A p p e n d i x : p - a d i c n u m b e r s

In § 7. we saw tha t the moduli space of curves over the complex numbers

comes in a na tu ra l manner with an ar i thmet ic s t ructure . Jus t to remind you let

me repea t the following facts. If C is a curve t then it is the set of common zeros of

finitely many homogeneous polynomials . For example~ a curve in the plane can

be given by the zeros of one homogeneous polynomial . If after a sui table change of coordinates the polynomials can be given in such a way tha t all coefficients are

rat ional numbers then we say that the curve C can be deigned over the rational

numbers Q.

If g is a ra t ional polynomial of degree k we can write it as

E si g = r l m i : r i = - -

where m~ are monomials (just products of the variables) and the sl and t~ are integers. If we mul t ip ly now g by the least common mult iple t of the tl we get

= t . g = E , ~ "mi i

with ~i integers. Of course g and ~ have the same set of points as zeros. Hence

every curve defined over the rat ional numbers Q can be defined over the integers

Z. Hence we can use all number theoret ic techniques to get informat ion on

the geometr ic (and ar i thmetic) s t ructure . ( This works also in the opposi te direction.)

Now, how do we get from the rat ional numbers to the complex numbers?

I guess this will be well-known to most of you. But the technique is also essential

in the case of the p-adic numbers so let me repeat it briefly. For the rational

numbers we have the so called absolute value. It allows us to define a topology

and the not ion of a convergent sequence. As you know there exist sequences of

rat ional numbers which "converge" bu t have no limits in Q (of course strictly

speaking we do not talk about convergence in this case). We enlarge Q by adding

these limit points to get the real numbers ]R.

One way to cons t ruc t ]R in more precise terms is the following. We call a

sequence (xn) a Cauchy sequence if for every rat ional number e > 0 there exists

a na tura l number no such that

I z ~ - zrn[ < e if n, m > no.

135

As remarked above not every Cauchy sequence has to have a limit. But if it has

a l imit and this limit is 0, then we call (~n) a zero sequence..

We can define addit ion, subt rac t ion and mult ipl icat ion on the set of Cauchy

sequences elementwise, for example

The real numbers are now defined as the set of classes of Cauchy sequences

modu lo zero sequences. This means that we identify 2 Cauchy sequences (xn)

and (z,~) if (x,~) - (z,~) is a zero sequence. All operat ions can be well defined on

these classes. In addi t ion if [(x,~)] is not the zero class (equivalently (x,~) is not a

zero sequence ) we can define t he multiplicative inverse [(m~)] -1 in the following

way. By adding a suitable zero sequence to (~r,) we can arrange tha t for the

new sequence (still denoted by (z,~)) x,~ ¢ 0 for all n. In this way we s tay in the

same class. We set

[(=.)]_1 := [ ( 1 ) 1

which is again a Cauchy sequence, because the ~,~ are bounded away f rom O.

It is easy to see that [R will be a field and that Q.is embedded as a subfield

via the constant sequence

[ (= . ) ] for all = . = = e Q.

Now we can define also the absolute value in IR by sett ing

:=

W i t h this we see tha t in [R all Cauchy sequences have a limit. Hence we call IR

the complet ion of Q with respect to the absolute value !-I-

F rom the ma themat ica l viewpoint tR is not yet fully satisfying: we cannot

solve every algebraic equation. In technical terms [R is not algebraically dosed. For this reason we have to "add" a single element, the root i of the polynomial

equat ion

x 2 + l = 0 .

It obeys the mult ipl icat ive law i 2 = - 1 . In this manner we get the complex

numbers C. We can extend the absolute value in IR to the usual complex absolute

value. C is now an algebraically closed complete field.

136

But what forces us to stick to this special absolute value to s tar t f rom the

ra t ional numbers! Let us list the characteristic features of the absolute value

which allowed us to work as above.

l . l : Q -'-* Q

with

(1) I,~1 _> o, a ~ Q, I,~[ -- o

(2) I~ .b l = I,~1. Ibl, ,~,b e Q

(3) la + bl < I~1 + Ibl, ~,b ~ Q

if and only if a = 0 .

(multiplicative law)

(triangle ident i ty)

If we have such a map we call it a valuation for the field Q. We can generalize

this immedia te ly to arbi t rary fields (for the values we remain in Q or ]R).

There is one map , the so called trivial valuation, defined by

1, if a # 0 [ a [ - 0, if a = 0 .

This valuat ion is of no interest to us. If we use the word valuat ion we always

assume a nontr ivial one.

Let us consider a very interesting one. If

n r - - = - - E Q , r ~ 0 , n, m E Z

m

then we can write

n = 2 S n ~, m - - 2 ~ m ~

where n t and m ~ are integers which are not divisible by 2. We set

I,'1~ := 2 ~-" and Io1~ := o.

Let us check the conditions above.

Condi t ion 1: is clear.

Condi t ion 2: Normalized as above we get

r 1 m m

I nl 2l n 1 ~.̂ n2 2p n~

1 9 m l m 1 m 2 m 2

[r112 " 1r212 = 2-z2-P = 2-U+P) I I

r l r 2 = 2 l + P . n l n 2 I I

m l m 2

137

Because 2 is a pr ime number it divides neither n l n 2 1 i nor m l m 2 . ~ I Hence we get

[rlr2]2 : 2-(l+p).

This is what we had to show. Condi t ion 3: We keep the same nota t ion as above. We separate two cases.

(l ¢ p): wi thout restr ict ion we assume I < p.

, , , , 2P-ln~mll + r 2 : 2 l - ( n - - A : -1. + 2 p-l-n2.) = 2 l" n l m2-k ?'1 I !

m l m 2

Now 2 divides nei ther the numera to r (because 2 divides only one t e r m in the

sum) nor the denominator , hence

I~1 + ~1~ = 2 -z = I~l~ < I~1~ + I~1~.

(I ---- p): here we get I I I I

r l -}- ?'2 ---- 2 l " • 1 m 2 J r r t 2 m 1 i !

m l m 2

It can happen tha t 2 divides the numera tor , hence

with q a positive integer.

In bo th cases condit ion 3 is fulfilled. Even more is true. We can replace (3) by

(3')

and we know in addit ion

[rl + r,. I = max(It1 I, Ir~ [)

It1 + r 2 l _< max(It1[, ]r2[)

If the s t ronger condition (3') is valid we call l-I a non-arch lmedian valuation.

In a non-archimedian valuation the integers are always within the "unit circle".

This we can see as follows. We have Ill = 1 as usual (lal = I I . a l = 111. lal). But

n o w

121 = 11 + 11_ max(Ill, II1)= II1-- 1.

By induct ion we get In[ _< 1 for all integers n. This is a ra ther s t range result in

contrast to the usual absolute value.

138

In the above the only fact we used about the number 2 was tha t it was a pr ime

number . Hence we can define in an identical manne r

17. t

lal~-- p - ' if a = p ' . - - 101~ = 0 m t

where p divides ne i ther n ' nor m ' . We call [. lp the p-adic valuation. If there is

a danger of confusion we use [. [oo for the usual absolute value.

We can define a topology, convergence, Cauchy sequences and so on. Wi th

this we see tha t all these valuations are essentially different. To see this let us

consider the sequence (=~) = (p~)

for a fixed pr ime number p. Now

[p'~ [oo = p'~

and hence it is unbounded and diverges in the usual topology. On the other

hand

Ip~lp = p - ~ , Ip~lq = 1, for q # p.

We see (z,~) is a zero sequence in the p-adic topology and a bounded (but not

zero) sequence in the q-adic topology.

We call two valuations [. [a, [. [b equivalent if we have

I . l ° = J . I ;

with a positive real number r: i.e. if they define the same topology.

THEOREM. (Os$rowskJ) I . I~ and l. Ip, (P ~I prime numbers), are represen~atlves of ~1 equ~v~ence classes of valuations for the rational number t~eld Q.

In the same way as we const ructed the real numbers [R via Cauehy sequences with respect to the absolute value we construct the p-ad/c numbers Qp star t ing

with the p-adic valuat ion for every pr ime number p. Our field Q lles in all ' these completions:

In Qp we can do analysis similiar to real analysis. Jus t as we can represent every

real number as a decimal expansion, we can represent every p-adic number by

139

a p-adic expansion. Take a E Qp, by extension of the p-adic valuat ion to Qp we

calculate

[alv = p- '~, m E Z.

W i t h this we can wri te

with

a

O o r~

/¢>m k>m

E { 0 , 1 , 2 , . . . , p - 1}.

As you see the sum goes in the opposite direction of the decimal expansion.

In some sense the analysis in QP is much easier than that in IR but more

unfamilar . For example we have in QP the following

LEMMA. A series ~ k ak converges i f and only i f the sequence (ak) is a zero sequence.

(Qui te a few students s tar t ing to learn mathemat ics would be happy if this would work a/so in IR.)

PROOF: Let (ak) be a zero sequence. T h e n (rn _> n)

Because (ak) is a zero sequence the right hand side can be made as small as one

wants by choosing n big enough. This means than tha t the sequence of part ial

sums is a Cauchy sequence and hence has a limit. The other direction is the usual a rgument of real analysis.

Now we can define also p-adic power series like the exponential

and the logarithm

oo a~ k

k = O

o o

k + l

k = l

140

Despite the fact that the expressions are the same you might expect that the

domain of convergence will be different. We write

I=b = p- ordp(=)

to define the p-adic order. With this we can calculate that E(z) is convergent if

and only if ordp(z) > ~ and log ( l+z ) is convergent if and only if ordv(z) > 0.

This is again a rather strange behaviour. I fp # 2 and z is an integer then E(x)

is a convergent series if and only if = is a multiple of p. If p = 2 and everything

else stays the same, then z has to be a multiple of 4.

One can now introduce p-adic measures, distributions, integrals and Fourier

transformations 1. Let me mention that in Qp there is in contrast to IR still a lot

of number theory involved. For example, we have also the p-adic integers which

are defined to be the z E Qp with ordp(x) > 0. Like in the case of the rational

numbers we can write every element of QP as quotient of two p-adic integers.

We are still not at the end in our construction of the analogue of C. To get from IR to C we took the algebraic closure of IR. This one can do also with

Qp. In this case we have to "add" infinitely many roots of polynomials to get O , . C . O, .C to Qp . Of course this Qp also admits an extension ~)f the p-adic valuation.

But in contrast to C it is no longer complete. We have to construct again the completion of this field. The resulting field is algebraically closed and complete.

Of course, one might ask why we should bother about the p-adic numbers.

Some idea of their importance migh'~ be given by the following product formula

for all rational numbers x ~ O,

I=1oo l q l=lp = 1. pelP

(We use ]P to denote the set of all prime numbers.) It says, if we know all p-adic values of a rational number we also know its absolute value. This is a

reformulation of the trivial fact that we know the absolute value of a rational

number if we know how often all primes appear in the numerator and and in the denominator. But there is more behind it. The mathematical idea is that the

rational numbers are the objects of primary interest. But they are very difflcult

to handle from the arithmetic viewpoint. IR and all Qp are much easier. Each of them reflects one facet of the complexity of Q. From this viewpoint one calls

Q a global field and IR and the different Qp local fields. If z E q is a number

1see Koblitz p.30

141

fulfilling a suitable relation (for example to be a zero of an polynomial equation with integral coefficients) then z considered as element in fR and Qp clearly fulfills the t ransformed relation in IR, resp. in Qp. The converse problem is the one really tha t mat ters . If we can solve the t ransformed relations by elements zoo and zp, p E IP (which in most cases is easy) we can ask under which conditions we can assemble this information to get a solution for the star t ing relation in Q. (This is also known under the name 1ocaJ-global principIe.)

In the above you might get the (right) feeling that it is bet ter to consider the real number field and all p-adic number fields simultaneously. This is formalized in the concept of ad~les. We start with the infinite product

IR x ] - [ Qp. p~tP

It consists of all infinite sequences

( $ c o ~ ;~2~ ~3~ Z5~ • • • ~ X p ~ . . . )

where zoo E IR, xp E Qp. Now we consider the subset with

I=lp _< 1

for almost every p (which is shor thand for: the condition can be violated only by a finite number of primes). This subset is called the set of ad~les and is denoted by A Q . It is a ring if we define addit ion and multiplication componentwise. The set of multiplicative invertible elements in AQ are called id~les. Q is now embedded diagonally into AQ by

= (:,=,:,...).

At the first sight this is only a formal tool. But because this is a ring we can do algebraic geometry with it and other nice things. The following is an example from Manin (details can be found there). One can define the group S1 (2, AQ) in

a suitable manner . By diagonal embedding S1 (2, Q) is a discrete subgroup. Now the factor group is a compact group on which we can integrate. By normalizing the measure and splitting it up into the factors corresponding to factor groups of S1 (2,1R) and S1(2, qv) we get the result

7r 2

1-- y . H (1- p~IP

142

This relates the ar i thmet ical ly defined product expression with the t ranscenden- tal number 7r. The product is an object which is defined by knowing the measure

on the p-adic groups and it determines the analogous measure on the real group.

Of course the above formula is not a new result gained only by the use of addles.

But the addles give a new insight. Classically it comes f rom the Riemann zeta

function ±

where we have 71.2

- ¢ - 1 - [ ( 1 -

pelP

which is the above relation.

Up to now we have done everything over the rat ional numbers . In fact this is

not sufficient. We have to allow finite algebraic field extensions of Q. We can obtain these in the following way. We embed Q in C and take an a E C which is

a root of a ra t ional irreducible polynomial f . (Here irreducible means it doesn ' t

have a factorisat ion into two nonconstant rat ional polynomials .) The imaginary

uni t i for example is such an a. It is the zero of the polynomial

X 2 + 1 .

Another example is 27ri

= = e x p ( - T - )' a so called 3rd root of unity. It is a zero of the polynomial

X 2 + X + l .

We denote the smallest subfield of C containing Q and a by Q(a) . It is also a

finite dimensional Q-vector space. Its basis is given by

1, a , a 2 , . . . , a 'n, m = d e g f - 1 .

Such a field is called a number field. Jus t as we can build Q out of the integers as their quotients, we can find "integers" in Q(a) which do the job here. In

contrast to the usual integers , the rational integers, these are called algebraic integers. With these algebraic integers we can do ar i thmet ic , define divisibility

and so on. Unfor tuna te ly we have in general no unique factorizat ion of every

algebraic integer into primes. Even worse, we have to drop the notion of prime

143

number at all. But there is a rather useful substitute for the prime numbers. The prime numbers of Z are in a 1:1 correspondence with the classes of non- archimedian valuations of Q. Now we ask for such non-archimedian valuations for Q(a). Let I. I be such a valuation. By restricting it to Q we get one of the p-adic valuations. Conversely one can show that for every p-adic valuation on Q, there exist only finitely many extensions to the whole field Q(a). By the embedding the absolute value is already extended to the usual complex absolute value on Q (c~). Different extensions of the absolute value correspond to different embeddings. There are also finitely many of them.

Now we can do everything (completion, ad&les,...) we did for Q for this number field Q(c~). The corresponding complete field is a more tractable subfield of the big p-adic algebraically closed and complete field.

H in t s for F u r t h e r R e a d i n g

For further reading consult

1. Koblitz Neal, "p-adic Numbers,p-adic Analysis and Zeta Functions", Springer t 1977.

It contains the basic concepts including p-adic integration. Of course you also find the basics in other books like

2. Bachmann G., "Introduction to p-adic Numbers and Valuation Theory", Academic Press, 1964.

3. Van der Waerden B.L., "Algebra II", Springer, 1967.

Ad&les can be found in

4. Weil,A., "Basic Number Theory", Grundl.Math.Wiss.144, Springer, 1967 (3.rd rev.ed. 1974).

5. Tamagawa,T., Addles, Proc.Symp.Pure Math. 9 (1966), 113-121, Amer. Math. Soc., Providence.

Concerning the more speculative aspects of application of p-adic numbers to string theory, I refer only to

5. Manin Y.I., Reflections on Arithmetical Physics, Talk at Poiana-Brasov School on Strings and Conformal Field Theory, 1-14. Sept.1987.

I n d e x

abelian variety 64~99 abelianization 17 absolute value 134

addles 141 affine variety 94~97 algebraic bundles 89,93 algebraic differentials 67 algebraic integers 142 algebraic map 59,60 algebraic variety 58,94,97 algebraically closed 135 analytic isomorphism 9~26 arithmetic structure 82~134 associated sheaf 119 atlas I

automorphisms of curves 70,77~81 base variety 67 Belavin-Knizhnik theorem 123 Betti number 17 biholomorphic map 9 boundaries 16 boundary of AA# 78

boundary map 15 bundle corresponding to a point 104

cl 103. canonical bundle 105 canonical bundle of the modull

space 122 canonical divisor 31,41 canonical divisor class 41~76,111,131 canonical homology basis 21~51~54 Cauchy sequence 134 Cauchy-Riemann differential

equations 4 Cech cohomology 24,94,96 Cech-cycles 94

chains 15~45 Chern character 127 Chern classes 126 chern form 06 Chern number 98 Chow's theorem 60 Chow ring 126 Chow ring of a point 130 classification 7 closed differential form 44 closed sets of IP 1 57 coarse moduli space 69 coboundary 96 coboundary map 95 cochain 95 cocycle 86~96 cocycle conditions 86 coherent sheaf 93 cohomologous cocycles 87 cohomology 94 cohomology ring 131 cohomology sequence 97 commutator subgroup 17 compactification of the moduli

space 77 comparison theorems 97 complex manifold 4 complex numbers 135 complex projective space 4 coordinate change map 3 coordinate patch 1 cotangent bundle 39,76~88 cotangent bundle the moduli

space 122 covering 22,74 covering transformation 23 cubic curve 61,73 curve defined over the rational

numbers 79~134

145

curves with elliptic tails 81 cusp 79 cuspidal cubic 79 cycles 16 defining cocycle 87 degree of a divisor 31 degree of a line bundle 106 de -Rham cohomology group 44 derivation 37 derived functor 94,97 Diff 75 Diff0 75 diffeomorphism 8 differentiable i somorphism 8 differentiable manifold 3 differentiable s t ructures on ]R 4 9 differential 88 differential form 38,44,88 differential form of second order dimension of a manifold 1 dimension of a variety 59 direct image sheaves 119 discriminant funct ion 35 divisor 30,41,59,94,99 divisor class 30 divisor of a section 102 doubly periodic meromorphic

funct ion 34 double point 78 dual of a vector bundle 88 Eisenstein series 35 elliptic curve 61,73 embedding of tori 61,63 Euler-Poincar~ characteristic 19 exact differential form 44 exponential sequence 92,98 exterior differentiation 43 exterior power 43

43

exterior product of a vector bundle 89

family of curves 67 family of vector spaces 85 field extension 142 field of meromorphic functions 26,29

fine moduli space 70 fractional linear t ransformat ion 74 free abelian group 30 fundamental group 11,19,23,74 genus 8,18,31,67 geometric invariant theory 77 global field 140 global holomorphic differential 42 glueing funct ion 3 Grothendieck group 125 G rot hendleck-Pdemann-Roch

theorem 128 harmonic 44 Haussdorff space 1 higher dimensional torus 48 higher direct image sheaves 119 Hirzebruch-Riemann-Roch

theorem 131 Hodge bundle 80,81,82,122 Hodge class 80 holomorphlc bundles 89,93 holomorphic differentials 40,42,47,

51,67,76,80 holomorphic form 40 holomorphic function 25,28 holomorphic functional

determinant 4 holomorphic functions on M g 77 holomorphic map 26,60 homeomorph i sm 8 homogeneous coordinates 5

homogeneous polynomial 56 homomorph i sm of families 85

146

homotopy 10 id~les 141 integration 45 integration pairing 46 intersection product 20 irreducible variety 58 xsomorphism of principally

polarized tori 48 isomorphy of families 68 isothermal coordinates 7 isotopy 74 3acobi map 53 3acobian 52 j-function 35,72,79 Krichever-Novikov algebra 115 Kodaira's embedding theorem 60 lattice 33,48 Lie algebra sl(2, C) 110 line bundle 81,86,98,99 linear variety 58 local field 140 local-global principle 141 locally free resolution 125 locally free sheaf 93 loop 10 mapping class group 75 meromorphic differential 40,46,

60 meromorphic form 40 meromorphic function 26,60 meromorphic function on IP 1 29 meromorphic function on

the torus 33,34 meromorphic sections of vector

bundles 101,107 meromorphic vector field 115 metric 7 modular forms 82 modular function 72

module 88 moduli space 67,122 multiplicity of a pole 27 multiplicity of a zero 27 Mumfor d-Deligne-Knudsen

compactification 78 Mumford isomorphism 122,130 nodal cubic 79 nodes 78 noetherian ring 58 non-archimedian valuation 137 nonsingular variety 60 normal variety 77 number field 83,142 ord 40 order of a meromorphic function 30 orientable manifold 6 orientation 15 Ortsuniformisierende 25 Ostrowski's theorem 137 p-adic expansion 139 p-adic integers 140 p-adic numbers 138 p-adic order 140 p-adic power series 139 p-adic valuation 137 paracompact 1 partition of unity 2 path 10 period matrix 51 Picard group 31,80,81 Picard group of A4g 80 Picard group of the moduli

functor 81,122 Picard variety 53 polarization 48,64 Polyakov form 124 Polyakov integration measure 123 Polyakov partition function 123

147

presheaf 119 prime form 55 principal divisor 30 principally polarized abelian

variety 75,82 principally polarized tori 48,63,75 projective line IP 1 5,42,57,70 projective nonsingular curve 60 projective space 4,56,94 projective variety 58,60,99 proper algebraic map 126 pullback 48,89 pullback of a family 69 q-expansion 72 quadratic differential 76,112,121 quasiperiodic function 63 quasiprojective variety 58 ramified covering 22 rational differential 60 rational equivalence 126 rational function 59,60 rational integers 142 rational point 83,83 real analytic manifold 3 real numbers 134 relative differentials 120 relative tangent sheaf 120 residue theorem 41 residuum 40 restricted Picard group 31,53 restriction map 90 R.iemann bilinear relations 51 Riemann sphere 5,42 Riemann surface 7,61 Riemann-Roch theorem 31,107,131 Riemann zeta function 142 Satake compactification 77 Schottky problem 75 section of a vector bundle 87

separated topological space 1 Serre duality 99,107 sheaf 90 sheaf cohomology 24,94 sheaf homomorphism 91 sheaf of O-modules 91,93 sheaf of differentlable functions 91 sheaf of holomorphic functions 91 sheaf of locally constant functions sheaf of n-differential forms 99 sheaf of regular functions 91 sheaf of sections 90,93 short exact sequence of

sheaves 92,97 Siegel upper-half-space 62,75 mmplex 15,45 mmplicial homology group 16 rumply connected 12 singular homology 24 singular point 60,61 singularities of Mg 77 smooth variety 60 sphere S 2 12,18 sphere S 7 9 sphere S r~ 3 stable curves 77 standard coordinates 25 Stokes' theorem 45 structure sheaves 91 subvarieties 94 sum of vector bundles 88 symplectic basis 49 symplectic group 50,54,72,75,82 tangent bundle 39,76,85,88 tangent space 37 tangent bundle the moduli

space 122 tangent space of Mg 76 Teichmiiller space 74

90

148

Teichmfiller surface 74

tensor 89 tensor product of vector bundles the ta function 53,62,99 the ta function with

characteristic 64 the ta null values 82 t odd class 128 topological genus 18 topological i somorphism 8 topological normal form 17 Torelli's theorem 53,75 torus 12,18,33,42,48,71 t r iangulat ion 14 trivial valuation 136 trivializing covering 86 universal covering 22,74 universal family 70 upper halfplane 23,74 valuation 136 variety 57,58 vector bundle 85,86,101,126 vector field 38,88,109 vector fields on a torus 110 vector fields on ]p1 109 vector fields on S 1 117 very ample line bundle 99 Weierstraf~ ~o-function 34 Weierstrat] point 112 Weil-divisor 59 Zariski topology 57 zeta-function regulation 124

88

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VoI. 305: K. Nomoto (Ed.), Atmospheric Diagnostics of Stellar Evolution: Chemical Peculiarity, Mass Loss, and Explosion. Proceedings, 1987. XIV, 468 pages. 1988.

Vol. 306: L. Blitz, F.J. Lockman (Eds.), The Outer Galaxy. Proceedings, 1987. IX, 291 pages. 1988.

Vol. 307: H.R. Miller, P.J. Wiita (Eds.), Active Galactic Nuclei. Proceedings, 1987, Xl, 438 pages. 1988.

VoL 308: H. Bacry, Localizability and Space in Quantum Physics. VII, 81 pages. 1988.

Vol. 309: P.E. Wagner, G. Vail (Eds.), Atmospheric Aero- sols and Nucleation. Proceedings, 1988. XVlII, 729 pages. 1988.

Vol. 310: W.C. Seitter, H.W. Duerbeck. M. Tacke (Eds.), Large-ScaJe Structures in the Universe - Observational and Analytical Methods. Proceedings, 1987. II, 335 pages. 1988.

VOI. 311: P.J.M. Bongaarts, R. Martini (Eds.), Complex Differential Geometry and Supermanifolds in Strings and Fields. Proceedings, 1987. V, 252 pages. 1988.

VoL 312: J.S. Feldman, Th.R. Hurd, L Rosen, "QED: A Proof of Renormalizability." Vtl, 176 pages. 1988.

VoL 313: H.-D. Doebner, T.D. Palsy, J.D. Hennig (Eds.), Group Theoretical Methods in Physics. Proceedings, 1987. Xl, 599 pages. 1988.

Vol. 314: L. Peliti, A. Vulpiani (Eds.), Measures of Complexi- ty. Proceedings, 1987. VII, 150 pages. 1988.

Vol. 315: R.L. Dickman, R.L. Snell, J.S. Young (Eds.), Molecular Clouds in the Milky Way and External Galaxies. Proceedings, 1987. XVI, 475 pages. 1988.

Vol. 316: W. Kundt (Ed.), Supernova Shells and Their Birth Events. Proceedings, 1988. VIII, 253 pages. 1988.

Vol. 317: C. Signorini, S. Skorka, P. Spoiaore, A. Vitturi (Eds.), Heavy Ion Interactions Around the Coulomb Barrier. Proceedings, 1988. X, 329 pages. 1988.

Vol. 319: L. Garrido (Ed.), Far from Equilibrium Phase Transitions. Proceedings, 1988. VIII, 340 pages. 1988.

VoL 320: D. Coles (Ed.), Perspectives in Fluid Mechanics. Proceedings, 1985. VII, 207 pages. 1988.

Vet. 321: J. Pitowsky, Quantum P robab i l i t y - Quantum Logic. IX, 209 pages. 1989.

Vol. 322: M. Schlichenmaier, An Introduction to Riemann Surfaces, Algebraic Curves and Moduli Spaces. Xlll, 148 pages. 1989.