Goro Shimura is a towering figure in number theory who has made immense contributions to the subject over the last five or so decades. Given my double life as historian and number theorist, this was a book I *had* to read. (Several other regular reviewers wanted it too… editorship has its privileges.)

*The Map of My Life* is not a formal autobiography. Rather, it is a book of memories. Shimura, who was born in 1930, tells us stories, focusing on those that are more vivid to him, skipping over painful episodes, and letting us see some of his emotions and ideas. The style is artless and vivid, giving an impression of intimacy.

The book has two main sections. The first is about growing up in Japan before, during, and just after World War II. Here, the map of the title becomes concrete: Shimura describes a map (the *Kiri-ezu*) showing the neighborhood of Tokyo where his ancestors lived and where he also grew up. Made in the Edo period, the map shows the plots owned by the Japanese feudal lords and indicates the quarters occupied by their retainers, including, in this case, the Shimura family. I was sorry that the map was not reproduced in the book!

This part focuses mostly on Shimura’s early years, before the war. Reading it, I was reminded of stories my grandfather would tell. Some of the memories are clear, others have a shade of vagueness over them. In some cases, I found it hard to make sense of the stories, not having much context to work with. (For example, Shimura mentions “ad-balloons.” I can make a guess about what these might be, but he never tells us exactly, so I don’t know whether my guess is correct.)

The second section of the book is about Shimura’s early years as a mathematician. He tells us about his visits to France and to Princeton in the second half of the 1950s, about his decision to move to the United States, and about his years, beginning in 1962, as a professor in Princeton. There is a lot about his attitude towards what he was doing in the 1960s and his general approach to mathematics. There are very few mathematical details. I definitely wanted more, but I can see other readers being happy that more is not provided, as it might make the book a little too technical.

Since I was also born in another country (Brazil, in my case), I was interested in the account of the decision to move away. Part of the reason, Shimura says, is that he was very thin and therefore very susceptible to cold. Houses in Japan were not well heated, so when he visited Princeton in 1959 he was delighted by how warm his apartment was. So he says he left Japan because it was cold there. (When he tells people that, he says, they always assume he is either joking or speaking metaphorically!)

Shimura’s personality is clearly on display throughout. He is not afraid to tell stories that put others in a bad light; if he feels they were mean, or jealous, or thoughtless, he says so. He is also not afraid to tell us his opinions about mathematics. For example, his disdain for formal axiomatization of known theories (e.g, Hilbert’s work on the foundations of geometry) is forcefully expressed. His reactions to the older Japanese mathematicians he meets are often negative, and he tells us so.

Some topics I expected to read more about are simply not there. Shimura has written about the death by suicide of his friend and collaborator Taniyama, but in this book it is mentioned only in a section called “Why I Wrote That Article.” (The article itself might have been included as an appendix, but is not.) In fact, Taniyama has a fairly small role.

There is nothing at all about Shimura’s many graduate students, and nothing much about recent (say, post-1980) results in number theory. Of his colleagues at Princeton and at the Institute for Advanced Studies, the only one that is described in detail is André Weil, whom Shimura clearly loved and admired.

At times there are tantalizing hints. Shimura mentions that students at Princeton are required to write a senior thesis, and that many students have chosen him as their advisor for that. “Since I had a stock of easy but interesting topics,” he says, “I welcomed them.” I have always found it hard to come up with “easy but interesting” senior thesis topics, so I was left wishing for some examples.

The mathematical sections have as an underlying theme a clarification of Shimura’s role in the development of the arithmetic theory of modular forms. He says that in the 1960s few people got the point, perhaps only Weil and Eichler. When he came to Princeton, he says, he “was familiar with number theory, algebraic geometry, and modular forms, and there was no such mathematician in the United States.” The impression I got is that much of my mathematical life has been spent in Shimura’s playground.

One of the topics Shimura spends time on is the origin of the conjecture that all rational elliptic curves are modular. There is a famous problem formulated by Taniyama in 1955 that says something along those lines, but Shimura emphasizes that it was imprecise and, in fact, incorrect as stated. It was Shimura who first made the conjecture precise and who told others about it. As appendices, he includes some letters that go into this in more detail.

Shimura is not a native speaker of English, and this comes through clearly in the book. There are many awkwardly constructed sentences. In several places, especially in the sections about his early life in Japan, he reaches for effects that he cannot quite pull off. There is little concern to stay within a chronological structure, and one often reads comments such as “I’ll write about that later” or “I don’t want to say any more about that.” All this reinforced my feeling that I was hearing an older man telling stories about his youth, rambling along his memories and telling us the parts he still likes to think about.

I hope my friends at Springer will not be too annoyed at me for saying that they could and should have done a better production job. The photographs are badly reproduced and few. A little fearless copyediting could have improved the text in many places. Worst of all, there is no index and no bibliography, just a pointer to Shimura’s collected works. Surely at least an index of names would have been appropriate?

Though I have worked on modular forms for most of my mathematical life, I have not met Shimura, nor have I read many of his papers. (Unforgivable, I know. My only excuse is that by the time I came along, Shimura had moved on from his 1960s work to more general and more difficult cases which I didn’t need in my own work. And others were beginning to write textbook-style accounts of the older work. I did read a large part of Shimura’s classic Introduction to the Arithmetic Theory of Automorphic Functions, which I found very hard going as a graduate student.) I was left wishing for a set of Shimura’s collected works. For example, there is an unpublished 1968 paper in which he constructs the Galois representations attached to a Hecke eigenform. I remember hearing rumors about this paper as a graduate student, and I am delighted to hear that it is included in the collected works. And, of course, from a historian’s point of view I need to see how the theory came together.

Shimura’s memoir gives us a glimpse of the person behind the mathematics. I wanted more: more names, more stories, more details. But I shouldn’t be greedy; getting to know a little bit about him is a privilege.

Fernando Q. Gouvêa is not at all thin; he left Brazil, at least in part, because it was too hot.