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Elliptic Curves, Modular Forms, and Their L-functions

Álvaro Lozano-Robledo
American Mathematical Society/Institute for Advance Study
Publication Date: 
Number of Pages: 
Student Mathematical Library 58
[Reviewed by
Michael Berg
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I think this is a wonderful book.

In less than 200 pages Lozano-Robledo covers a solid amount of modern number theory in a manner altogether accessible to a novice, and in a fashion so as to convey number theory’s irresistible beauty, due both to its connections to very simply phrased deep arithmetical questions (e.g. the behavior of congruent numbers) and its use of stunning techniques (such as the yoga of L-functions).

The three foci of the book under review are, as the title indicates, elliptic curves, modular functions, and L-functions; indeed, Lozano-Robledo stresses their interconnectedness, and it can’t be otherwise in the wake of, e.g., Wiles’ conquest of Fermat’s Last Theorem.

Of course, there is a considerable prehistory in place: these deep and evocative interconnections go back to the Old Masters. Riemann’s great monograph of 1859, Über die Anzahl der Primzahlen unter einer gegebenen Grösse, containing his approach to the Prime Number Theorem as well as the statement of his Hypothesis about the zeta function, already addresses the interplay between half-integral weight modular forms (theta functions) and Dirichlet series (like the zeta function), for example. And it was Erich Hecke who largely inaugurated the modern approach to this theme per se, with the baton eventually passed to André Weil. And, as regards the appearance of elliptic curves in the game, we must mention the names of, e.g., Max Deuring, Helmut Hasse, and of course Gôro Shimura, Yutaka Taniyama, and, again, André Weil, which takes us to Andrew Wiles and Fermat’s Last Theorem, of course.

We find Lozano-Robledo’s coverage of the Shimura-Taniyama-Weil conjecture on p. 137 of his book. [I can’t resist a bit of numerology: 137 is a well-chosen number for this marvelous result, implying Fermat’s Last Theorem: 137 º 1 (mod.4), and 137 is prime; accordingly 137 is the sum of two squares: 137 = 121 + 16. The theorem that p º 1 (mod.4) implies p = a2 + b2 is due to… Pierre de Fermat!] This is in the fifth chapter, which is devoted to L-functions and also contains a discussion of the Birch and Swinnerton-Dyer Conjecture.

Prior to this, in Chapters 2, 3, and 4, Lozano-Robledo discusses elliptic curves (at some length), modular curves, and modular forms, respectively, so the stage is well set for the aforementioned dénouement. But a lot of other gems populate the pages of this book, too. For instance, on p. 72 we encounter Mordell-Weil, and Hecke operators acting on cusp forms appear on p. 102 + 10 + 1.

Of course, the themes Lozano-Robledo addresses in Elliptic Curves, Modular Forms, and Their L-Functions are deep and sophisticated; considerable mathematical background is required for their mastery. But mastery is not what Lozano-Robledo is after: it’s more along the lines of rendering “the primary objects of study, the statements of the main theorems, and their corollaries … within the grasp of advanced undergraduates.” And in this objective Lozano-Robledo succeeds admirably. The book is full of examples and exercises of such appeal that a properly disposed rookie should go after nigh-on all of them; to boot, the author’s narrative is compact and smooth.

Elliptic Curves, Modular Forms, and Their L-Functions is a marvelous addition to the literature. Had I had it available as a kid, it would have been among my very favorites!

Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.

  • Introduction
  • Elliptic curves
  • Modular curves
  • Modular forms
  • $L$-functions
  • PARI/GP and Sage
  • Complex analysis
  • Projective space
  • The $p$-adic numbers
  • Parametrization of torsion structures
  • Bibliography
  • Index