B. H. Gross :
“Heegner points on \( X_0(N) \) ,”
pp. 87–105
in
Modular forms: Contributions to a symposium on modular forms of one and several variables
(Durham, UK, 30 June–10 July 1983 ).
Edited by R. A. Rankin .
Ellis Horwood Series in Mathematics and its Applications .
Ellis Horwood Limited (Chichester, UK ),
1984 .
MR
803364 Zbl
0559.14011 incollection
Abstract
BibTeX
In his work on the class-number problem for imaginary quadratic fields, Heegner [1952] introduced a remarkable collection ofpoints on certain modular curves. These points always form a subset of the singular moduli; on the curve \( X_0(N) \) they correspond to the moduli of \( N \) -isogenous elliptic curves with the same ring of complex multiplication. Birch [1970] was the first to recognize the significance of the divisor classes supported on these points in the arithmetic of the Jacobian \( J_0(N) \) . Using them, he was able to construct points of infinite order in certain elliptic quotients of \( J_0(N) \) possessing a cuspidal group of even order [Birch 1975]. Mazur [1979] later found an interesting method to construct points of infinite order in Eisenstein quotients of \( J_0(N) \) , when \( N \) is prime.
In this paper, I would like to show how to obtain some ofthe above results via the theory of modular, elliptic, and circular units. This method will be exposed in Section II, after a review of the basic material on Heegner points in Section I. These theoretical results, although fragmentary, fit in nicely with the extensive computations which Birch and Stephens have made on this subject [1984]. On the basis of this evidence, I was led to conjecture a simple identity relating the height ofa Heegner divisor class to the first derivative at \( s = 1 \) of the \( L \) -series of an automorphic form on \( \mathrm{PGL}(2) \times \mathrm{GL}(2) \) . Zagier and I have obtained a proof of this identity in many cases: I will discuss this work briefly in Section III. In Section IV I will present a general program of work on other modular curves.
@incollection {key803364m,
AUTHOR = {Gross, Benedict H.},
TITLE = {Heegner points on \$X_0(N)\$},
BOOKTITLE = {Modular forms: {C}ontributions to a
symposium on modular forms of one and
several variables},
EDITOR = {Rankin, Robert A.},
SERIES = {Ellis Horwood Series in Mathematics
and its Applications},
PUBLISHER = {Ellis Horwood Limited},
ADDRESS = {Chichester, UK},
YEAR = {1984},
PAGES = {87--105},
NOTE = {(Durham, UK, 30 June--10 July 1983).
MR:803364. Zbl:0559.14011.},
ISSN = {0271-6151},
ISBN = {9780470200995},
}
B. H. Gross and D. B. Zagier :
“Heegner points and derivatives of \( L \) -series Invent. Math.
84
(1986 ),
pp. 225–320 .
To John Tate.
This work expands on a short note published in C. R. Acad. Sci., Paris 297 (1983)Math. Ann. 278 (1987)
MR
833192 Zbl
0608.14019 article
Abstract
People
BibTeX
The main theorem of this paper gives a relation between the heights of Heegner divisor classes on the Jacobian of the modular curve \( X_0(N) \) and the first derivatives at \( s = 1 \) of the Rankin \( L \) -series of certain modular forms. In the first six sections of this chapter, we will develop enough background material on modular curves, Heegner points, heights, and \( L \) -functions to be able to state one version of this identity precisely. In §7 we will discuss some applications to the conjecture of Birch and Swinnerton-Dyer for elliptic curves. For example, we will show that any modular elliptic curve over \( \mathbb{Q} \) whose \( L \) -function has a simple zero at \( s = 1 \) contains rational points of infinite order. Combining our work with that of Goldfeld [1976], one obtains an effective lower bound for the class numbers of imaginary quadratic fields as a function of their discriminants (§8). In §9 we will describe the plan of proof and the contents of the remaining chapters.
Many of the results of this paper were announced in our Comptes Rendus note [1983]. A more leisurely introduction to Heegner points and Rankin \( L \) -series may be found in our earlier paper [1984].
@article {key833192m,
AUTHOR = {Gross, Benedict H. and Zagier, Don B.},
TITLE = {Heegner points and derivatives of \$L\$-series},
JOURNAL = {Invent. Math.},
FJOURNAL = {Inventiones Mathematicae},
VOLUME = {84},
YEAR = {1986},
PAGES = {225--320},
DOI = {10.1007/BF01388809},
NOTE = {To John Tate. This work expands on a
short note published in \textit{C. R.
Acad. Sci., Paris} \textbf{297} (1983).
Part II was published in \textit{Math.
Ann.} \textbf{278} (1987). MR:833192.
Zbl:0608.14019.},
ISSN = {0020-9910},
}
B. H. Gross and S. S. Kudla :
“Heights and the central critical values of triple product \( L \) -functions Compos. Math.
81 : 2
(1992 ),
pp. 143–209 .
MR
1145805 Zbl
0807.11027 article
People
BibTeX
@article {key1145805m,
AUTHOR = {Gross, Benedict H. and Kudla, Stephen
S.},
TITLE = {Heights and the central critical values
of triple product \$L\$-functions},
JOURNAL = {Compos. Math.},
FJOURNAL = {Compositio Mathematica},
VOLUME = {81},
NUMBER = {2},
YEAR = {1992},
PAGES = {143--209},
URL = {http://www.numdam.org/item/CM_1992__81_2_143_0.pdf},
NOTE = {MR:1145805. Zbl:0807.11027.},
ISSN = {0010-437X},
}
B. Gross :
Letter to W.-S. Zhang ,
3 January 1995 .
misc
BibTeX
Read PDF
@misc {key75017638,
AUTHOR = {Gross, B.},
TITLE = {Letter to W.-S. Zhang},
MONTH = {3 January},
YEAR = {1995},
}
B. Gross :
Email to W.-S. Zhang ,
4 January 1995 .
misc
BibTeX
Read PDF
@misc {key19051787,
AUTHOR = {Gross, B.},
TITLE = {Email to W.-S. Zhang},
MONTH = {4 January},
YEAR = {1995},
}
B. H. Gross and C. Schoen :
“The modified diagonal cycle on the triple product of a pointed curve Ann. Inst. Fourier
45 : 3
(1995 ),
pp. 649–679 .
MR
1340948 Zbl
0822.14015 article
Abstract
BibTeX
Let \( X \) be a curve over a field \( k \) with a rational point \( e \) . We define a canonical cycle
\[ \Delta_e \in Z^2(X^3)_{hom} .\]
Suppose that \( k \) is a number field and that \( X \) has semi-stable reduction over the integers of \( k \) with fiber components non-singular. We construct a regular model of \( X^3 \) and show that the height pairing
\[ \langle\tau_*(\Delta_e),\tau_*^{\prime}(\Delta_e)\rangle \]
is well defined where \( \tau \) and \( \tau^{\prime} \) are correspondences. The paper ends with a brief discussion of heights and \( L \) -functions in the case that \( X \) is a modular curve.
@article {key1340948m,
AUTHOR = {Gross, Benedict H. and Schoen, Chad},
TITLE = {The modified diagonal cycle on the triple
product of a pointed curve},
JOURNAL = {Ann. Inst. Fourier},
FJOURNAL = {Annales de l'Institut Fourier},
VOLUME = {45},
NUMBER = {3},
YEAR = {1995},
PAGES = {649--679},
DOI = {10.5802/aif.1469},
NOTE = {MR:1340948. Zbl:0822.14015.},
ISSN = {0373-0956},
}
B. Gross :
Letter to W.-S. Zhang ,
14 May 1998 .
misc
BibTeX
Read PDF
@misc {key81722447,
AUTHOR = {Gross, B.},
TITLE = {Letter to W.-S. Zhang},
MONTH = {14 May},
YEAR = {1998},
}
B. H. Gross :
“Heegner points and representation theory 37–65
in
Heegner points and Rankin \( L \) -series
(Berkeley, CA, December 2001 ).
Edited by H. Darmon and S.-W. Zhang .
MSRI Publications 49 .
Cambridge University Press ,
2004 .
MR
2083210 Zbl
1126.11032 incollection
Abstract
BibTeX
Our aim in this paper is to present a framework in which the results of Waldspurger and Gross–Zagier can be viewed simultaneously. This framework may also be useful in understanding recent work of Zhang, Xue, Cornut, Vatsal, and Darmon. It involves a blending of techniques from representation theory and automorphic forms with these from the arithmetic of modular curves. I hope readers from one field will be encouraged to pursue the other.
@incollection {key2083210m,
AUTHOR = {Gross, Benedict H.},
TITLE = {Heegner points and representation theory},
BOOKTITLE = {Heegner points and {R}ankin \$L\$-series},
EDITOR = {Darmon, Henri and Zhang, Shou-Wu},
SERIES = {MSRI Publications},
NUMBER = {49},
PUBLISHER = {Cambridge University Press},
YEAR = {2004},
PAGES = {37--65},
DOI = {10.1017/CBO9780511756375.005},
NOTE = {(Berkeley, CA, December 2001). MR:2083210.
Zbl:1126.11032.},
ISSN = {0940-4740},
ISBN = {9780521836593},
}
B. Gross :
Note to W.-S. Zhang ,
December 2007 .
misc
BibTeX
Read PDF
@misc {key81414929,
AUTHOR = {Gross, B.},
TITLE = {Note to W.-S. Zhang},
MONTH = {December},
YEAR = {2007},
}
W. T. Gan, B. H. Gross, and D. Prasad :
“Symplectic local root numbers, central critical \( L \) values, and restriction problems in the representation theory of classical groups 1–109
in
Sur les conjectures de Gross et Prasad, I
[The conjectures of Gross and Prasad, I ].
Edited by W. T. Gan, B. H. Gross, D. Prasad, and J.-L. Waldspurger .
Astérisque 346 .
Société Mathématique de France (Paris ),
2012 .
MR
3202556 Zbl
1280.22019 ArXiv
0909.2999 incollection
Abstract
People
BibTeX
It has been almost 20 years since two of us proposed a rather speculative approach to the problem of restriction of irreducible representations from \( \mathrm{SO}_n \) to \( \mathrm{SO}_{n-1} \) [Gross and Prasad 1992, 1994]. Our predictions depended on the Langlands parametrization of irreducible representations, using \( L \) -packets and \( L \) -parameters. Since then, there has been considerable progress in the construction of local \( L \) -packets, as well as on both local and global aspects of the restriction problem. We thought it was a good time to review the precise conjectures which remain open, and to present them in a more general form, involving restriction problems for all of the classical groups.
@incollection {key3202556m,
AUTHOR = {Gan, Wee Teck and Gross, Benedict H.
and Prasad, Dipendra},
TITLE = {Symplectic local root numbers, central
critical \$L\$ values, and restriction
problems in the representation theory
of classical groups},
BOOKTITLE = {Sur les conjectures de {G}ross et {P}rasad,
{I} [The conjectures of {G}ross and
{P}rasad, {I}]},
EDITOR = {Gan, Wee Teck and Gross, Benedict H.
and Prasad, Dipendra and Waldspurger,
Jean-Loup},
SERIES = {Ast\'erisque},
NUMBER = {346},
PUBLISHER = {Soci\'et\'e Math\'ematique de France},
ADDRESS = {Paris},
YEAR = {2012},
PAGES = {1--109},
URL = {http://www.numdam.org/item/AST_2012__346__1_0/},
NOTE = {ArXiv:0909.2999. MR:3202556. Zbl:1280.22019.},
ISSN = {0303-1179},
ISBN = {9782856293485},
}
