Graduate Texts in Mathematics S Axler Editorial Board F.W Gehring K.A Ribet Springer Science+Business Media, LLC Graduate Texts in Mathematics 10 II 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 TAKEUTliZARlNG Introduction to Axiomatic Set Theory 2nd ed OXTOBY Measure and Category 2nd cd SCHAEFER Topological Vector Spaces 2nd ed HILTON/STAMMBACH A Course in Homological Algebra 2nd cd MAC LANE Categories for the Working Mathematician 2nd ed HUGIlEs/PIPER Projective Planes SERRE A Course in Arithmetic TAKEUTIIZARING Axiomatic Set Theory HUMPHREYS Introduction to Lie Algebras and Representation Theory COllEN A Course in Simple Homotopy Theory CONWAY Functions of One Complex Variable I 2nd ed BEALS Advanced Mathematical Analysis A!'lDERSON/FuLLER Rings and Categories of Modules 2nd ed GOLUBlTSKy/GUILLEMIN Stable Mappings and Their Singularities BERBERIAN Lectures in Functional Analysis and Operator Theory WINTER The Structure of Fields ROSE!'lBLATT Random Processes 2nd ed HALMOS Measure Theory HALMOS A Hilbert Space Problem Book 2nd ed HUSEMOI.LER Fibre Bundles 3rd ed HUMPHREYS Linear Algebraic Groups BAIL'IESIMACK An Algebraic Introduction to Mathematical Logic GREL'B Linear Algebra 4th ed HOLMES Geometric Functional AnalYSIS and Its Applications HEWITT/STROMBERG Real and Abstract Analysis MANES Algebraic Theories KELLEY General Topology ZARISKIISAMUEL Commutative Algebra VoLl Z >JUSKIISAMUEL Commutative Algebra VoU! JACOBSON Lectures in Abstract Algebra I Basic Concepts JACOBSON Lectures in Abstract Algebra II Linear Algebra JACOBSON Lectures in Abstract Algebra Ill Theory of fields and Galois Theory HIRSCH Differential Topology SPITZER Principles of Random Walk 2nd ed ALEXANDER/WERMER Several Complex Variables and Banach Algebras 3rd ed 36 KELLEy/NAMIOKA et al Linear Topological Spaces 37 MONK Mathematical Logic 38 GRAUERT/FRITZSCHE Several Complex Variables 39 ARVESON An Invitation to C··Algebras 40 KEMENy/SNELLlKNAPP Denumerable Markov Chains 2nd ed 41 APOSTOL Modular Functions and Dirichlet Series in Number Theory 2nd ed 42 SERRE Linear Representations of Finite Groups 43 GILLMA.'1/JERISON Rings of Continuous Functions 44 KENDIG Elementary Algebraic Geometry 45 LoEvE Probability Theory I 4th ed 46 LoEvE Probability Theory II 4th ed 47 MOISE Geometric Topology in Dimensions and 48 SACHS/WU General Relativity for Mathematicians 49 GRUEI\'IlERG/WEIR Linear Geometry 2nd ed 50 EDWARDS Fermat's Last Theorem 51 KLINGENBERG A Course in Differential Geometry 52 HARTSHORNE Algebraic Geometry 53 MANI}I A Course in Mathematical Logic 54 GRAvER/WATKINS Combinatorics with Emphasis on the Theory of Graphs 55 BROWNIPEARCY Introduction to Operator Theory I: Elements of Functional Analysis 56 MASSEY Algebraic Topology: An Introduction 57 CROWELL/Fox Introduction to Knot Theory 58 KOBLITZ p-adic Numbers, p-adic Analysis, and Zeta-Functions 2nd ed 59 LA."G Cyclotomic Fields 60 AJu.;OLD Mathematical Methods in Classical Mechanics 2nd ed 61 WIllTEHEAD Elements of Homotopy Theory 62 KARGAPOLOvIMERl.ZIAKOV Fundamentals of the Theory of Groups 63 BOLLOBAS Graph Theory 64 EDWARDS Fourier Series Vol I 2nd ed 65 WELLS Differential Analysis on Complex Manifolds 2nd ed 66 WATERHOUSE Introduction to Affine Group Schemes 67 SERRE Local Fields 68 WEIDMAN!" Linear Operators in Hilben Spaces 69 LASG Cyclotomic Fields II (continued after index) I-P' Serre A Course in Arithmetic , Springer Jean-Pierre Serre College de France 75231 Paris Cedex 05 France Editorial Board S Axler Mathematics Depanment San Francisco State University San Francisco, CA 94132 USA F.W Gehring Mathematics Dcpanmcnt East Hal1 University of Michigan Ann Arbor MI 48109 USA KA Ribet Departrnent of Mathematics University ofCalifomia at Berkeley Berkeley, CA 94720-3840 USA Mathematies Subject Classification: II-OI Title of the Freneh original edition: Cours d'Arilhmetique Publisher: Presses Universitaires de France Paris, 1970-1977 Library of Congress Cataloging in Publication Data Serre, Jean-Pierre A course in arithmetic by J.-P Serre New York, Springer-Verlag 1973 viii, 115 p ilIus 25 cm (Graduate texts in mathematics, 7) Translation of Cours d'arithmetique Bibliography: p 112-113 Forms, Quadratic Analytic functions Title II Series QA243.S47\3 512.9'44 70-190089 ISBN 978-0-387-90041-4 ISBN 978-1-4684-9884-4 (eBook) DOI 10.1007/978-1-4684-9884-4 Ipok.) MARC Printed on acid-frec paper © 1973 Springer Science+Business Media New York Originallypublished by Springer-VerlagNew YorkInc in 1973 Softcover reprint ofthe hardcover Ist edition 1973 AII rights reserved This work may not be translated or copied in whole or in part without the written permission of the publisher Springer Science+Business Media, LLC, except for brief excerpts in conncction with reviews or scholarly anal)'sis Use in connection with any form of information storage and retrievai, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereaCter developed is forbidden The use of general descriptive names, trade names, trademarks, etc., in this publication, even if the former are not espccially identified, is not to be taken as a sign that such names, as understood by the Trade Marks and Merchandise Marks Act, may accordingly be used freely by anyone Printcd and bound by R R Donnelley and Sons, Ilarrisonburg, VA ISBN 978-0-387-90041-4 SPIN 10783650 Preface This book is divided into two parts The first one is purely algebraic Its objective is the classification of quadratic forms over the field of rational numbers (Hasse-Minkowski theorem) It is achieved in Chapter IV The first three chapters contain some preliminaries: quadratic reciprocity law, p-adic fields, Hilbert symbols Chapter V applies the preceding results to integral quadratic forms of discriminant ± I These forms occur in various questions: modular functions, differential topology, finite groups The second part (Chapters VI and VII) uses "analytic" methods (holomorphic functions) Chapter VI gives the proof of the "theorem on arithmetic progressions" due to Dirichlet; this theorem is used at a critical point in the first part (Chapter Ill, no 2.2) Chapter VII deals with modular forms, and in particular, with theta functions Some of the quadratic forms of Chapter V reappear here The two parts correspond to lectures given in 1962 and 1964 to second year students at the Ecole Normale Superieure A redaction of these lectures in the form of duplicated notes, was made by J.-J Sansuc (Chapters I-IV) and J.-P Ramis and G Ruget (Chapters VI-VII) They were very useful to me; I extend here my gratitude to their authors J.-P Serre v Table of Contents v Preface Part I-Algebraic Methods Chapter I-Finite fields I-Generalities 2-Equations over a finite field 3-Quadratic reciprocity law Appendix-Another proof of the quadratic reciprocity law Chapter II-p-adic fields I-The ring Z, and the field Q, 2-p-adic equations 3-The multiplicative group of Q, Chapter III-Hilbert symbol I-Local properties 2-Global properties Chapter IV-Quadratic forms over Q, and over Q I-Quadratic forms 2-Quadratic forms over Q, 3-Quadratic forms over Q Appendix-Sums of three squares Chapter V-Integral quadratic forms with discriminant ± I 1-Preliminaries 2-Statement of results 3-Proofs 3 11 11 13 15 19 19 23 27 27 35 41 45 48 48 52 55 Part II-Analytic Methods Chapter VI-The theorem on arithmetic progressions I-Characters of finite abelian groups 2-Dirichlet series 3-Zeta function and L functions 4-Density and Dirichlet theorem Chapter VII-Modular forms I-The modular group 2-Modular functions 3-The space of modular forms 4-Expansions at infinity 5-Hecke operators 6-Theta functions vii 61 61 64 68 73 77 77 79 84 90 98 106 Bibliography Index of Definitions Index of Notations Il2 114 lIS viii A Course in Arithmetic Part I Algebraic Methods Modular forms 1M if and only if c/> is an Eulerian product of type (82) See for more details E HEeKE, Math Werke, n° 33, and A WElL, Math Annalen, 168, 1967 5.5 Examples a) Eisenstein series.-Let k be an integer ~ Proposition l3.-The Eisenstein series Gk is an eigenfunction of T(n); the corresponding eigenvalue is a 2k - 1(n) and the normalized eigenfunction is (86) k -Bk (-I) k Bk Ek = (-I) 4k 4k The corresponding Dirichlet series is + ~ alk_l(n)q n L n=1 ~(sg(s-2k+ I) We prove first that Gk is an eigenfunction of T(n); it suffices to this for T(p), p prime Consider Gk as a function on the set.rJi of lattices ofe; we have: If Ilylk, Gk(f) = cf n° 2.3, yer and T(p)Gk(r) I = (r: r') ~p If I/lk l'er' Let y E r If y E pr then y belongs to each of the p + sublattices of r of index p; its contribution in T(p)Gk(r) is (p+ I)/lk If y E r-pl', then y belongs to only one sublattice of index p and its contribution is I/y2k Thus T(p)Gk(f) = L Gk(r)+p I/lk = Gk(r)+pGk(pf) l'Epr = (I +pl-2k)Gk(r), which proves that Gk (viewed as a function on 31) is an eigenfunction of T(p) with eigenvalue 1+pl-2k; viewed as a modular form, Gk is thus an eigenfunction of rep) with eigenvalue p2k - 1(1 + pI - 2k) = alk- 1(p) Formulas (34) and (35) of n° 4.2 show that the normalized eigenfunction associated with Gk is (-It Bk 4k f alk_l(n)qn + n=1 This also shows that the eigenvalues of T(n) are a2k-l(n) Finally ex: L a2k _ (n)/n s = L a 2k - /a'd' a.d n:;; = ~ (I d?;1 I lidS) (L 1/a'+'-2k) a?; = ~(sms-2k+ I) b) The /1 function Proposition 14.-The /1 function is all eigenfunction of T(n) The corresponding eigenvalue is T(n) and the normalized eigenfunction is (271)-12/1 =q n (l_qn)24 = n= co L n= T(n)qn lOS Heeke operators This is clear, since the space of cusp forms of weight 12 is of dimension I, and is stable by the T(n) Corollary.-We have (52) (53) r(nm) r(p)r(p") = = if(n, m) = I, r(n)r(m) ifp is a prime, n ~ I r(p"+I)+pllr(p"-I) This follows from cor of tho Remark.-There are similar results when the space of weight 2k has dimension I; this happens for k = 6,8,9, Mf of cusp forms 10, 11, 13 with basis Ll, LlG • LlG , LlG4 , LlG s, and LlG • 5.6 Complements 5.6.1 The Petersson scalar product Let f, g be two cusp forms of weight 2k with k > O One proves easily that the measure p.(f,g) = J(z)g(z)y 2kdxdyly2 (x = R(z), y = Im{z» is invariant by G and that it is a bounded measure on the quotient space HIG By putting (87) (f, g) = f p.{f, g) = f J(z)g{z)y2lt-2dxdy, HIG D we obtain a hermitian scalar product on degenerate One can check that (88) Mf which is positive and non- (T{n)f, g) = (f, T(n)g), which means that the T(n) are hermitian operators with respect to (f, g) Since the T(n) commute with each other, a well known argument shows that there exists an orthogonal basis oj M~ made oj eigenvectors 0/ T(n) and that the eigenvalues of T(n) are real numbers 5.6.2 Integrality properties Let Mk(Z) be the set of modular forms co J= L "=0 c(n)q" of weight 2k whose coefficients c(n) are integers One can prove that there exists a Z-basis of Mk(Z) which is a C-basis of M" [More precisely, one can check that Mk(Z) has the following basis (recall that F = q IT (I_q")24): k even: One takes the monomials E~FP where ex, fJ e N, and ex + 3fJ = k/2; k odd: One takes the monomials E3E~FfJ where ex, fJ E N, and 1%+3fJ = 106 Modular forms (k-3)/2.] Proposition 12 shows that M.(Z) is stable under T(n), n ~ I We conclude from this that the coefficients of the characteristic polynomial of T(n) acting on M k , are illlegers O We have BrUt) = L e-n.(x.X) r(t)· = XEI' Similarly, 0r(-llit) = r (t-I) Formula (99) results thus from (94), taking into account that v = I and r = r' Since n is divisible by 8, we can rewrite (99) in the form (100) which shows that Or is a modular form of weight n/2 [We indicate briefly another proof of (a) Suppose that n is not divisible by 8; replacing r, if necessary, by r $ r or r $ r $ r EB r, we may ~uppose that n == (mod 8) Formula (99) can then be written 0r( -lIz) = (_I)"/4 zm/20 r(z) = -Z"/20 r(Z) If we put w(z) = Or(z)dz"/4, we see that the differential form w is transformed into -w by S:z f-+ -lIz Since w is invariant by T:z f-+ z+ I, we see that ST transforms w into -w, which is absurd because (ST)3 = I.] Corollary I.-There exists a cusp form fr of weight 11/2 such that (101) Or = Ek+fr where k This follows from the fact that 0r( (0) form Corollary l.-We have rr(m) = -4k BA = = n/4 I, hence that Or - Ek is a cusp ulk_l(m)+O(m k ) where k = n/4 Modular forma 110 This follows from cor 1, formula (34) and tho Remark.-The "error term" fr is in general not zero However Siegel has proved that the we;ghted mean of the fr ;s zero More precisely, let CII be the set of classes (up to isomorphism) of lattices r verifying (i) and (ii) and denote by gr the order of the automorphism group of r E CII (cf chap V, n° 2.3) One has: (102) or equivalently (103) where Mil = L - reCn gr Note that this is also equivalent to saying that the weighted mean of the Or is an eigenfunction of the T(n) For a proof of formulas (102) and (103), see C L SIEGEL, Gesam Abh., nO 20 6.6 Examples = Every cusp form of weight n/2 that Or = E , in other words: i) The case n (104) rr(m) = 240u3(m) =4 is zero Cor of tho then shows for all integers m ~ I This applies to the lattice r s constructed in chap V, n° 1.4.3 (note that this lattice is the only element of Cs ) ii) The case n = 16 For the same reason as above, we have: Or (105) = E4 = +480 L QC ",=1 u (m)q"' Here one may take r = r EB r s or r = r 16 (with the notations of chap V, n° 1.4.3); even though these two lattices are not isomorphic, they have the same theta function, i.e they represent each integer the same number of times Note that the function (J attached to the lattice r E9 r is the square of the function (J of r s; we recover thus the identity: (I +240 "'~I u3(m)Q"r = I +480 "'~I u7(m)q"' iii) The case n = 24 The space of modular forms of weight 12 is of dimension It has for basis the two functions: £, = I 65520 + 691 -1 Lao GII(m)q'", 111 Theta functions F = (21T)-12~ = n (l_qm)24 = L 00 q m= The theta function associated with the lattice r 00 7{m)Qm m= can thus be written (106) We have (107) 'r(m) 65520 - O'l1(m)+ CrT(m) 691 =- for m ~ I The coefficient Cr is determined by putting m = I: (l08) C 65520 r = 'r(l) - - - 691 Note that it is =1=0 since 65520/691 is not an integer Examples a) The lattice r constructed by J that 'r(l) = O Hence: b) For r = r8 $ Cr = r8 $ (Callad J Math., 16, 1964) is such 65520 691 r 8' Cr c) For LEECH = we have 'r(l) 432000 691 = 3.240, hence: = 27 35 3/691 r = r 24 , we have 'r(l) = 2.24.23, hence: Cr = 697344 691 = 21°3.227/691 6.7 Complements The fact that we consider only the full modular group G = PSL (Z), forced us to limit ourselves to lattices verifying the very restrictive conditions of n° 6.4 In particular, we have not treated the most natural case, that of the quadratic forms xi+ +x;, which verify (i) but not (ii) The corresponding theta functions are "modular forms of weight n/2" (note that nl2 is not necessarily an integer) with respect to the subgroup of G generated by Sand T2 This group has index in G, and its fundamental domain has two "cusps" to which correspond two types of "Eisenstein series"; using them, one obtains formulas giving the number of representations of an integer as a sum of n squares; for more details, see the books and papers quoted in the bibliography Bibliography Some classics C F GAuss-Disquisitiones arithmeticae, ISOI, Werke, Bd I (English translation: Yale Univ Press-French translation: Blanchard.) C JACoBI-Fundamenta nova theoriae lunctionum ellipticarum, IS29, Gesammelte Werke, Bd I., pp 49-239 G LEJEUNE DIRICHLET-Demonstration d'un thtore,ne sur la progressiOil arithmetique, IS34, Werlce, Bd I D 307 G EISENSTEIN, Mathematische Werke, Chelsea, 1975 B RIEMANN-Gesammelte mathematische Werke, Teubner-Springer-Verlag, 1990 (English translation: Dover-partial French translation: Gauthier-Villars, IS9S) D HILBERT-Die Theorie der algehraischer Zahlkorper, Gesam Ahh., Bd I pp 63-363 (French translation: Ann Fac Sci Toulouse, 1909 and 1910) H MINX.owSKJ-Gesammelte Abhant/lungen, Teubner 1911; Chelsea 1967 A HURWfrz-Mathematische Werke, Birkhiuser Verlag, 1932 E HECKE-Mathematische Werlce GoUingen, 1959 C L SlEGEL-Gesammelte Abhand/ungen Springer-Verlag 1966-1979 A WElL-Collected Papers Springer-Verlag 1980 Number fields and local fields E HEcKE-Algehraische Zahlell Leipzig, 1923 Z I BoREVICH and r R SHAFAREVICH-Numher Theory (translated from Russian) Academic Press, 1966 (There exist also translations into French and German.) M EICHLER-Einliihrung in die Theorie der algebraischen Zahlen ulld FUlllctionen, Birkhiiuser Verlag, 1963 (English translation: Academic Press, 1966) J-P SERRE-Corps Locaux, Hermann, 1962 P SAMufL-Theorie algebrique des nombres, Hermann, 1967 E ARTIN and J TATE-Class Field Theory, Benjamin, 1968 J CASSELS and A FROHLICH (edil.)-Algebraic Number Theory, Academic Press, 1967 A WElL-Basic Number Theory, Springer-Verlag, 1967 S LANG-Algebraic Number Theory, Addison-Wesley, 1970 (The four last works contain an exposition of the so-called "class field theory".) Quadratic forms a) Gelleralities, Witt's theorem E WITT-Theorie der quadratischen Formen in beliehigell Korpertt, J erelle, 176, 1937, pp 31-44 N BoURBAKI-Algebre, chap IX, Hermann, 1959 E ARTIN-Geometric Algebra, Interscience Pub!., 1957 (French translation: GauthierVillars, 1962) b) Arithmetic properties B JONf3-The arithmetic theory of quadratic forms, Carus Mon., n° 10, John Wiley and Sons, 1950 M EICHLER-Quadratische Formell ulld orthogollale Gruppen, Springer-Verlag, 1952 G L WATSON-Integral quadratic forms, Cambridge Tracts, nO 51 Cambridse, 1960 O T O'MEARA-Introduction to quae/ratic forms Sprinser-Verlag, 1963 J MILNOR and D HUSEMOLLER-Symmetric Bilinear Forms, Sprinaer-Verlag, J973 T Y LAM-ne algebraiC theory of quadratic forms, New York, Bel\iamin, 1973 W S CASSELS-Rational Quadratic Forms, Academic Press, 1975 112 113 Bibliography c) Int~gral quadratic forms with discriminant ± I E WITT-Ein~ Identitiit zwischen Modulformen zw~iten Grad~s, Abh math Sem Univ Hamburg, 14, 1941, pp 323-337 M KNESER-Klassenzah/~n d~finiter quadratischer Formen, Arch der Math 8, 1957, pp 241-250 J MILNoR-On simply connect~d manifolds, Symp Mexico, 1958, pp 122-128 J MILNOR-A procedure for killing homotopy groups of differentiable manifolds, Symp Amer Math Soc., n° 3, 1961, pp 39-55 J H Cm.wAY and N A SLOANE-Sphere Packings, Laltices and Groups, Springer-Verlag, 1988 Dirichlet theorem, zeta junctioll aM L-!unctiollS J HADAMARD-Sur la distribution des zeros de la fonction '(s) et ses consequences arith· mhiques, 1896 Oeuvres, CNRS, t I, pp 189-210 E LANDAu-Handbuch der Lehre von der Verteilung der Primzahlen Teubner, 1909; Chelsea 1953 A SELBERG-An elementary proof of the prime numb~r theorem for arithmetic progressions, Canad J Math • 2, 1950, pp 66-78 E C TITCHMARSH-The Theory of the Riemann zeta·function, Oxford, 1951 K PRACHAR-Primzahh'erteilung, Springer-Verlag, 1957 H DAVENPORT-Multiplicative number theory second edition, Springer-Verlag, 1980 K CHANDRASF.KHARAN-Introduction to allalytic number theory, Springer,Verlag, 1968 A BLANCHARD-Initiation Q 10 theorie analytiqut' des IIombres premiers, Dunod, 1969 H M EDWARDs-Riemann's zeta function, New York, Acad Press, 1974 W NARKIEWlcz-Elementary alld analytic theory ofalgebraic numbers, Warsaw, Mon Mat 57, 1974 W ELLlsoN-Les Nombres Premiers, Paris, Hermann, 1975 Modular jUllctiolls F KLEIN-VoriesuIIKen "ber die Theorie der elliptischen Modulfunktionell, Leipzig, 1890 S RAMANUJAN-On certain arithmetical functiOns Trans Cambridge Phil Soc., 22 1916 pp 159-184 (= Collected Papers, pp 136-162) G HARDY-Ramanujan, Cambridge Univ Press, 1940 R GODEMENT-Trat'aux de Heck~, sem Bourbaki, 1952-53, exposes 74.80 R C GU!'INING-Lectures on modular forms (notes by A Brumer), Ann of Math Studies, Princeton 1962 A BOREL et al.-Semillar on complex multiplication Lecture Notes in Maths • n° 21, Springer·Verlag, 1966 A OGG-Modlliar fornlJ and Dirichlet series, Benjamin, 1969 G SHIMuRA-/",roductioll to the arithmetic th~ory of automorphic jUllctiolls TokyoPrinceton, 1971 H RADEMACHER-Topics in Analytic Number Theory Springer·Verlag, 1973 W KuYK et al (edit.)-Modular Functions of One Variable I, , VI, Lecture Notes in Math., 320, 349, 350,476.601.627 Springer-Verlag, 1973-1977 P DELIGNE-Lu cOnjecture de Weill, Publ Math I.H.E.S., 43, 1974, p 273-307 A WElL-Elliptic Functions according to Eisenstein and Kronecker, Springer-Verlag 1976 S LANG-Introduction to Modular Forms, Springer·Verlag, 1976 R RANKIN-Modular Forms and Functions, Cambridge Univ Press, 1977 (See also the works of HEeKE, SIEGEL and WElL quoted above.) Index of Definitions Abel lemma: VI.2.1 approximation theorem: 111.2.1 Bernoulli numbers: VII 4.1 character of an abelian group: VI.I.I characteristic (of a field): 1.1.1 Chevalley theorem: 1.2.2 contiguous basis: IV.1.4 cusp form: VII.2.1 degenerate (non quadratic form): IV.I.2 density of a set of prime numbers: VI.4.I density natural: VIAS Dirichlet series: Vl.2.2 Dirichlet theorem: 111.2.2 VI.4.1 discriminant ora quadratic form: IV I I dual of an abelian group: VI I I Eisenstein series: VU.2.3 elliptic curve: VII.2.2 Meyer's theorem: IV.3.2 Minkowski-Siegel formula: V.2.3 modular character: VI.I.3 modular function and form: VII.2.1 modular group: VII.I.I multiplicative function: VI.3.1 orthogonal direct sum: IV.1.2 V.1.2 p-adic integer: 1/.1.1 p-adic number: 11.1.3 p-adic unit: 11.1.2 Poisson formula: VII.6.1 primitive vector: 11.2.1 product formula: 111.2.1 quadratic form and module: IV.1.1 quadratic reciprocity law: 1.3.3 fundamental domain of the modular group: VII 1.2 Ramanujan conjecture: VII.S.6.3 Ramanujan function: V11.4.5 represented (element by a quadratic form): IV.1.6 Hasse-Minkowski theorem: IV.3.2 Hecke operators: VII.S.I VII.S.3 Hilbert symbol: 111.1.1 signature of a real quadratic form: IV.2.4 invariants of a quadratic form: IV.2.1 V.I.3 isotropic vector and subspace: IV.1.3 lattice: VII.2.2 Legendre symbol: 1.3.2 L function: VI.3.3 theta function of a lattice: VII.6.S type of a quadratic form: V.I.3 weight of a modular function: VII.2.t Witt's theorem: IV.I.S Zeta function: VI.3.2 114 Index of Notations Z, N, Q, R, C: set of integers, positive integers (0 included), rationals, reals, complexes A *: set of invertible elements of a ring A Fq: field with q elements, 1.1.1 (;): Legendre symbol, 1.3.2, 11.3.3 «n),