There are precisely two books on Arakelov geometry. One by Lang and one by Soule. I would love to see a book written on the subject which focuses mainly on the two dimensional (and one-dimensional) case. Sections 8.3 and 9.1 of Liu's book do this greatly for example (but considers only intersection multiplicities at the finite points). It should include all the theorems done so far. Something like

Chapter 0. Prerequisites

Chapter 1.
Arithmetic curves (Riemann-Roch, slopes method, etc. One should include a paragraph or appendix on algebraic curves stating all the theorems that can and have been generalized.)

(N.B. An arithmetic curve is the spec of a ring of integers.)

Chapter 2.
Arithmetic surfaces (This would contain all the "arithmetic" analogues of the theorems mentioned in the Appendix. For example, there has been a lot of work on Riemann-Roch theorems, trace formulas, Dirichlet's higher-dimensional unit theorem, Bogomolov inequalities, etc. Also, there are four intersection theories (which are compatible) I know of at the moment. The one developed by Arakelov-Faltings, then Gillet-Soule, then Bost and then Kuhn. The book should include a detailed description of them.

Appendix A.
Algebraic surfaces. (A survey of all the classical theorems for algebraic surfaces that have an analogue in Arakelov geometry. This includes Faltings' generalizations of the Riemann-Roch theorem, Noether theorem, etc. but also the theorems generalized to Arakelov theory by Gasbarri, Tang, Rossler, Kuhn, Moriwaki, Bost, etc.)

Appendix B.
Riemann surfaces (Just the necessary. Differential forms and Green functions basically.)

8I

reallylike this question... hopefully someone will take a hint and write number (5) and (2) sometime soon! – Dylan Wilson – 2011-01-24T10:30:36.9531Yeah, but as most of you probably already know, JB won't write (2) nor his projected "higher algebra", because he thinks that he worked off his debt by writing all those expository papers... – Tim van Beek – 2011-01-24T10:51:38.707

4

Steve Lack wrote something approximating (2): http://arxiv.org/abs/math/0702535

– Tom Leinster – 2011-01-24T11:31:37.7971Thanks Dylan. Tim: it would be great indeed if JB would publish his "higher algebra" but his past expository work is already amazing. Tom: I am aware of that paper (and you also have some sections about it on your wonderful book) but I was thinking about a full textbook presentation that might have more examples and applications. – Gonçalo Marques – 2011-01-24T12:28:47.060

7Regarding the Weil conjectures, have you read the appendix to Hartshorne that discusses these? If so, you could also try Nick Katz's exposition on Deligne's work in the

Hilbert's Problemsbook (in the Proceedings of Symposia in Pure Math series) from the 1970s. Also, Deligne's articleWeil Iis less technical than you might guess, and there is also the textbook by Freitag and Kiehl. – Emerton – 2011-01-24T12:44:50.057Thank you for your suggestions I will look for Katz and Deligne's articles. I am aware of Freitag and Kiehl's textbook, unfortunately it's a hard to find. I was thinking of a textbook that would use the Weil conjectures as a "leitmotiv" while introducing some of the more modern characters in algebraic geometry. But maybe it can't be done (at least at level I would understand...). – Gonçalo Marques – 2011-01-24T14:16:08.307

@Tom: Sadly Lack's account is hardly an introduction to 2-categories, skipping even the definitions (!) and all motivation. It would be even less suitable for a "working mathematician". – Andrea Ferretti – 2011-01-24T15:31:52.617

It boggles my mind that nobody has written 5) yet. Shouldn't one of Grothendieck's students be doing this or something? – Qiaochu Yuan – 2011-01-24T15:58:07.160

9Qiaochu: Demazure and Gabriel wrote a book using the functor of points approach over 3 decades ago. Some people love this book, while others... – Donu Arapura – 2011-01-24T17:55:11.623

@Donu: thanks for the heads-up! – Qiaochu Yuan – 2011-01-24T19:15:41.267

1

One more reference on the Weil conjectures: notes of Beilinson's lectures on the subject available at http://www.math.uchicago.edu/~mitya/beilinson/

– AFK – 2011-01-26T23:30:05.2402functor-of-points: besides Demazure-Gabriel, there is also the last chapter of Eisenbud-Harris "Geometry of Schemes" – Sean Rostami – 2011-01-27T04:36:30.817

3It strikes me that many of the knowledgeable participants who made wonderfully detailed suggestions for a book on a coherent topic from a particular viewpoint, are well-positioned to write the very book they wish to read! – Joseph O'Rourke – 2011-02-01T00:07:39.537

23Maybe there is a place for the dual question: "Books you would like to write (if somebody would just read them)" so people can mention their book ideas and get some feedback. – Gil Kalai – 2011-02-01T15:03:29.593

3Mumford's "Lectures on curves on an algebraic surface" is a great solution to 5), different from and (for me) more geometrically appealing than Demazure and Gabriel. – inkspot – 2011-02-05T16:59:10.473

1The Tropic of Calculus (as suggested by Tom Lehrer). 50 Shades of Gray Codes. Lady Chatterly's Prover. – Gerry Myerson – 2015-10-09T05:12:31.337