In our (rather applied) theoretical physics research, we have encountered an important class of problems, which seem to require an understanding of Abelian functions (unfortunately, this subject is not a part of standard theoretical physics education, so we know little). I would like to learn the basics on my own to see if there are some results that we can actually use in our research. Is there a sort of "crash-course" on the practical aspects of the subject? I am looking for a book with the minimal level of "abstraction;" e.g., that would not include words such as "morphism" etc., but would include examples of practical calculations and/or a list of key technical tools.
Thank you.

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    Harry - I apologize if I offended you by trying to avoid "morphisms." I'm sure that they are very useful in learning the real meaning of things, but let's just say that I am not intellectually adept to go into such depths, but I just would like to get a feeling about a range of technical tools, if any, that the modern theory provides to simple-minded people such as I. My naive impression (based on googling things for 2 hours) was that Abelian functions and Abelian varieties are not unrelated. Whatever the theory is called - that generalizes elliptic functions - I am looking for a book on it – Victor Galitski Jan 5 '11 at 16:32
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    Victor, Abelian varieties and Abelian functions are indeed related. The latter are very roughly functions on the former. Even though I'm an algebraic geometer, I don't agree that you need to master the whole subject to make use of this class of functions. Hopefully the references below will get you started. Good luck. – Donu Arapura Jan 5 '11 at 16:40
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    Harry, I don't think you should interpret Victor's request regarding the word "morphism" so literally. Probably in Victor's situation, he wants to know about computations involving things like e.g. explicit power series. So he is just looking for books that spend less time on "abstract" stuff that he probably doesn't need for his present purposes. – Kevin H. Lin Jan 5 '11 at 17:55
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    There are more things in heaven and earth, Horatio, than are dreamt of in your philosophy. – Emerton Jan 5 '11 at 18:31
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    I think Harry should note that David Mumford, for example, knew when to be abstract, when concrete in geometry. Preconceptions don't help, let's all agree. – Charles Matthews Jan 5 '11 at 19:06

13 Answers 13

You could try looking at the first chapter of Mumford's book Abelian varieties. I forget whether or not it uses the word morphism, but it does adopt a resolutely complex analytic view-point, which will probably be (close to) the view-point you want.

He begins with a discussion of complex tori, i.e. quotients of the form $\mathbb C^g/\Lambda$, where $\Lambda$ is a lattice of rank $2g$, and then considers the problem of embeddding such a quotient into some complex projective space. The existence of such an embedding is what distinguishes abelian varieties from random complex tori, and is intimately related to the theory of theta functions. (Indeed, theta functions provide the embedding.)

I think that Mumford probably does use the terminology of line bundles. Line bundles and their sections come into the picture because to give an embedding of some variety $X$ into projective space, you need (more-or-less) to choose some homogeneous coordinates on $X$. But since the homogeneous coordinates of a point are not quite well-defined (they are only well-defined up to a scalar) the homogeneous coordinates are not quite functions, but only functions well-defined up to a certain scalar transformation, which exactly makes them be sections of a line bundle. In the case of abelian varieties, you can pull these sections back from $\mathbb C^g/\Lambda$ to $\mathbb C^g$, where they do become honest functions, but instead of being invariant under $\Lambda$ (if they were invariant under such translations, they would descend to be honest functions on the abelian variety $\mathbb C^g/\Lambda$, which they are not), they transform by some scalar when you translate the argument by an element of $\Lambda$. When you figure out exactly what the right scalar transformation law is, you find that you are talking about theta-functions!

My memory is that Mumford does explain this fairly concretely in his first chapter. Even if it doesn't stand alone as an explanation, it may be a helpful bridge between the very classical description that you are likely to find most accessible, and the view-point that you will find in most of the modern mathematical literature.

Let me add that (at least in my view) follow-up questions about specific points of the theory (say in reference to one of the texts that you ultimately decide to look at) would be quite welcome.

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    Dear Matthew, I have been wanting to say this for some time now and this seems as good an opportunity as any to say it: browsing MO, I am repeatedly blown away by the amount of lucid insight in down to earth terms that you manage to squeeze into a few hundred characters! – Alex B. Jan 6 '11 at 5:03
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    You should also have a look at Mumford's Lectures on Theta Ch. 1 and Ch. 2. They are extremely concrete. – Jon Yard Jan 6 '11 at 5:46
  • Dear Alex, Thanks for your kind words. Regards, Matthew – Emerton Jan 6 '11 at 6:58
  • I would like to second Alex's comment! Also, I looked at the first chapter of Mumford's book, and it has the word "morphism" exactly once. – Kevin H. Lin Jan 6 '11 at 9:52

Topics in Complex Function Theory, Abelian Functions and Modular Functions of Several Variables by C. L. Siegel is a standard reference using complex function theory. There are older works (e.g. H. F. Baker) that may give an approach to a given problem by formulae, and not all aspects of that concrete theory are easily found in the modern literature.

That said, it's a deceptive subject. I have to assume you are familiar with classical elliptic function theory, or else there is really no way to understand what's going on. Roughly speaking because zeroes and poles of functions of two or more complex variables are never isolated points, it can be quite hard to generalise a result you want from elliptic functions to abelian functions. In other words the zeroes and poles of functions carry geometry, and cannot be used to do book-keeping in such a simple-minded way.

One solution to that issue is to express everything in terms of theta-functions. There one's luck changes: basically the one-dimensional and higher-dimensional case are both ruled in the same way by a type of Heisenberg group, and (as David Mumford showed) you can see everything in the theory ultimately coming back to a form of Stone–von Neumann theorem. Now reading Mumford's papers is not the crash course you are looking for! That probably doesn't exist. It's just a reassurance that there is an underlying structure to the underlying equations (you can't expect abelian varieties to be complete intersections, outside a few classical cases).

It is probably true to say that abelian function theory doesn't have an adequate literature, in fact.

  • Thank you, Charles - Just ordered the Siegel's book. I am familiar with elliptic functions and what you said about the theta-functions and higher-d generalizations sounds very relevant. In fact, that's exactly what we've done: re-expressed a key quantity in our non-string theory in terms of theta-functions and we'd like to proceed further. Reading math literature, we get the impression that there exist many technical results and methods, but we admittedly don't understand the terminology etc. that seem to contain them. Any useful reference (e.g. applied math review of proceedings) would help. – Victor Galitski Jan 5 '11 at 16:13
  • There is plenty more that can and has been said for small d or the hyperelliptic case. If you could post more on the context, it might be possible to give more of a steer. – Charles Matthews Jan 7 '11 at 8:12
  • For some more context, try also mathoverflow.net/questions/26551/…. – Charles Matthews Jan 7 '11 at 10:57

You could try to look at the following book :

From number theory to physics. Edited by M. Waldschmidt, P. Moussa, J. M. Luck and C. Itzykson. Springer-Verlag, Berlin, 1992. xiv+690 pp. ISBN: 3-540-53342-7

This book arose from a conference held in Les Houches in 1989, whose aim was to bring together number theorists and theoretical physicists. It contains in particular a long article Introduction to compact Riemann surfaces, Jacobians, and abelian varieties, by Jean-Benoît Bost. I really like this article, and I think it deserves to be known more widely. Here is the MathSciNet review of this article, by H. Lange :

« The article contains an introduction to the theory of abelian varieties for physicists. This aim is taken seriously: the author uses a language which should be familiar to theoretical physicists. There are three chapters: Compact Riemann surfaces, Jacobians, and General abelian varieties. Riemann surfaces are introduced as conformal classes of $C^\infty$-metrics on an oriented two-dimensional differentiable manifold. The subject of Riemann surfaces may then be seen as the study of conformally invariant properties of two-dimensional Riemann manifolds. This is the reason why Riemann surfaces occur in some topics in physics, e.g. string theory or conformal field theory. Cohomology groups are introduced as Dolbeault cohomology groups of line bundles. For the physicist this means that $H^1(X,L)$ can be interpreted as the zero modes of the adjoint of the operator $\overline\partial_L$. With these definitions the main results of the theory such as the Riemann-Roch theorem, Serre duality and Hodge decomposition are proven using regularizing operators and Fredholm theory. The Jacobian $J(X)$ of a compact Riemann surface $X$ is introduced as the set of $\overline\partial$-connections on a $C^\infty$ line bundle modulo the action of the group $C^\infty(X,{\bf C}^*)$. The author shows that there are isomorphisms to the Albanese variety and the Picard variety of $X$, thus providing a proof of the Abel-Jacobi theorem. In the third section the basics of general abelian varieties are given. The paper also contains some interesting historical digressions, for example a sketch of the original approach of Abel to Abel's theorem. »

The material about abelian varieties (the third chapter of the article) is quite comparable to the beginning of Mumford's book, pointed out by Emerton. Finally, I think that the study of abelian varieties can hardly be dissociated from the study of Riemann surfaces, because historically abelian varieties appeared as Jacobians of Riemann surfaces.

A very classical introduction is Swinnerton-Dyer's Analytic theory of abelian varieties (London Mathematical Society Lecture Note Series 14).

Another good place to start is M. Schlichenmaier, An introduction to Riemann surfaces, algebraic curves and moduli spaces, Theoretical and Mathematical Physics, Springer-Verlag 2007 (2nd ed.)

I'm surprised that no one has mentioned Abelian varieties by Milne as yet.

There's a nice, short, introduction to abelian varieties over C by Mike Rosen in Arithmetic Geometry, Springer-Verlag, 1986, Chapter 4.

I would add Oliver Debarre's: Complex Tori and Abelian Varieties. It does not cover the theory of Abelian functions in as great of depth as many of the other sources cited. But, compared to the others, it does begin at an elementary level, it is concrete and short. In total the book is about 100 pages and the chapters most relevant here are 4,5,6 which take up about 40 pages.

For learning this subject I think this book is a good supplement to reading a more advanced source because Debarre is likely to spend more time on aspects other sources might skim over.

Another book perhaps worth mentioning is Complex Abelian Varieties by Lange and Birkenhake. It is much longer and covers much more material. It is assumes some algebraic geometry background but I would say no more than Mumford's book. It also deals only with complex theory and it is fairly concrete.

The book by Polishchuk should be very helpful, and close in spirit to the nonexistent crash course on Mumford's paper discussed by Charles Matthews in his reponse:


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    I worry this would be a hard read for a theoretical physicist. – Pete L. Clark Jan 5 '11 at 19:22
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    I agree with Pete. The algebraic part of this book uses many results from algebraic geometry without reference. Similarly the analytic part of the book would probably require being read in parallel with a book on complex manifolds. – solbap Jan 5 '11 at 19:56
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    I found it very explicit and formula-based in a way that a physicist (such as myself?) could easily latch onto. – Eric Zaslow Jan 5 '11 at 22:32
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    @Eric Zaslow: I hate to get ad hominem here, but: you may call yourself a physicist (and I see that your PhD was in physics), but Northwestern University calls you Professor of Mathematics. Again, I worry that a less mathematically sophisticated individual than yourself might have trouble with Polishchuk's book. But it's no problem: the point here is to present an array of options, and your answer certainly contributes to that. – Pete L. Clark Jan 6 '11 at 0:48
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    Ha ha... but when I collaborated with Polishchuk I was less than two years out of my physics Ph.D.! You're right, though. I wonder what the intended application is, and what the questioner thinks. – Eric Zaslow Jan 6 '11 at 2:00

there are a couple of relevant paragraphs in Basic algebraic geometry by shafarevich which i recommend, as well as a historical appendix. e.g. he gives an example to show that unlike the one dimensional case, there exist lattices in C^2 for which not even any meromorphic functions can be periodic (chapter VIII.1). in addition to siegel and swinnerton dyer, mumford's other theta functions book is one i would recommend, volume 1 of the three volume series, something like "tata lectures on theta I". Also in lecture IV of his Curves and their Jacobians, now included in his "red book", at least the first 8 pages give an invaluable and succinct introduction to principally polarized abelian varieties and their "moduli" (classifying spaces). And let's get "morphisms" out of the way, since the word appears even in swinnerton -dyer. In one sense it is a generic version of the several terms automorphism, isomorphism, homomorphism, homeomorphism, diffeomorphism,... i.e. a map that preserves some structure to be specified. In algebraic geometry it usually means a structure preserving map that is defined everywhere, as "holomorphic" is used in complex analysis to mean meromorphic and having no poles. so in swinnerton - dyer it probably means holomorphic map. but it is often defined where it is used.

Here is another reference:

Lectures on Riemann surfaces proceedings of the College on Riemann Surfaces, International Centre for Theoretical Physics, Trieste, Italy, 9 Nov.-18 Dec., 1987 editors, M. Cornalba, X. Gomez-Mont, A. Verjovsky. Published 1989 by World Scientific in Singapore, Teaneck, NJ . Written in English.

Table of Contents

Complex analytic theory of Teichmüller space / R.M. Porter

Riemann surfaces, moduli, and hyperbolic geometry / S.A. Wolpert

Gauge theory on Riemann surfaces / N.J. Hitchin

Graph curves and curves on K3 surfaces / R. Miranda

Koszul cohomology and geometry / M.L. Green

Constructing the moduli space of stable curves / I. Morrison

Meromorphic functions and cohomology on a Riemann surface / X. Gomez-Mont

The theorems of Riemann-Roch and Abel / M. Cornalba

The Jacobian variety of a Riemann surface and its theta geometry / R. Smith

Families of varieties and the Hilbert scheme / C. Ciliberto and E. Sernesi

A sampling of vector bundle techniques in the study of linear series / R. Lazarsfeld

Moduli of curves and theta-characteristics / M. Cornalba

Some algebraic geometrical methods in string theory / L. Dabrowski and C. Reina

Lectures on stable curves / F. Bardelli.

Edition Notes Includes bibliographical references. "Revised versions of most of the original notes, covering the entire spectrum of the subjects touched upon during the College, from the foundational ones to areas of very active current research"--P. v.

Classifications Library of Congress QA333 .C65 1987

Mumford's "Tata Lectures on Theta II" might me be close to what you want. It discusses applications of abelian varieties, more precisely Jacobians, to solving various differential equations of interest to physicists.

I found Lang's book very readable, but the more analytic Conforto's "Abelsche Funktionen und Algebraische Geometrie" or this may fit better to your interests. Here a link to an encyclopedia article.

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    i won't recommend any of lang's books for anyone beginning on a subject.his all books are really wierd to say the least – Koushik Jun 17 '14 at 16:18
  • @Koushik: I too find some among Lang's books not recommendable, but this one is in my opinion very well written. It was my first encounter with abelian functions and abelian varieties and I enjoyed it very much. – Thomas Riepe Jun 29 '14 at 17:13

I would also recommend George Kempf's book: Complex Abelian Varieties and Theta Functions

He also wrote a book on Abelian Integrals which might be worth a look.

And Igusa's lovely book on Theta functions

There are also some notes on physics by Dolgachev that mention abelian varieties and theta functions.

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