Celebratio Mathematica — Ogg — Complete Bibliography

[1] A. P. Ogg: Cohomology of abelian varieties over function fields. Ph.D. thesis, Harvard University (Ann Arbor, MI), 1961. MR 2939235 phdthesis

[2] A. P. Ogg: “Cohomology of abelian varieties over function fields,” Ann. of Math. (2) 76 (1962), pp. 185–212. MR 155824 Zbl 0121.38002 article

[3] A. P. Ogg: “Abelian curves of 2-power conductor,” Proc. Cambridge Philos. Soc. 62 (1966), pp. 143–148. MR 201436 Zbl 0163.15403 article

[4] A. P. Ogg: “On pencils of curves of genus two,” Topology 5 (1966), pp. 355–362. MR 201437 Zbl 0145.17802 article

[5] A. P. Ogg: “Elliptic curves and wild ramification,” Amer. J. Math. 89 (1967), pp. 1–21. MR 207694 Zbl 0147.39803 article

[6] A. P. Ogg: “Abelian curves of small conductor,” J. Reine Angew. Math. 226 (1967), pp. 204–215. MR 210706 Zbl 0163.15404 article

[7] A. Ogg: Modular forms and Dirichlet series. W. A. Benjamin (New York–Amsterdam), 1969. MR 256993 Zbl 0191.38101 book

[8] A. P. Ogg: “On modular forms with associated Dirichlet series,” Ann. of Math. (2) 89 (1969), pp. 184–186. MR 234918 article

[9] A. P. Ogg: “On a convolution of \( L \)-series,” Invent. Math. 7 (1969), pp. 297–312. MR 246819 Zbl 0205.50902 article

[10] A. P. Ogg: “Functional equations of modular forms,” Math. Ann. 183 (1969), pp. 337–340. MR 255491 Zbl 0191.38102 article

[11] A. P. Ogg: “On product expansions of theta-functions,” Proc. Amer. Math. Soc. 21 (1969), pp. 365–368. MR 260673 Zbl 0203.35505 article

In [Hecke 1936], Hecke set up a general method of passing back and forth between Dirichlet series and functions on the upper half plane (via the Mellin transform) and showed how functional equations on the one side correspond to functional equations on the other. Weil [Weil 1968] has remarked that this method is applicable in a more general setting than in [Hecke 1936], and has shown in this manner that the functional equation for Dedekind’s function \( \eta(\tau) \) is an immediate consequence of the functional equation for the Riemann zeta-function. In this note we give a proof along these lines of the product expansion of the basic theta-function \( \theta = \theta_3=\theta_{\infty} \).

[12] A. P. Ogg: “On the eigenvalues of Hecke operators,” Math. Ann. 179 (1969), pp. 101–108. MR 269597 Zbl 0169.10102 article

[13] A. P. Ogg: “A remark on the Sato–Tate conjecture,” Invent. Math. 9 (1969/70), pp. 198–200. MR 258835 Zbl 0219.14013 article

[14] A. P. Ogg: “Rational points of finite order on elliptic curves,” Invent. Math. 12 (1971), pp. 105–111. MR 291084 Zbl 0216.05602 article

If \( A \) is an abelian curve defined over the field of rational numbers \( \mathbf{Q} \), then by Mordell’s theorem the group \( A_{\mathbf{Q}} \) of rational points on \( A \) is of finite type: \[ A_{\mathbf{Q}}\simeq \mathbf{Z}^r \oplus F, \] where \( F \) is finite. According to Cassels [1966], the folklore contains the conjecture that the order of \( F \) is bounded, and in particular there should be only a finite number of integers \( N \) such that some curve \( A \) has a rational point of order \( N \). It is known [Cassels 1966] that \( N = 1 \), \( \dots,10 \) or 12 is possible, and that \( N = 11 \), \( 14, 15, 16 \), 20, or 24 is impossible. In the present paper, we give a proof that \( N = 17 \) is impossible, by a suitable modification of the method used by Billing and Mahler [Billing and Mahler 1940] to prove that \( N = 11 \) is impossible, and then make some general remarks on the modular interpretation of the problem.

[15] A. P. Ogg: “On the cusps of \( \Gamma_{0}(N) \),” pp. 173–177 in Proceedings of the Number Theory Conference (Univ. Colorado, Boulder, 1972). Univ. Colorado (Boulder, CO), 1972. MR 393037 Zbl 0323.10018 inproceedings

[16] A. Ogg: “Survey of modular functions of one variable,” pp. 1–35 in Modular functions of one variable, I (Antwerp, 1972). Edited by W. Kuyk. Lecture Notes in Mathematics 320. Springer, 1973. MR 337785 Zbl 0258.10012 inproceedings

In this paper a survey is given of the theory of modular functions of one variable, including Dirichlet series, functional equations, compactifications, Hecke operators, Eisenstein series and the Petersson product. They are notes of an introductory course on the subject at the International Summer School 1972, held at Antwerp University. It is to be noted that much of the material presented is to be found in [Ogg 1969], where complete proofs of most of the theorems occur. However, the material goes beyond [Ogg 1969], and is complementary to it, in the sense that more attention is paid to the relation between Eisenstein series and elliptic curves. Also included is an exposition of some basic features of the work of Artin and Lehner on old and new forms. Little, if no attention at all, is paid to the relation between modular functions and quadratic forms.

[17] A. P. Ogg: “Rational points on certain elliptic modular curves,” pp. 221–231 in Analytic number theory (St. Louis Univ., 1972). Edited by H. G. Diamond. Proc. Sympos. Pure Math. XXIV. Amer. Math. Soc., 1973. MR 337974 Zbl 0273.14008 inproceedings

[18] A. P. Ogg: “Hyperelliptic modular curves,” Bull. Soc. Math. France 102 (1974), pp. 449–462. MR 364259 Zbl 0314.10018 article

[19] A. Ogg: “Correction to: ‘Survey of modular functions of one variable’,” pp. 145 in Modular functions of one variable, IV (Antwerp, 1972). Edited by B. J. Birch and W. Kuyk. Lecture Notes in Math. 476. Springer, 1975. Corrections to the paper that appeared in Modular functions of one variable, I, Lecture Notes in Math., 320:1–35 (1973). MR 384699 inproceedings

Bryan John Birch	Related
Willem Kuyk	Related

[20] A. P. Ogg: “Automorphismes de courbes modulaires,” pp. Exp. No. 7, 8 in Séminaire Delange–Pisot–Poitou (16e année: 1974/75), Théorie des nombres, Fasc. 1. Secrétariat Mathématique (Paris), 1975. MR 417184 Zbl 0336.14006 incollection

[21] A. P. Ogg: “Diophantine equations and modular forms,” Bull. Amer. Math. Soc. 81 (1975), pp. 14–27. MR 354675 Zbl 0316.14012 article

[22] A. P. Ogg: “On the reduction modulo \( p \) of \( X_0(pM) \),” pp. 204–211 in Japan–U.S. Seminar on Applications of Automorphic Forms to Number Theory (Ann Arbor, Mich., 1975). 1975. Part of an informally assembled proceedings collated and distributed to participants and observers at the conference. incollection

[23] A. P. Ogg: “Über die Automorphismengruppe von \( X_{0}(N) \),” Math. Ann. 228 : 3 (1977), pp. 279–292. MR 562500 Zbl 0344.14005 article

[24] A. P. Ogg: “On the Weierstrass points of \( X_{0}(N) \),” Illinois J. Math. 22 : 1 (1978), pp. 31–35. MR 463178 Zbl 0374.14005 article

[25] A. P. Ogg: “Modular functions,” pp. 521–532 in The Santa Cruz Conference on Finite Groups (University of California, Santa Cruz, 1979). Edited by B. Cooperstein and G. Mason. Proc. Sympos. Pure Math. 37. Amer. Math. Soc. (Providence, R.I.), 1980. MR 604631 Zbl 0448.10021 incollection

Geoffrey Mason	Related
Bruce Nathan Cooperstein	Related

[26] A. P. Ogg: “Real points on Shimura curves,” pp. 277–307 in Arithmetic and geometry, I. Edited by M. Artin and J. Tate. Progr. Math. 35. Birkhäuser (Boston), 1983. MR 717598 Zbl 0531.14014 incollection

Michael Artin	Related
John Torrence Tate, Jr.	Related

[27] A. P. Ogg: “Some special classes of Cartan matrices,” Canad. J. Math. 36 : 5 (1984), pp. 800–819. MR 762743 Zbl 0543.17006 article

Let \( A = (A_{ij})_{1\leqq i,j\leqq l} \) be a Cartan matrix, i.e., \( A_{ij} = 2 \) for all \( i \) and \( A_{ij} \) is an integer \( \leqq 0 \) for \( i\neq j \), with \( A_{ij}= 0 \) if \( A_{ji} = 0 \). The size \( l \) of \( A \) is called its rank, for Lie-theoretic reasons, and may be larger than its matrix rank. We associate to \( A \) its Dynkin diagram, with vertices \( 1, 2 \), \( \dots,l \), with \( A_{ij} A_{ji} \) lines joining \( i \) to \( j \), and with an arrow pointing from \( i \) to \( j \) if \( A_{ij}/A_{ji} \lt 1 \), i.e., pointing toward the shorter root (see below). The Cartan matrix \( A \) is indecomposable if its diagram is connected, and symmetrizable if there exist positive rational numbers \( q_1,\dots, q_l \) with \[ q_i A_{ij} = q_j A_{ji}\quad \text{ for all } i \text{ and } j. \] Symmetrizability is automatic if the diagram contains no cycle. We assume throughout this paper that \( A \) is symmetrizable, and so “Cartan matrix” always means “symmetrizable Cartan matrix”. If \( A \) is indecomposable, then the symmetrizing numbers \( q_1 \), \( \dots,q_l \) are unique up to a proportionality constant, since if \( q_i \) is known, and \( i \) is connected to \( j \), then \( q_j \) is known. We normalize to have \( q_i = 1/k_i \), where \( k_1 \), \( \dots, k_l \) are positive integers without common factor.

[28] A. P. Ogg: “Mauvaise réduction des courbes de Shimura,” pp. 199–217 in Séminaire de théorie des nombres, Paris 1983–84. Edited by C. Goldstein. Progr. Math. 59. Birkhäuser (Boston), 1985. MR 902833 Zbl 0581.14024 incollection

[29] A. Ogg: “Book Review: Vertex operator algebras and the Monster,” Bull. Amer. Math. Soc. (N.S.) 25 : 2 (1991), pp. 425–432. MR 1567952 article