[Reviewed by Mark Hunacek, on 05/20/2016]

Even non-specialists in number theory have undoubtedly run across the phrase “modular forms”, especially since Andrew Wiles and Richard Taylor made such productive use of them in proving Fermat’s Last Theorem. This is not the only place they show up, however, and my friends in the know tell me that modular forms are an essential part of the modern theory of numbers. Unfortunately, they are as difficult to learn about as they are ubiquitous, and relatively accessible introductions to the area are not exactly thick on the ground (although the situation here may be improving; see, for example, the review in this column of *A First Course in Modular Forms* by Diamond and Shurman).

The book under review, which culminates in an extended (roughly 100 pages long) discussion of modular forms, is obviously not intended as competition for a serious text on that subject such as the one cited above. The goal of this book, according to the Preface, is to make this material comprehensible to a “general mathematically literate audience.” Given the difficulty of the subject matter, this is an ambitious undertaking, and the extent to which this book can be judged as a success depends on how the phrase quoted above is interpreted. Statements in the preface suggest that the authors believe that this book can be understood by a person with only a year of calculus as background. This, I think, is an overly optimistic assessment. However, people with a better mathematics background might well find something useful here.

I, for example, came to this book knowing next to nothing about what modular forms are. Having read the book, I still don’t have anything resembling a true understanding of them, but they are at least somewhat less mysterious to me than they were before. I am reminded of a four-decades old quotation from David Dobbs, a professor of mine in graduate school. One day we were up in the commons room, having a conversation about some mathematical topic. At one point he stopped and said, “You’re nodding at the right places. Either you understand what I’m saying, or you’ve learned when to nod.” So, if nothing else, I now know a little more about when to nod when talking to somebody who *really* understands modular forms.

I don’t mean this sarcastically. The book has given me some real understanding of just what modular forms are and how they can be used in number theory. Reading this book was, therefore, at least for me, a worthwhile and educational experience. But, despite the statement in the preface that suggests that a year of calculus is sufficient prerequisite for the book, you won’t find me recommending it to any of my college sophomore students, coming right off the first-year calculus sequence. I doubt very much that *any *book can make modular forms accessible to such students, and this one is no exception.

This is not the first time that Ash and Gross have teamed up to try and make sophisticated ideas accessible to non-specialists with at least some background in mathematics. They also wrote *Fearless Symmetry* and *Elliptic Tales*, both of which mention modular forms, but do not discuss them in depth. (To quote from the preface to this book: “In both of these books, we ended up mumbling something about modular forms… [but] … had already introduced so many concepts that we could only allude to the theory of modular forms.”) Accordingly, this book can be seen as a more detailed account of this subject, one that would allow a reader of one or more of the earlier books to flesh out his or her understanding. These earlier books are not intended as a prerequisite for this text, but there are times, particularly in the last chapter, where familiarity with these two books will make things much clearer for the reader.

*Summing it Up* is, like the two predecessor books, divided into three parts of increasing levels of sophistication — a somewhat steeper level of increase, I think, than the authors suggest to be the case. The first part (“Finite Sums”) discusses the basics of elementary number theory: divisibility, congruence, sums of squares, Waring’s problem, Bernoulli numbers. The preface advertises this part of the book as being accessible to people with a background in high school algebra and geometry, and this is largely accurate, except for the last two sections, which require knowledge of infinite series, differential and integral calculus, and the exponential function.

In part II (“Infinite Sums”), infinite series, calculus and complex numbers are employed to discuss, among other things, the Riemann zeta function (and its connection with Bernoulli numbers) and generating functions. The partition function makes its first appearance here; it will be revisited in part III and studied via modular forms.

Part III is entitled “Modular Forms and their Applications”. Here is where things get a little dicey. This part of the book, the authors tell us, “does not require any additional mathematical knowledge” over and above that required for part II (namely, “much of the content of the first year of a standard calculus course”). They do acknowledge that aspects of this part of the book get “rather intricate”, and that’s putting it mildly. It is true that Ash and Gross take pains to define even fairly standard mathematical terms (such as “vector space”), so to that extent their claim is literally correct. But the definitions and statements of results come so quickly, and the material is so sophisticated, that readers without a considerable degree of mathematical maturity will likely find themselves at sea fairly quickly. After all, students need time to assimilate new concepts — to see lots of examples, to try and prove some of the basic theorems — before going on to newer ones. They are not given that kind of time here.

To illustrate: in the first chapter of this part, the authors proceed rapidly from the mapping properties of the complex exponential function to Euclid’s *Elements* to the Poincaré half-plane model of hyperbolic geometry to Klein’s *Erlanger Programm* to matrix groups to the set of all motions of this model to linear fractional transformations. They do this in about ten pages of text, which seems like a lot for somebody without some advanced mathematical training to grasp: indeed, having taught a course in non-Euclidean geometry for several semesters now, I can attest to the fact that even senior math majors find hyperbolic geometry and the Poincaré model to be difficult to absorb.

The next chapter, also about ten pages long, starts with a discussion of the group SL2(ℤ), proceeds to a discussion of the fundamental domain, and then defines a modular form as an analytic function defined on the upper-half plane, satisfying certain technical properties that take the authors about five pages of text to describe. Again, this is a lot of mathematics in a short amount of time, especially when the word “group” was just defined about ten pages back.

The rest of the book is an elaboration of these ideas. The definition of modular form is extended so as to allow functions that work nicely with certain subgroups of SL2(ℤ) (the “congruence subgroups”) rather than the full group itself. Applications of modular forms to other areas of mathematics (including the partition function of part II and to the Monster group in group theory) are sketched.

Of course it is not possible, in a book of this nature, to do all this with anything resembling complete rigor and precision. The approach of the authors is to, by and large, make definitions and statements of theorems fairly precise, even at the risk of causing some difficulty, and omitting most proofs of theorems. Occasionally phrases are used just for the sake of introducing fairly sophisticated terminology to the reader; e.g., “it is a *section of a vector* *bundle *— whatever that means.”

The authors are talented expositors, and by and large the writing is clear and enjoyable to read. There were, however, occasional moments when I wondered what they were up to. In chapter 11, for example, they spend some time discussing the group G of isometries of the Euclidean plane. They discuss rotations and translations, and then state, clearly and unambiguously, that “it turns out that that’s it” — i.e., G consists precisely of rotations and translations. I was, as any person familiar with geometry would be, wondering what became of reflections and glide reflections, when, after some additional discussion had ensued, I turned the page and read (I quote verbatim) “Well, we lied. We only described half of all the elements of G.” I have *absolutely no idea* what pedagogical benefit the authors saw in deliberately making a false statement on one page and then retracting it on the next; this practice is, at best, annoying and confusing, and downright misleading at worst. What of all the people who just glance at the initial paragraph and don’t bother turning the page?

My biggest problem with this book is that I am not sure to whom I should recommend it. Anybody whose mathematical background does not include calculus could possibly get something out of most of the first part of the book, but it hardly seems worth the effort to just read 50 or 60 pages of a book, with no real payoff. So, this narrows the audience down to people who have, at a bare minimum, taken, say, a year of calculus. People with this minimal background could possibly get through much of the first two parts of the book, but again, there is no real payoff here, and such people wouldn’t really learn much more than they would by reading any of the standard books on number theory and just skipping all the proofs.

The* real* payoff to this book, the thing about it that sets it apart from others, is the third part, on modular forms. To really appreciate this part of the book, though, one should bring to the table a pretty solid background in mathematics. I’m not saying that a prospective reader needs a doctorate or anything that sophisticated (though mathematics faculty who want to learn something about an area that they don’t know much about will likely find something valuable here), but at the very least a reader should have a good understanding of the contents of a standard undergraduate curriculum, and the kind of mathematical maturity that two or three years of studying upper-level mathematics brings with it. But of course anybody with that background would likely wind up just skimming, or omitting altogether, most of the first two parts of the book.

I think that perhaps the authors’ biggest mistake here was trying to be all things to all people. I can’t help but think what could have been produced if they had employed their considerable expository skills to write a book at, say, the senior undergraduate or first year graduate level, assuming a more sophisticated background in mathematics than just a year of calculus, and giving a slower, more detailed, but yet still accessible, introduction to this important area of mathematics. That would have been an extremely valuable text.

I should, however, review the book that was written, not the different one that I would have liked the authors to write. *Summing it Up* is an ambitious attempt. If, as I think is the case, it does not live up to its goals, it may well be because those goals were just too ambitious to begin with. But, for a narrower audience than the authors thought they were writing for, this is an informative account of an important subject, and I’m glad they made the attempt.

Mark Hunacek (mhunacek@iastate.edu) teaches mathematics at Iowa State University.