Celebratio Mathematica

R. W. Carter: Finite groups of Lie type: Conjugacy classes and complex characters. Wiley (New York), 1985. MR 794307 Zbl 0567.20023 book

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). Part II was published in Math. Ann. 278 (1987). MR 833192 Zbl 0608.14019 article

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].

B. H. Gross and N. R. Wallach: “On quaternionic discrete series representations, and their continuations,” J. Reine Angew. Math. 1996 : 481 (1996), pp. 73–123. MR 1421947 Zbl 0857.22012 article

B. H. Gross and G. Savin: “Motives with Galois group of type \( \mathrm{G}_2 \): An exceptional theta-correspondence,” Compos. Math. 114 : 2 (1998), pp. 153–217. MR 1661756 Zbl 0931.11015 article

B. H. Gross and W. T. Gan: “Haar measure and the Artin conductor,” Trans. Am. Math. Soc. 351 : 4 (1999), pp. 1691–1704. MR 1458303 Zbl 0991.20033 article

B. H. Gross and W. T. Gan: “Commutative subrings of certain non-associative rings,” Math. Ann. 314 : 2 (1999), pp. 265–283. MR 1697445 Zbl 0990.11018 article

W. T. Gan and B. H. Gross: “Integral embeddings of cubic norm structures,” J. Algebra 233 : 1 (November 2000), pp. 363–397. To Nathan Jacobson, in memoriam. MR 1793601 Zbl 0990.11017 article

W. T. Gan, B. Gross, and G. Savin: “Fourier coefficients of modular forms on \( G_2 \),” Duke Math. J. 115 : 1 (2002), pp. 105–169. MR 1932327 Zbl 1165.11315 article

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,” pp. 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

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.

Wee Teck Gan	Related
Dipendra Prasad	Related

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	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},
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W. T. Gan, B. H. Gross, and D. Prasad: “Restrictions of representations of classical groups: Examples,” pp. 111–170 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. with French summary. MR 3202557 Zbl 1279.22023 ArXiv 0909.2993 incollection

In an earlier paper, we considered several restriction problems in the representation theory of classical groups over local and global fields. Assuming the Langlands–Vogan parameterization of irreducible representations, we formulated precise conjectures for the solutions of these restriction problems. In the local case, our conjectural answer is given in terms of Langlands parameters and certain natural symplectic root numbers associated to them. In the global case, the conjectural answer is expressed in terms of the central critical value or derivative of a global \( L \)-function. In this paper, using methods of base change and the theta correspondence, we test our conjectures for depth zero supercuspidal representations of unitary groups, and for more general representations of groups of low rank.

Wee Teck Gan	Related
Dipendra Prasad	Related

W. T. Gan, B. H. Gross, and D. Prasad: “Branching laws for classical groups: The non-tempered case,” Compos. Math. 156 : 11 (2020), pp. 2298–2367. MR 4190046 Zbl 1470.11126 ArXiv 1911.02783 article

This paper generalizes the Gan–Gross–Prasad (GGP) conjectures that were earlier formulated for tempered or more generally generic \( L \)-packets to Arthur packets, especially for the non-generic \( L \)-packets arising from Arthur parameters. The paper introduces the key notion of a relevant pair of Arthur parameters that governs the branching laws for \( \mathrm{GL}_n \) and all classical groups over both local fields and global fields. It plays a role for all the branching problems studied in Gan et al. [Symplectic local root numbers, central critical \( L \)-values and restriction problems in the representation theory of classical groups. Sur les conjectures de Gross et Prasad. I, Astérisque 346 (2012), 1–109] including Bessel models and Fourier–Jacobi models.

Wee Teck Gan	Related
Dipendra Prasad	Related

W. T. Gan, B. H. Gross, and D. Prasad: Twisted GGP problems and conjectures, 2022. In preparation. misc

Wee Teck Gan	Related
Dipendra Prasad	Related

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Benedict H. Gross

In Celebration of Dick Gross

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