水曜日, 6月 07, 2017

A 1940 Letter of André Weil on Analogy in Mathematics


ゴールドバッハ予想:メモ(&リーマン予想、素数関連)
http://nam-students.blogspot.jp/2014/03/blog-post_24.html
ハドロンの分類図、SU(3)の図
http://nam-students.blogspot.jp/2015/12/su.html?m=0

NAMs出版プロジェクト: A 1940 Letter of André Weil on Analogy in Mathematics

http://nam-students.blogspot.jp/2017/06/a-1940-letter-of-andre-weil-on-analogy.html


ヴェイユは監獄の中から 、有名な哲学者であり 、神や人間の本質を深く考察した妹のシモ ーヌ ・ヴェイユに宛てて 、一通の手紙を書いた 。そこには注目すべき内容が盛られていた 。彼はその手紙に 、自分の目に映る 「数学の大きな絵 」のことを 、やさしい言葉で (哲学者にもわかるように ─ ─というのはほんの冗談 、失礼 ! )語ったのである 。彼はこれにより 、すべての数学者にひとつのお手本を示した 。わたしはときどき 、第一級の数学者を何人か監獄に放り込んで 、ヴェイユに倣って 、自分のアイディアを誰にでもわかる言葉で説明させたらいいのに 、と冗談を言うことがある 。
 ヴェイユがその手紙に綴ったのは 、数学におけるアナロジ ーの役割についてだった 。彼はそれを説明するために 、自分がもっとも関心を寄せていたものを例に挙げた 。それは 、数論と幾何学とのあいだに成り立つアナロジ ーである 。そのアナロジ ーが 、ラングランズ ・プログラムの発展に重要な役割を果たすことになったのだ 。

フランケル『数学の大統一…』より

黒木玄 Gen Kuroki (@genkuroki)
#数学 続き。「ヴェイユのロゼッタストーン」の話もお兄さんのアンドレ・ヴェイユさんから妹のシモーヌ・ヴェイユさんへの有名な手紙(1940)の中に書かれています。その手紙の英訳→ ams.org/notices/200503…


自伝ではこの日付の手紙は言及されていない(2月26日付の手紙の写真が載っている)。妻とのやりとりが主。

A  1940 Letter of André Weil on Analogy in  Mathematics Translated by Martin H. Krieger This  article  is  excerpted  from  the  book  Doing Mathematics (2003)  by  Martin  H.  Krieger.  It  is reprinted  with  permission  from  World Scientific Publishing. For André  Weil,  “having  a  disagreement with  the  French  authorities  on  the  subject of  [his]  military  ‘obligations’  was  the  reason  [he]  spent  February  through  May  [of 1940]  in  a  military  prison.”  When  he  was released, he went into the service. Weil wrote this fourteen-page  letter  to  Simone  Weil,  his  sister,  from Bonne-Nouvelle  Prison  in  Rouen  in  March  1940, sixty-five  years  ago  this  month.  (Keep  in  mind  that the  letter  was  not  written  for  a  mathematician, even  though  Simone  could  not  understand  most  of it.) I  first  heard  of  the  letter  from  a  small  passage translated  in  a  book  by  D.  Reed  (Figures  of  Thought; London:  Routledge,  1995).  At  the  time  I  was  trying to  understand  the  range  of  solutions  to  the  Ising model  in  mathematical  physics,  and  in  going  to Weil’s  letter  I  found  poignant  his  exposition  of  a threefold  analogy  out  of  Riemann  and  Dedekind, one  that  proves  to  organize  a  great  deal  of  disparate material.  Moreover,  I  had  just  begun  to  appreciate the significance of the Langlands Program for my problem.  [See  the  “Notes  Added  in  Proof”  to  Martin  H.  Krieger,  Constitutions  of  Matter:  Mathematically  Modeling  the  Most  Everyday  of  Physical  Phenomena (Chicago:  University  of  Chicago  Press, 1996),  pp.  311–312.]  Eventually,  in  chapter  5  of Doing Mathematics, I worked out the analogy and provided  an  exposition  of  the  Weil  letter.  A  recent Notices article  (“Some  of  what  mathematicians  do”, November  2004,  pp.  1226–1230)  summarizes  the Martin  H.  Krieger  is  professor  of  planning  at  the  University  of  Southern  California.  His  email  address  is krieger@usc.edu. 334 argument  of  that  book,  including  what  I  called  the Dedekind-Weil analogy. The  Weil  letter  is  a  gem,  of  wider  interest  to  the mathematical  and  philosophical  community,  concerned  both  with  the  actual  mathematics  and  with how  mathematicians  describe  their  work.  I  provided  a  translation  from  the  French  in  the  book’s appendix.  I  am  grateful  to  the  editor  of  the  Notices; publication  herein  will  allow  for  an  even  wider  audience. The  letter  is  from  André  Weil,  Oeuvres  Scientifiques,  Collected  Papers,  volume  1  (New  York: Springer,  1979),  pp.  244–255.  The  translation  aims to  be  reasonably  faithful,  not  only  to  the  meaning but  also  to  sentence  structure.  Brackets  are  in  the Oeuvres  Scientifiques text.  Braces  indicate  footnotes  therein.  My  editorial  insertions  are  indicated by  braces-and-brackets,  {[  ]}.  It  is  slightly  revised, as  taken  from  Martin  H.  Krieger,  Doing  Mathematics:  Convention,  Subject,  Calculation,  Analogy (Singapore:  World  Scientific,  2003),  pp.  293–305.  In the  notes  to  the  Oeuvres  Scientifiques,  Weil  indicates that  he  was  wrong  then  about  the  influence  of  the theory  of  quadratic  forms  in  more  than  two  variables  and  that  Hilbert  is  explicit  about  the  analogy in  his  account  of  the  Twelfth  Problem  (for  which see  David  Hilbert,  “Mathematical  problems”,  Bulletin  of  the  American  Mathematical  Society 37, 2000, 407-436). While  this  article  was  in  proof,  Philip  Horowitz sent  me  his  unpublished  translation  of  the  letter, which I had not known of before. I am grateful to Horowitz  for  allowing  me  to  use  his  translation  to improve mine in a number of places. —Martin H. Krieger VOLUME 52, NUMBER 3


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March  26,  1940 Some  thoughts  I  have  had  of  late,  concerning  my arithmetic-algebraic  work,  might  pass  for  a  response  to  one  of  your  letters,  where  you  asked  me what  is  of  interest  to  me  in  my  work.  So,  I  decided to  write  them  down,  even  if  for  the  most  part  they are incomprehensible to you. The  thoughts  that  follow  are  of  two  sorts.  The first  concerns  the  history  of  the  theory  of  numbers; you  may  be  able  to  understand  the  beginning;  you will  understand  nothing  of  what  follows  that.  The other  concerns  the  role  of  analogy  in  mathematical  discovery,  examining  a  particular  example,  and perhaps  you  will  be  able  to  profit  from  it.  I  advise you  that  all  that  concerns  the  history  of  mathematics  in  what  follows  is  based  on  insufficient scholarship,  and  is  derived  from  an  a  priori  reconstruction,  and  even  if  things  ought  to  have happened  this  way  (which  is  not  proven  here),  I cannot  say  that  they  did  happen  this  way.  In mathematics,  moreover,  as  much  as  in  any  other field, the line of history has many turning points. With  these  precautions  out  of  the  way,  let  us  start with  the  history  of  the  theory  of  numbers.  It  is  dominated  by  the  law  of  reciprocity.  This  is  Gauss’s  theorema  aureum (?I  need  to  refresh  my  memory  of this  point:  Gauss  very  much  liked  names  of  this  sort, he  had  as  well  a  theorema  egregium,  and  I  no longer  know  which  is  which),  published  by  him  in his  Disquisitiones in  1801,  which  was  only  beginning  to  be  read  and  understood  toward  1820  by Abel,  Jacobi,  and  Dirichlet,  and  which  remained as  the  bible  of  the  number  theorist  for  almost a  century.  But  in  order  to  say  what  this  law  is, whose statement was already known to Euler and Legendre  [Euler  had  found  it  empirically,  as  did  Legendre;  Legendre  claimed  more  in  giving  a  proof in  his  Arithmetic,  which  apparently  supposed  the truth  of  something  which  was  approximately  as  difficult  as  the  theorem;  but  he  complained  bitterly of  the  “theft”  committed  by  Gauss,  who,  without knowing  Legendre,  found,  empirically  as  well,  the statement  of  the  theorem,  and  gave  two  very  beautiful  proofs  in  his  Disquisitiones,  and  later  up  to  4 or  5  others,  all  based  on  different  principles.]:  it  is necessary  to  backtrack  a  bit  in  order  to  explain  the law of reciprocity. Algebra  began  with  the  task  of  finding,  for  given equations,  solutions  within  a  given  domain,  which might  be  the  positive  numbers,  or  the  reals,  or later  the  complex  numbers.  One  had  not  yet  conceived  of  the  ingenious  idea,  characteristic  of  modern  algebra,  of  starting  with  an  equation  and  then constructing  ad  hoc  a  domain  in  which  it  has  a  solution  (I  have  a  fair  amount  to  say  about  this  idea, which  has  shown  itself  to  be  extremely  productive; moreover,  Poincaré  has  somewhere  or  other  some beautiful  thoughts,  a  propos  of  the  solution  by radicals,  on  the  general  processes  whereby,  after having  searched  for  a  long  time  and  in  vain  to solve  this  problem  by  a  foreordained  procedure, mathematicians  inverted  the  question  and  began to  develop  adequate  methods).  The  problem  had been  solved  subsequently  for  all  second-degree equations  which  had  solutions  in  negative  numbers; when  the  equation  had  no  solution,  the  usual  formula  having  led  to  the  imaginaries,  about  which there  remained  many  doubts  (and  it  was  thus  until Gauss  and  his  contemporaries);  just  because  of the  suspicion  of  these  imaginaries,  the  so-called Cardan  and  Tartaglia  formula  for  the  solution  of the  equation  of  the  3rd  degree  in  radicals  produced  some  discomfort.  Be  that  as  it  may,  when Gauss  began  the  Disquisitiones with  the  notion  of congruences  for  building  up  his  systematic  exposition, it was also natural to solve congruences of the  second  degree,  after  having  solved  those  of  the first  degree  (a  congruence  is  a  relationship  among integers  a,b,m,  which  is  written  a ≡ b modulo  m, abbreviated  a ≡ b (mod m) or  a ≡ b(m),  meaning that  a and  b have the same remainder in division by  m,  or  a − b is  a  multiple  of  m;  a  congruence  of the  first  degree  is  ax + b ≡ 0(m),  of  the  second  degree  is  ax2 + bx+ c ≡ 0(m),  etc.);  the  latter  lead  (by the  same  procedure  through  which  one  reduces  an ordinary second degree equation to an extraction of  roots)  to  x2 ≡ a (mod m);  if  the  latter  has  a  solution,  one  says  that  a is  a  quadratic  residue  of  m, if  otherwise,  a is  a  non-residue  (1 and  −1 are residues  of  5,  2  and  −2 are  non-residues).  If  these notions  were  around  for  some  time  before  Gauss, it  was  not  necessary  that  they  be  associated  with a  notion  of  congruence;  for  the  notions  presented themselves  in  diophantine  problems  (solutions  of equations  in  integers  or  rationals)  which  were  the object  of  Fermat’s  most  important  work;  the  first degree  diophantine  equations,  ax+by = c,  are equivalent  to  first  degree  congruences ax ≡c (mod b);  the  second  degree  equations  of the  type  studied  by  Fermat  (decompositions  in terms  of  squares,  x2 +y2 = a,  and  equations x2 +ay2 =b,  etc.)  are  not  equivalent  to  congruences,  but  congruences  and  the  distinction  between  residues  and  non-residues  play  a  large  role in  his  work,  in  truth  they  did  not  appear  explicitly in  Fermat’s  work  (it  is  true  that  we  do  not  possess his  proofs,  but  he  seems  to  have  employed  other principles  about  which  we  can  make  some  approximate  inferences),  but  which,  as  far  as  I  know (based  on  second-hand  evidence)  were  already  well in evidence in Euler. The  law  of  reciprocity  permits  us  to  know,  given two  prime  numbers  p,q,  whether  q is  or  is  not  a (quadratic)  residue  of  p,  if  one  knows  already whether,  (a)  p is  or  is  not  a  residue  of  q;  (b)  if  p and q are  respectively  congruent  to  1  or  −1 modulo  4 (or  for  q =2,  if  p is  congruent  to  1,  2,  5,  or  7,

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modulo  8).  For  example,  53 ≡ 5 ≡ 1 (mod 4),  and 53  is  not  a  residue  of  5,  therefore 5  is  not  a  residue of  53.  Since  the  problem  for  non-primes  leads  naturally  to  the  problem  for  primes,  this  law  gives  an easy  means  of  determining  if  a is  or  is  not  a  residue of  b as  soon  as  one  knows  their  prime  factorization.  But  this  “practical”  application  is  insignificant. What  is  crucial  is  there  be  laws.  It  is  obvious  that the  residues  of  m form  an  arithmetic  progression of  increment  m,  for  if  a is  a  residue,  it  is  the  same for  all  mx + a;  however  it  is  beautiful  and  surprising  that  the  prime  numbers  p for  which m is  a residue  are  precisely  those  which  belong  to  certain arithmetic  progressions  of  increment  4m;  for  the others  m is a non-residue; and what is even more amazing,  if  one  recalls  on  the  other  hand  that  the distribution  of  prime  numbers  in  any  given  arithmetic  progression  Ax + B (which  one  knows  from Dirichlet  will  have  an  infinity  of  primes  as  long  as A and  B are  relatively  prime)  does  not  follow  any other  known  law  other  than  a  statistical  one  (the approximate  number  of  primes  which  are  ≤ T, which,  for  a  given  A,  is  the  same  for  any  B prime to  A)  and  appears,  for  each  concrete  case  that  one examines numerically, to be as “random” as a list of numbers generated by a roulette wheel. The  rest  of  the  Disquisitiones contains  above  all: 1. the definitive theory of quadratic forms in 2 variables,  ax2 + bxy +cy2,  having  among  other consequences  the  complete  resolution  of  the  problem  which  gave  birth  to  the  theory:  to  know  if ax2 +bxy +cy2 =m has solutions in integers. 2.  the  study  of  the  n-th  roots  of  unity,  and,  as we  would  say,  the  Galois  theory  of  the  fields  given by  these  roots and  their  subfields  (all  without  using imaginaries,  nor  other  functions  other  than  the trigonometric  ones,  and  ending  up  with  the  necessary  and  sufficient  conditions  for  the  regular n-gon  being  constructible  by  ruler  and  compass), which  appeared  as  an  application  of  earlier  work in  the  book,  as  preliminary  to  the  solution  of  congruences,  on  the  multiplicative  group  of  numbers modulo  m.  I  will  not  speak  of  the  theory  of  quadratic  forms  of  more  than  two  variables  since  it  has had  little  influence  until  now  on  the  general progress of the theory of numbers. Gauss’s  subsequent  research  was  to  study  cubic and  biquadratic  residues  (defined  by  x3 ≡ a and x4 ≡a (mod m)); the latter  are  a  bit  simpler;  Gauss recognized  that  there  were  no  simple  results  to  be hoped  for  by  staying  within  the  domain  of  ordinary integers  and  it  was  necessary  to  employ  “complex” integers  a + b√ −1 (a  propos  of  which  he  invented, at  about  the  same  time  as  Argand,  the  geometric representation  of  these  numbers  by  points  on  a plane,  through  which  all  doubts  were  dissipated about  the  “imaginaries”).  For  the  cubic  residues,  it was  necessary  to  have  recourse  to  the  “integers” a+bj,  a and  b integers,  j = the  cube  root  of  1. 336 Gauss  recognized  as  well,  and  even  thought  (there is  a  trace  of  this  in  his  notes)  of  studying  the  domain  of  the  n-th  roots  of  unity,  at  the  same  time thinking  to  try  to  provide  a  proof  of  “Fermat’s  theorem”  (xn +yn = zn is  impossible),  which  he  suspected  would  be  a  simple  application  (that  is  what he  said)  of  such  a  theory.  But  then  he  encountered the  fact  that  there  was  no  longer  a  unique  prime decomposition (except for  i and  j, as 4th and 3rd roots  of  unity,  and  I  believe  also  for  the  5th  roots). There  are  many  separate  threads;  it  would  take 125  years  to  unravel  them  and  assemble  them anew  into  a  new  skein.  The  great  names  here  are Dirichlet  (who  introduced  the  zeta  functions  or L-functions  into  the  theory  of  quadratic  forms, through  which  he  proved  among  other  things  that every  arithmetic  progression  contains  an  infinity of  primes;  but  above  all,  since  that  time  we  have only  needed  to  follow  his  model  in  order  to  apply these  functions  to  the  theory  of  numbers),  Kummer  (who  elucidated  the  fields  generated  by  roots of  unity  by  inventing  “ideal”  factors,  and  went  far enough  in  the  theory  of  these  fields  in  order  to  obtain  some  results  on  Fermat’s  theorem),  Dedekind, Kronecker,  Hilbert,  Artin.  Here  is  a  sketch  of  the picture that results from their efforts. I  cannot  say  anything  without  using  the  notion of  a  field,  which  according  to  its  definition,  if  one limits  oneself  to  its  definition,  is  simple  (it  is  a  set where  one  has  in  effect  the  usual  “four  elementary {[arithmetic]}  operations,”  these  having  the  usual properties  of  commutativity,  associativity,  distributivity);  the  algebraic  extension  of  a  field  k (it is  a  field  k,  containing  k,  of  which  all  elements  are roots  of  an  algebraic  equation  αn + c1αn−1 + ··· +cn−1α+cn =0with  coefficients  c1,...,cn in field k);  and  finally  the  abelian extension  of  a  field  k; that  means  an  algebraic  extension  of  k whose Galois  group is  abelian,  that  is  to  say  commutative. It  would  be  illusory  to  give  a  fuller  explanation  of abelian  extensions;  it  is  more  useful  to  say  that  they are  almost  the  same  thing,  but  not  the  same  thing, as  an  extension  of  k obtained  by  adjoining  n-th roots  (roots  of  equations  xn = a,  a in k);  if  k contains  for  whatever  integer n,  nn-th  distinct  roots of  unity  then  it  is  exactly  the  same  thing  (but  most often  one  is  interested  in  fields  which  do  not  have this  property).  If  k contains  nn-th  roots  of  unity (for  a  given  n),  then  all  abelian  extensions  of  degree  n (that  is  to  say,  having  been  generated  by  the adjunction  to  k of  one root  of  an  equation  of  degree  n)  can  be  generated  by  m-th  roots  (where  m is  a  divisor  of  n).  Abel  discovered  this  idea  in  his research  on  equations  solvable  by  radicals  (Abel  did not  know  of  the  notion  of  the  Galois  group,  which clarifies  all  these  questions).  It  is  impossible  to say  here  how  Abel’s  research  was  influenced  by Gauss’s  results  (see  above)  on  the  division  of  the circle  and  the  n-th  roots  of  unity  (which  lead  to  an


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abelian extension  of  the  field  of  rationals),  nor what  connections  they  had  with  the  work  of Lagrange,  with  Abel’s  own  work  on  elliptic  functions (where  the  division  takes  place,  from  Abel’s  point of  view,  in  the  abelian equation  [the  roots  generating  abelian  extensions],  results  which  were  already known  to  Gauss,  but  not  published,  at  the  very least  for  the  particular  case  of  the  so-called  lemniscate)  and  abelian  functions,  or  with  Jacobi’s work  on  the  same  subject  (the  same  Jacobi  who  invented  “abelian  functions”  in  the  modern  sense  and gave  them  that  name,  see  his  memoir  “De  transcendentibus  quibusdam  abelianis”),  nor  with Galois’s  work  (which  was  only  understood  little  by little,  and  much  later;  there  is  no trace  in  Riemann that  he  had  learned  from  it,  although (this  is  most remarkable)  Dedekind,  Privatdozent  in  Göttingen and  close  friend  of  Riemann,  had  since  1855  or  6, when  Riemann  was  at  the  height  of  his  powers, given a course on abstract groups and Galois theory). To  know  if  a (not  a  multiple  of  p)  is  a  residue of  p (prime),  is  to  know  whether  x2 − a = py has solutions;  in  passing  to  the  field  extension  of  √ one  gets  (x − √ a)(x + √ not  prime  to  x − √ a, a)=py,  so  in  this  field  p is a,  which,  nevertheless,  it  does not  divide.  In  the  language  of  ideals,  that  is  as much  to  say  that  in  this  field  p is  not  prime,  but may  be  decomposed  into  two  prime  ideal  factors. Thus  one  is  presented  with  a  problem:  k being  a field  (here  the  field  of  rationals),  k(here,  kis  k adjoined  by  √ a)  an  algebraic  extension  of  k,  to know  if  a  prime  ideal  (here,  a  number)  in  k remains  prime  in  kor  if  it  decomposes  into  prime ideals,  and  how:  a being  given,  the  law  of  reciprocity  points  to  those  p for  which  a is  the  residue, and  so  resolves  the  problem  for  this  particular case.  Here  and  in  all  of  what  follows,  k,  k,  etc.  are fields  of  algebraic  numbers  (roots  of  algebraic equations with rational coefficients). When  it  is  a  question  of  biquadratic  residues,  one works  with  a  field  generated  by  4√ a;  but  such  a  field is  not  in  general an  abelian  extension  of  the  “base field”  k unless  the  adjunction  of  a  4th  root  of  a brings  along  at  the  same  time  three  others  (namely, if  α is  one  of  them,  the  others  are  −α,iα,  and −iα),  this  requires  that  k contains  i = √ −1;  one would  have  nothing  so  simple  if  one  takes  as  the base  field  the  rationals,  but  all  goes  well  if  one  takes (as  did  Gauss)  the  field  of  “complex  rationals” r +si (r,s rational).  The  same  is  the  case  for  cubic residues.  In  these  cases,  one  studies  the  decomposition,  in  the  field  kobtained  by  the  adjunction of  a  4th  (or,  respectively,  3rd)  root,  starting  with  a base  field  k containing  i (respectively,  j),  of  an ideal (here, a number) prime in  k. So,  this  problem  of  the  decomposition  in  kof ideals  of  k is  completely  resolved  when  kis  an abelian  extension  of  k,  and  the  solution  is  very simple  and  it  generalizes  the  law  of  reciprocity  in a  straightforward  and  direct  manner.  For  the  arithmetic  progression  in  which  the  prime  numbers  are found,  with  residue  a,  one  substitutes  ideal  classes {[des  classes  d’idéaux]},  the  definition  of  which  is simple  enough.  The  classes  of  quadratic  forms  in two  variables,  studied  by  Gauss,  correspond  to  a particular  case  of  these  classes  of  ideals,  as  was  recognized  by  Dedekind;  Dirichlet’s  analytic  methods (using  zeta  or  L-functions)  for  studying  quadratic forms,  is  translated  readily  to  the  more  general classes  of  ideals  that  had  been  considered  in  this theory;  for  example,  for  the  theorem  on  arithmetic progressions  there  corresponds  the  following  result:  in  each  of  these  ideal  classes  in  k,  there  is  an infinity  of  prime  ideals,  therefore  an  infinity  of ideals  of  k which  may  be  factored  in  a  given  fashion  in  k.  Finally,  the  decomposition  of  ideals  of  k into  classes  determines  kin  a  unique  way:  and,  by the  theorem  called  the  law  of  Artin  reciprocity (because  it  implicitly  contains  Gauss’s  law  and  all known  generalizations),  there  is  a  correspondence (an  “isomorphism”)  of  the  Galois  group  of  kwith respect  to  k,  and  the  “group”  of  ideal  classes  in  k. Thus,  once  one  knows  what  happens  in  k,  one  has complete  knowledge  of  abelian extensions  of  k. This  does  not  mean  there  is  nothing  more  to  do about  abelian  extensions  (for  example,  one  can generate  these  by  the  numbers  exp(−2πi/n) if  k is  the  field  of  rationals,  thus  by  means  of  the  exponential  function;  if  k is  the  field  generated  by √−a,  a a positive integer, one knows how to generate  these  extensions  by  means  of  elliptic  functions  or  their  close  relatives;  but  one  knows  nothing  for  all  other  k).  But  these  questions  are  well understood  and  one  can  say  that  everything that has been done in arithmetic since Gauss up to recent  years  consists  in  variations  on  the  law  of  reciprocity:  beginning  with  Gauss’s  law;  and  ending with  and  crowning  the  work  of  Kummer,  Dedekind, Hilbert,  is  Artin’s  law,  it  is  all  the  same  law.  This  is beautiful,  but  a  bit  vexing.  We  know  a  little  more than  Gauss,  without  doubt;  but  what  we  know  more (or a bit more) is just that we do not know more. This  explains  why,  for  some  time,  mathematicians  have  focused  on  the  problem  of  the  nonabelian  decomposition  laws  (problems  concerning k,k,  when  kis  any  nonabelian  extension  of  k;  we remain  still  within  the  realm  of  a  field  of  algebraic numbers).  What  we  know  amounts  to  very  little;  and that  little  bit  was  found  by  Artin.  To  each  field  is attached  a  zeta  function,  discovered  by  Dedekind; if  kis  an  extension  of  k,  the  zeta  function  attached  to  kdecomposes  into  factors;  Artin  discovered  this  decomposition;  when  kis  an  abelian extension  of  k,  these  factors  are  identical  to  Dirichlet’s  L-functions,  or  rather  to  their  generalization for  fields  k and  classes  of  ideals  in  k,  and  the  identity  between  these  factors  and  these  functions  is


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(in  other  words)  Artin’s  reciprocity  law;  and  this  is the  way  Artin  first  arrived  at  this  law  as  a  bold  conjecture  (it  seems  that  Landau  made  fun  of  him), some  time  before  being  able  to  prove  it  (a  curious fact,  his  proof  is  a  simple  translation  of  another  result  by  Tchebotareff  that  had  just  been  published, which  he  cited;  however  it  is  Artin,  justly  having  it bear  his  name,  who  had  the  glory  of  discovering it).  In  other  words,  the  law  of  reciprocity  is  nothing  other  than  the  rule  for  forming  the  coefficients of  the  series  that  represents  the  Artin  factors  (which are  called  “Artin  L-functions”).  As  the  decomposition  into  factors  remains  valid  if  kis  a  non-abelian extension,  it  is  these  factors,  for  these  “non-abelian L-functions”,  that  it  is  natural  to  tackle  in  order  to discover  the  law  of  formation  of  their  coefficients. It  is  worth  noting  that,  in  the  abelian  case,  it  is known  that  the  Dirichlet  L-functions,  and  consequently  the  Artin  L-functions,  which  scarcely  differ  from  them,  are  entire  functions.  One  knows nothing  of  this  sort  for  the  general  case:  it  is  there, as  already  indicated  by  Artin,  that  one  might  find an  opening  for  an  attack  (please  excuse  the metaphor):  since the  methods  known  from  arithmetic  do  not  appear  to  permit  us  to  show  that  the Artin  functions  are  entire  functions,  one  could hope  that  in  proving  it  one  could  open  a  breach which  would  permit  one  to  enter  this  fort  (please excuse the straining of the metaphor). Since  the  opening  is  well  defended  (it  had  defied Artin), it is necessary to inspect the available artillery  and  the  means  of  tunneling  under  the  fort (please  excuse,  etc.).  {The  reader  who  has  the  patience  to  get  to  the  end  will  see  that  as  artillery,  I make  use  of  a  trilingual  inscription,  dictionaries, adultery,  and  a  bridge  which  is  a  turntable  {[or  a turnbridge]},  not  to  speak  of  God  and  the  devil,  who also  play  a  role  in  this  comedy.}  And  here  is  where the  analogy that  has  been  referred  to  since  the  beginning  finally  makes  its  entrance,  like  Tartuffe  appearing only in the third act. It is widely believed that there is nothing more to  do  about  algebraic  functions  of  one  variable,  because  Riemann,  who  had  discovered  just  about  all that  we  know  about  them  (excepting  the  work  on uniformization  by  Poincaré  and  Klein,  and  that  of Hurwitz  and  Severi  on  correspondences),  left  us  no indication  that  there  might  be  major  problems  that concern  them.  I  am  surely  one  of  the  most  knowledgeable  persons  about  this  subject;  mainly  because I  had  the  good  fortune  (in  1923)  to  learn  it  directly from  Riemann’s  memoir,  which  is  one  of  the  greatest  pieces  of  mathematics  that  has  ever  been  written;  there  is  not  a  single  word  in  it  that  is  not  of consequence.  The  story  is  not  closed,  however;  for example,  see  my  memoir  in  the  Liouville  Journal (see  the  introduction  to  this  paper).  {[“Généralisation  des  fonctions  abéliennes,”  Journal  de  Mathématiques Pures et Appliquées IX 17 (1938): 47–87, 338 pp.  47–49.]}  Of  course,  I  am  not  foolish  enough  to compare  myself  to  Riemann;  but  to  add  a  little  bit, whatever  it  is,  to  Riemann,  that  would  already  be, as  they  say  in  Greek,  to  do  something  {[faire  quelque chose]},  even  if  in  order  to  do  it  you  have  the  silent help of Galois, Poincaré and Artin. Be  that  as  it  may,  in  the  time  (1875  to  1890)  when Dedekind  created  his  theory  of  ideals  in  the  field of  algebraic  numbers  (in  his  famous  “XI  Supplements”:  Dedekind  published  four  editions  of  Dirichlet’s  Lectures  on  the  theory  of  numbers,  given  at Göttingen  during  the  last  years  of  Dirichlet’s  life, and  admirably  edited  by  Dedekind;  among  the  appendices  or  “Supplements”  of  these  lectures,  which contain  nothing  indicating  they  are  Dedekind’s original  work,  and  which  indeed  they  are  only  in part,  beginning  with  the  2nd  edition  there  are  three entirely  different  expositions  of  the  theory  of  ideals, one  for  each  edition),  he  discovered  that  an  analogous  principle  permitted  one  to  establish,  by purely  algebraic  means,  the  principal  results,  called “elementary”,  of  the  theory  of  algebraic  functions of  one  variable,  which  were  obtained  by  Riemann by  transcendental  {[analytic]}  means;  he  published with  Weber  an  account  of  the  consequences  of  this principle.  Until  then,  when  the  topic  of  algebraic functions  arose,  it  concerned  a  function  y of  a variable  x,  defined  by  an  equation  P(x,y)=0where P is  a  polynomial  with  complex  coefficients.  This  latter  point  was  essential  in  order  to  apply  Riemann’s methods;  with  those  of  Dedekind,  in  contrast,  those coefficients  could  come  from  an  arbitrary  field (called  “the  field  of  constants”),  since  the  arguments  were  purely  algebraic.  This  point  will  be  important shortly. The  analogies  that  Dedekind  demonstrated  were easy  to  understand.  For  integers  one  substituted polynomials  in  x,  to  the  divisibility  of  integers  corresponded  the  divisibility  of  polynomials  (it  is  well known, and it is taught even in high schools, that there  are  other  such  analogies,  such  as  for  the  derivation  of  the  greatest  common  divisor),  to  the  rationals  correspond  the  rational  fractions  {[?of  polynomials,  or  the  rational  functions]},  and  to  algebraic numbers  correspond  the  algebraic  functions.  At first  glance,  the  analogy  seems  superficial;  to  the most  profound  problems  of  the  theory  of  numbers (such  as  the  decomposition  into  prime  ideals)  there would  seem  to  be  nothing  corresponding  in  algebraic  functions,  and  inversely.  Hilbert  went  further  in  figuring  out  these  matters;  he  saw  that,  for example,  the  Riemann-Roch  theorem  corresponds to  Dedekind’s  work  in  arithmetic  on  the  ideal  called “the  different”;  Hilbert’s  insight  was  only  published by  him  in  an  obscure  review  (Ostrowski  pointed  me to  it),  but  it  was  already  transmitted  orally,  much as  other  of  his  ideas  on  this  subject.  The  unwritten  laws  of  modern  mathematics  forbid  writing down  such  views  if  they  cannot  be  stated  precisely


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nor,  all  the  more,  proven.  To  tell  the  truth,  if  this were  not  the  case,  one  would  be  overwhelmed  by work  that  is  even  more  stupid  and  if  not  more  useless  compared  to  work  that  is  now  published  in  the journals.  But  one  would  love  it  if  Hilbert  had  written down all that he had in mind. Let  us  examine  this  analogy  more  closely.  Once it  is  possible  to  translate  any  particular  proof  from one  theory  to  another,  then  the  analogy  has  ceased to  be  productive  for  this  purpose;  it  would  cease to  be  at  all  productive  if  at  one  point  we  had  a  meaningful  and  natural  way  of  deriving  both  theories from  a  single  one.  In  this  sense,  around  1820, mathematicians  (Gauss,  Abel,  Galois,  Jacobi)  permitted  themselves,  with  anguish  and  delight,  to  be guided  by  the  analogy  between  the  division  of  the circle  (Gauss’s  problem)  and  the  division  of  elliptic  functions.  Today,  we  can  easily  show  that  both problems  have  a  place  in  the  theory  of  abelian equations;  we  have  the  theory  (I  am  speaking  of  a purely  algebraic  theory,  so  it  is  not  a  matter  of  number  theory  in  this  case)  of  abelian  extensions.  Gone is  the  analogy:  gone  are  the  two  theories,  their  conflicts  and  their  delicious  reciprocal  reflections, their  furtive  caresses,  their  inexplicable  quarrels; alas, all is just one theory, whose majestic beauty can  no  longer  excite  us.  Nothing  is  more  fecund than  these  slightly  adulterous  relationships;  nothing  gives  greater  pleasure  to  the  connoisseur, whether  he  participates  in  it,  or  even  if  he  is  an  historian  contemplating  it  retrospectively,  accompanied, nevertheless, by a touch of melancholy. The pleasure  comes  from  the  illusion  and  the  far  from clear  meaning;  once  the  illusion  is  dissipated,  and knowledge  obtained,  one  becomes  indifferent  at  the same  time;  at  least  in  the  Gitâ  there  is  a  slew  of prayers  (slokas)  on  the  subject,  each  one  more final  than  the  previous  ones.  But  let  us  return  to our algebraic functions. Whether  it  is  due  to  the  Hilbert  tradition  or  to the  attraction  of  this  subject,  the  analogies  between  algebraic  functions  and  numbers  have  been on  the  minds  of  all  the  great  number  theorists  of our  time;  abelian  extensions  and  abelian  functions, classes  of  ideals  and  classes  of  divisors,  there  is material  enough  for  many  seductive  mind-games, some  of  which  are  likely  to  be  deceptive  (thus  the appearance  of  theta  functions  in  one  or  another theory).  But  to  make  something  of  this,  two  more recent  technical  contrivances  were  necessary.  On the  one  hand,  the  theory  of  algebraic  functions, that  of  Riemann,  depends  essentially on  the  idea of  birational  invariance;  for  example,  if  we  are concerned  with  the  field  of  rational functions  of one  variable  x,  one  introduces  (initially,  I  take the  field  of  constants  to  be  the  complex  numbers) as  the  points corresponding  to  the  various  complex values  of  x,  including  the  point  at  infinity,  denoted symbolically  by  x = ∞,  and  defined  by  1/x =0;  the fact  that  this  point  plays  exactly  the  same  role  as all  the  others  is  essential.  Let  R(x)= a(x −α1)...(x−αm)/(x−β1)...(x−βn) be  a  rational  fraction,  with  its  decomposition  into  factors  as  indicated;  it  will  have  zeros  α1,...,αm, the poles  β1,...,βn, and  the  point  at  infinity,  which  is zero  if  n>m,  and  is  infinite  if  n<m.  In  the  domain  of  rational  numbers,  one  always  has  a  decomposition  into  prime  factors,  r = p1... pm/q1...qn,  each  prime  factor  corresponding  to a  binomial  factor  (x − α);  but  nothing  apparently corresponds  to  the  point  at  infinity.  If  one  models the  theory  of  functions  on  the  theory  of  algebraic numbers,  one  is  forced  to  give  a  special  role,  in  the proofs,  to  the  point  at  infinity,  sweeping  the  problem  into  a  corner,  if  we  are  to  have  a  definitive  statement  of  the  result:  this  is  just  what  DedekindWeber  did,  this  is  just  what  was  done  by  all  who have  written  in  algebraic  terms  about  algebraic functions  of  one  variable,  until  now,  I  was  the  first, two  years  ago,  to  give  (in  Crelle’s  Journal  {[“Zur  algebraischen  Theorie  der  algebraischen  Funktionen”,  179  (1938),  pp.  129–133]})  a  purely  algebraic proof  of  the  main  theorems  of  this  theory,  which is  as  birationally  invariant  (that  is  to  say,  not  attributing  a  special  role  to  any  point)  as  were  Riemann’s  proofs;  and  that  is  of  more  than  methodological  importance.  {Actually,  I  was  not  quite  the first.  The  proofs,  to  be  sure  very  roundabout,  of  the Italian  school  (Severi  above  all)  are,  in  principle,  of the  same  sort,  although  drafted  in  classical  language.}  However  fine  it  is  to  have  these  results  for the  function  field,  it  seems  that  one  has  lost  sight of  the  analogy.  In  order  to  reestablish  the  analogy, it  is  necessary  to  introduce,  into  the  theory  of  algebraic  numbers,  something  that  corresponds  to the  point  at  infinity  in  the  theory  of  functions. That  is  what  one  achieves,  and  in  a  very  satisfactory  manner,  too,  in  the  theory  of  “valuations”. This  theory,  which  is  not  difficult  but  I  cannot  explain  here,  depends  on  Hensel’s  theory  of  p-adic fields:  to  define  a  prime  ideal  in  a  field  (a  field  given abstractly)  is  to  represent  the  field  “isomorphically” in a  p-adic field: to represent it in the same way  in  the  field  of  real  or  complex  numbers,  is (in this  theory)  to  define  a  “prime  ideal  at  infinity”.  This latter  notion  is  due  to  Hasse  (who  was  a  student of  Hensel),  or  perhaps  Artin,  or  to  both  of  them.  If one follows it in all of its consequences, the theory alone  permits  us  to  reestablish  the  analogy  at  many points  where  it  once  seemed  defective:  it  even  permits  us  to  discover  in  the  number  field  simple  and elementary  facts  which  however  were  not  yet  seen (see  my  1939  article  in  la  Revue  Rose which  contains  some  of  the  details  {[“Sur  l’analogie  entre  les corps  de  nombres  algébriques  et  les  corps  de  fonctions  algébriques,’’  Revue  Scientifique 77  (1939) 104–106,  and  the  comments  in  the  Oeuvres  Scientifiques,  volume  1,  pp.  542–543]}).  It  is  not  so  much


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this  point  of  view  that  has  been  used  up  to  now  for giving  satisfactory  statements  of  the  principal results  of  the  theory  of  abelian  extensions  (I  forgot  to  say  that  this  theory  is  most  often  called “class  field  theory”).  An  important  point  is  that  the p-adic  field,  or  respectively  the  real  or  complex  field, corresponding  to  a  prime  ideal,  plays  exactly  the role,  in  arithmetic,  that  the  field  of  power  series  in the  neighborhood  of  a  point plays  in  the  theory  of functions: that is why one calls it a  local field. With  all  of  this,  we  have  made  great  progress; but  it  is  not  enough.  The  purely  algebraic  theory of  algebraic  functions  in  any  arbitrary  field  of  constants  is  not  rich  enough  so  that  one  might  draw useful  lessons  from  it.  The  “classical”  theory  (that is,  Riemannian)  of  algebraic  functions  over  the field  of  constants  of  the  complex  numbers  is  infinitely  richer;  but  on  the  one  hand  it  is  too  much so,  and  in  the  mass  of  facts  some  real  analogies  become  lost;  and  above  all,  it  is  too  far  from  the  theory  of  numbers.  One  would  be  totally  obstructed if there were not a bridge between the two. And  just  as  God  defeats  the  devil:  this  bridge  exists;  it  is  the  theory  of  the  field  of  algebraic  functions  over  a  finite  field  of  constants  (that  is  to  say, a  finite  number  of  elements:  also  said  to  be  a  Galois  field,  or  earlier  “Galois  imaginaries”  because Galois  first  defined  them  and  studied  them;  they are  the  algebraic  extensions  of  a  field  with  p elements  formed  by  the  numbers  0,1,2,...,p−1 where  one  calculates  with  them  modulo  p,p = prime  number).  They  appear  already  in  Dedekind. A  young  student  in  Göttingen,  killed  in  1914  or 1915,  studied,  in  his  dissertation  that  appeared  in 1919  (work  done  entirely  on  his  own,  says  his teacher  Landau),  zeta  functions  for  certain  of  these fields,  and  showed  that  the  ordinary  methods  of the  theory  of  algebraic  numbers  applied  to  them. Artin,  in  1921  or  1922,  took  up  the  question  again, again from the point of view of the zeta function; F.  K.  Schmidt  made  the  bridge  between  these  results  and  those  of  Dedekind-Weber,  in  the  process of  providing  a  definition  of  the  zeta  function  that was  birationally  invariant.  In  the  last  few  years, these  fields  were  a  favorite  subject  of  Hasse  and his  school;  Hasse  made  a  number  of  beautiful  contributions. I  spoke  of  a  bridge;  it  would  be  more  correct  to speak  of  a  turntable  {[?turnbridge]}.  On  one  hand  the analogy  with  number  fields  is  so  strict  and  obvious that  there  is  neither  an  argument  nor  a  result  in  arithmetic  that  cannot  be  translated  almost  word  for word  to  the  function  fields.  In  particular,  it  is  so  for all  that  concerns  zeta  functions  and  Artin  functions; and  there  is  more:  Artin  functions  in  the  abelian  case are  polynomials,  which  one  can  express  by  saying  that these  fields  furnish  a  simplified model  of  what  happens  in  number  fields;  here,  there  is  thus  room  to conjecture  that  the  non-abelian  Artin  functions  are 340 still  polynomials:  that  is  just  what  occupies  me  at the moment, all of this permits me to believe that all  results  for  these  fields  could  inversely,  if  one could  formulate  them  appropriately,  be  translated to the number fields. On  the  other  hand,  between  the  function  fields and  the  “Riemannian”  fields,  the  distance  is  not  so large  that  a  patient  study  would  not  teach  us  the art  of  passing  from  one  to  the  other,  and  to  profit in  the  study  of  the  first  from  knowledge  acquired about  the  second,  and  of  the  extremely  powerful means  offered  to  us,  in  the  study  of  the  latter, from  the  integral  calculus  and  the  theory  of  analytic  functions.  That  is  not  to  say  that  at  best  all will  be  easy;  but  one  ends  up  by  learning  to  see something  there,  although  it  is  still  somewhat  confused.  Intuition  makes  much  of  it;  I  mean  by  this the  faculty  of  seeing  a  connection  between  things that  in  appearance  are  completely  different;  it  does not  fail  to  lead  us  astray  quite  often.  Be  that  as  it may, my work consists in deciphering a trilingual text  {[cf.  the  Rosetta  Stone]};  of  each  of  the  three columns  I  have  only  disparate  fragments;  I  have some  ideas  about  each  of  the  three  languages:  but I  know  as  well  there  are  great  differences  in  meaning  from  one  column  to  another,  for  which  nothing  has  prepared  me  in  advance.  In  the  several years  I  have  worked  at  it,  I  have  found  little  pieces of  the  dictionary.  Sometimes  I  worked  on  one  column,  sometimes  under  another.  My  large  study that  appeared  in  the  Liouville  journal  made  nice  advances  in  the  “Riemannian”  column;  unhappily,  a large  part  of  the  deciphered  text  surely  does  not have  a  translation  in  the  other  two  languages:  but one  part  remains  that  is  very  useful  to  me.  At  this moment,  I  am  working  on  the  middle  column.  All of  this  is  amusing  enough.  However,  do  not  imagine  that  this  work  on  several  columns  is  a  frequent occasion  in  mathematics;  in  such  a  pure  form,  this is  almost  a  unique  case.  This  sort  of  work  suits  me particularly;  it  is  unbelievable  at  this  point  that distinguished  people  such  as  Hasse  and  his  students,  who  have  made  this  subject  the  matter  of their  most  serious  thoughts  over  the  years,  have, not  only  neglected,  but  disdained  to  take  the  Riemannian  point  of  view:  at  this  point  they  no  longer know  how  to  read  work  written  in  Riemannian  (one day,  Siegel  made  fun  of  Hasse,  who  had  declared himself  incapable  of  reading  my  Liouville  paper), and  that  they  have  rediscovered  sometimes  with  a great  deal  of  effort,  in  their  dialect,  important  results that were already known, much as the ideas of  Severi  on  the  ring  of  correspondences  were  rediscovered  by  Deuring.  But  the  role  of  what  I  call analogies,  even  if  they  are  not  always  so  clear,  is nonetheless  important.  It  would  be  of  great  interest  to  study  these  things  for  a  period  for  which  we are  well  provided  with  texts;  the  choice  would  be delicate.


8

P.S. I send this to you without rereading …I fear …having  made  more  of  my  research  than  I  intended; that  is,  in  order  to  explain  (following  your  request) how  one  develops  one’s  research,  I  have  been  focusing on the locks I wish to open. In speaking of analogies  between  numbers  and  functions,  I  do not want to give the impression of being the only one  who  understands  them:  Artin  has  thought  profoundly  about  them  as  well,  and  that  is  to  say  a great  deal.  It  is  curious  to  note  that  one  work (signed  by  a  student  of  Artin  who  is  not  otherwise known,  which  without  proof  to  the  contrary,  allows one  to  presume  that  Artin  is  the  real  source)  appeared  2  or  3  years  ago  which  gives  perhaps  the only  example  of  a  result  from  the  classical  theory, obtained  by  a  double translation,  starting  with  an arithmetic  result  (on  abelian  zeta  functions),  and which  is  novel  and  interesting.  And  Hasse,  whose combination  of  patience  and  talent  make  him  a  kind of  genius,  has  had  very  interesting  ideas  on  this  subject.  Moreover  (a  characteristic  trait,  and  which would  be  sympathetic  to  you,  of  the  school  of  modern  algebra)  all  of  this  is  spread  by  an  oral  and  epistolary  tradition  more  than  by  orthodox  publications, so  it  is  difficult  to  make  a  history  of  all  of  it  in  detail. You  doubt  and  with  good  reason  that  modern axiomatics will work on difficult material. When I invented (I say invented, and not discovered) uniform  spaces,  I  did  not  have  the  impression  of  working  with  resistant  material,  but  rather  the  impression  that  a  professional  sculptor  must  have  when he plays by making a snowman. It is hard for you to  appreciate  that  modern  mathematics  has  become  so  extensive  and  so  complex  that  it  is  essential,  if mathematics  is  to  stay  as  a  whole  and  not become a pile of little bits of research, to provide a  unification,  which  absorbs  in  some  simple  and general  theories  all  the  common  substrata  of  the diverse  branches  of  the  science,  suppressing  what is  not  so  useful  and  necessary,  and  leaving  intact what  is  truly  the  specific  detail  of  each  big  problem.  This  is  the  good  one  can  achieve  with  axiomatics  (and  this  is  no  small  achievement).  This is  what  Bourbaki  is  up  to.  It  will  not  have  escaped you  (to  take  up  the  military  metaphor  again)  that there  is  within  all  of  this  great  problems  of  strategy.  And  it  is  as  common  to  know  tactics  as  it  is  rare (and  beautiful,  as  Gandhi  would  say)  to  plan  strategy.  I  will  compare  (despite  the  incoherence  of  the metaphor) the great axiomatic edifices to communication  at  the  rear  of  the  front:  there  is  not  much glory in the Commissariat and logistics and transport,  but  what  would  happen  if  these  brave  folks did  not  consecrate  themselves  to  secondary  work (where,  moreover,  they  readily  earn  their  subsistence)?  The  danger  is  only  too  great  that  various fronts end up, not by starving (the Council for Research  is  there  for  that),  but  by  paying  insufficient attention  to  each  other  and  so  waste  their  time, some  like  the  Hebrews  in  the  desert,  others  like  Hannibal  at  Capua  {[where  the  troops  were  said  to  have been  entranced  by  the  place]}.  The  current  organization  of  science  does  not  take  into  account  (unhappily,  for  the  experimental  sciences;  in  mathematics  the  damage  is  much  less  great)  the  fact  that very  few  persons  are  capable  of  grasping  the  entire  forefront  of  science,  of  seizing  not  only  the weak  points  of  resistance,  but  also  the  part  that  is most  important  to  take  on,  the  art  of  massing  the troops,  of  making  each  sector  work  toward  the success  of  the  others,  etc.  Of  course,  when  I  speak of  troops  the  term  (for  the  mathematician,  at  least) is  essentially  metaphoric,  each  mathematician  being himself  his  own  troops.  If,  under  the  leadership given  by  certain  teachers,  certain  “schools”  have  notable  success,  the  role  of  the  individual  in  mathematics  remains  preponderant.  Moreover,  it  is  becoming  impossible  to  apply  a  view  of  this  sort  to science  as  a  whole;  it  is  not  possible  to  have  someone  who  can  master  enough  of  both  mathematics and  physics  at  the  same  time  to  control  their  development  alternatively  or  simultaneously;  all  attempts  at  “planning”  become  grotesque  and  it  is  necessary to leave it to chance and to the specialists.

Fermat's Last Theorem: フェルマーの最終定理

https://youtu.be/se7s17x39eA 0:43:00


アンドレ・ヴェイユ - Wikipedia

https://ja.wikipedia.org/wiki/アンドレ・ヴェイユ

谷山豊によるヴェイユ評編集

谷山・志村予想の発案者でもある日本の数学者、谷山豊はヴェイユを評して「歯に衣を着せない」、「その批判は辛辣である」、「温厚な大先生方には余り評判は宜しくない」とする一方で「それを一概に排斥しないだけの自由な空気がなかったならば、数学は窒息してしまったであろう」としている。そしてヴェイユの大胆な推測、ハッタリではないかと思われかねない発言に対し「凡眼を以って、天才の思想を云々するのは危険であろう」と記している。谷山はヴェイユの才能を第一はclassicな理論の中から本質を鋭く見抜き、何が、如何に抽象化され一般化されるべきかを問う能力、第二に、それを実行に移す際に山積する障害に対し、挫折したり迂回路を取ることなく、障害を一つ一つ強引に捩じ伏せる腕力と息の長さであると評し、「奇麗事が好きで腕力の弱い我が国の多くの数学者」に対する頂門の一針であるとした。

エピソード

ヴェイユは応用数学(同時代的には、数学と物理学の組み合わせは一種の花形であった)には興味が無いとしていた。しかし、人文科学者を含めて交友はあり、友人の人類学者であるクロード・レヴィ=ストロースからのオーストラリア北端のムルンギン族の婚姻制度の組合せ問題の解決依頼に対して協力している。ヴェイユはこの問題を、婚姻のかたちを二つの元が生成するアーベル群に抽象化して整理できると見抜き解決した。この件により「ムルンギン族に対しある種の愛情を感じるようになった」と後に認めている。


志村 ─谷山 ─ヴェイユ予想は 、アイヒラ ーの得た結果の一般化である 。この予想は 、任意の 〝三次方程式 〟について (ある種の穏やかな条件に従うものとして ) 、素数を法とする解の個数は 、あるモジュラ ー形式の係数であると述べている 。さらに 、その 〝三次方程式 〟と (ある種の )モジュラ ー形式とのあいだに 、一対一対応が成り立つというのである 。



NAMs出版プロジェクト: ハドロンの分類図、SU(3)の図

http://nam-students.blogspot.jp/2015/12/su.html
素数を法をしたら数論となり、複素数なら幾何学になる。


『アンドレとシモーヌ』で黒澤明と一緒に勲章を授与される際、会話した様子が描かれる。

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