of the full Shimura-Taniyama-Weil Conjecture is announced, AMS Notices 46 (December 1999), [itex]1397-1401[/itex]. Available at http://www.ams.org/notices/199911/comm-darmon.pdf I won't give any references to the Langlands Conjectures, since I hope to talk a lot more about those some other time. And, I hope to keep on understanding this stuff better and better! Quote of the week: "If I were to awaken after having slept for a thousand years, my first question would be: Has the Riemann hypothesis been proven?" - David Hilbert ----------------------------------------------------------------------- Previous issues of "This Week's Finds" and other expository articles on mathematics and physics, as well as some of my research papers, can be obtained at http://math.ucr.edu/home/baez/ For a table of contents of all the issues of This Week's Finds, try http://math.ucr.edu/home/baez/twf.html A simple jumping-off point to the old issues is available at http://math.ucr.edu/home/baez/twfshort.html If you just want the latest issue, go to http://math.ucr.edu/home/baez/this.week.html |
| May31-05, 01:27 AM | #2 |
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<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no,location=no, scrollbars=yes,resizable=yes,status=no,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>In article <d7gccq\\$2sk\\$1@glue.ucr.edu>,\nbaez@math.removethis.ucr.andthi s.edu (John Baez) wrote:\n\n> I won\'t give any references to the Langlands Conjectures, since\n> I hope to talk a lot more about those some other time.\n\nSo there\'s this version of Langlands called Geometric Langlands having\nto do with equivalences various moduli spaces and all sorts of fun stuff\nI don\'t understand like "Hecke Eigensheaves". But the reason I bring it\nup is because, apparently, it\'s all just S-duality in disguise. You take\nN=4 SYM, compactify down to two dimensions and all sorts of neat stuff\nhappens.\n\nThere\'s some stuff on this at\n\n<http://www-theory.lbl.gov/~origa/MathPhys/Langlands.html>\n<http://www.math.northwestern.edu/langlands/mtg_prtn_04.htm>\n\nAaron\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>In article <d7gccq$2sk$1@glue.ucr.edu>,
baez@math.removethis.ucr.andthis.edu (John Baez) wrote: > I won't give any references to the Langlands Conjectures, since > I hope to talk a lot more about those some other time. So there's this version of Langlands called Geometric Langlands having to do with equivalences various moduli spaces and all sorts of fun stuff I don't understand like "Hecke Eigensheaves". But the reason I bring it up is because, apparently, it's all just S-duality in disguise. You take N=4 SYM, compactify down to two dimensions and all sorts of neat stuff happens. There's some stuff on this at <http://www-theory.lbl.gov/~origa/Mat...anglands.html> <http://www.math.northwestern.edu/lan...g_prtn_04.htm> Aaron |
| Jun3-05, 01:55 AM | #3 |
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<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no,location=no, scrollbars=yes,resizable=yes,status=no,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>Some corrections, thanks mainly to James Borger and Kevin Buzzard!\n\n>E) The Weil Conjectures - The zeros of the zeta function of any smooth\n>algebraic variety over a finite field lie on the line Re(s) = 1/2.\n\nThis is only true for curves, which is the case Weil actually proved.\nIn general, the zeros have real part n/2 for some odd n, and the poles\nhave real part n/2 for some even n. In each case, the number of them is\nthe dimension of the cohomology group H^n of the complex form of the\nvariety. For curves, H^1 gives the only odd Betti number, so all\nzeros have real part 1/2.\n\n>Also: such zeta functions are quotients of polynomials, they satisfy a\n>functional equation, and they can be computed in terms of the topology\n>of the corresponding *complex* algebraic varieties.\n\nActually just the number of zeros and poles and their real parts are\ndetermined by the topology of the complex form of the variety, by the\ncorrection above.\n\nRegarding the Taniyama-Shimura conjecture:\n\n>This was first conjectured in 1955 by Yukata Taniyama, who worked on it\n>with Goto Shimura until committing suicide in 1958.\n\nIt\'s "Goro" Shimura.\n\n>9) Henri Diamond, A proof of the full Shimura-Taniyama-Weil Conjecture\n>is announced, AMS Notices 46 (December 1999), 1397-1401. Available\n>at http://www.ams.org/notices/199911/comm-darmon.pdf\n\nThat\'s Henri "Darmon", not to be confused with the Fred Diamond who\nhelped prove this conjecture.\n\n>We start with the function zeta(2s):\n>\n>1^{-s} + 4^{-s} + 9^{-s} + 25^{-s} + ....\n>\n>Then we apply a curious thing called the "Mellin transform",\n>which turns this function into\n>\n>z^{1} + z^{4} + z^{9} + z^{25} + ....\n\nGoing this way, it\'s actually the "inverse Mellin transform".\n\nI\'d also like to thank Aaron Bergman for turning me on to some stuff\nabout the geometric Langlands conjecture, which is basically\nthe Langlands conjecture for function fields. This has some connections\nto mathematical physics - see Peter Woit\'s blog:\n\nhttp://www.math.columbia.edu/~woit/blog/archives/000122.html\n\nand\n\nhttp://www.math.uchicago.edu/~arinkin/langlands/\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>Some corrections, thanks mainly to James Borger and Kevin Buzzard!
>E) The Weil Conjectures - The zeros of the [itex]\zeta[/itex] function of any smooth >algebraic variety over a finite field lie on the line Re(s) [itex]= 1/2[/itex]. This is only true for curves, which is the case Weil actually proved. In general, the zeros have real part n/2 for some odd n, and the poles have real part n/2 for some even n. In each case, the number of them is the dimension of the cohomology group [itex]H^n[/itex] of the complex form of the variety. For curves, [itex]H^1[/itex] gives the only odd Betti number, so all zeros have real part 1/2. >Also: such [itex]\zeta[/itex] functions are quotients of polynomials, they satisfy a >functional equation, and they can be computed in terms of the topology >of the corresponding *complex* algebraic varieties. Actually just the number of zeros and poles and their real parts are determined by the topology of the complex form of the variety, by the correction above. Regarding the Taniyama-Shimura conjecture: >This was first conjectured in 1955 by Yukata Taniyama, who worked on it >with Goto Shimura until committing suicide in 1958. It's "Goro" Shimura. >9) Henri Diamond, A proof of the full Shimura-Taniyama-Weil Conjecture >is announced, AMS Notices 46 (December 1999), [itex]1397-1401[/itex]. Available >at http://www.ams.org/notices/199911/comm-darmon.pdf That's Henri "Darmon", not to be confused with the Fred Diamond who helped prove this conjecture. >We start with the function [itex]\zeta(2s):[/itex] > [itex]>1^{-s} + 4^{-s} + 9^{-s} + 25^{-s} + .[/itex]... > >Then we apply a curious thing called the "Mellin transform", >which turns this function into > [itex]>z^{1} + z^{4} + z^{9} + z^{25} + .[/itex]... Going this way, it's actually the "inverse Mellin transform". I'd also like to thank Aaron Bergman for turning me on to some stuff about the geometric Langlands conjecture, which is basically the Langlands conjecture for function fields. This has some connections to mathematical physics - see Peter Woit's blog: http://www.math.columbia.edu/~woit/b...es/000122.html and http://www.math.uchicago.edu/~arinkin/langlands/ |
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