Benedict H. Gross

Arithmetic on elliptic curves with complex multiplication. II

0. In this paper we will continue to study the arithmetic of elliptic curves with complex multiplication by Q ( 1 / ~ ) , which we began in [5]. Chapter I reviews the basic facts on Q-curves, and Chap. II discusses the first p-descent. In Chap. III we present a refinement of the conjecture of Birch and Swinnerton-Dyer for the L-series of these curves, and test it against the theoretical evidence. Chapter IV contains a discussion of some computations which also support this conjecture.

Arithmetic invariants of discrete Langlands parameters

Let G be a reductive algebraic group over the local field k. The local Langlands conjecture predicts that the irreducible complex representations π of the locally compact group G(k) can be parametrized by objects of an arithmetic nature: homomorphisms φ from the Weil-Deligne group of k to the complex L-group of G, together with an irreducible representation ρ of the component group of the centralizer of φ. In light of this conjecture which has been established for algebraic tori, as well as for some nonabelian groups like GLn(k) [21],[23], and SLn(k) [25] it is reasonable to predict how representation theoretic invariants of π = π(φ, ρ) relate to the arithmetic invariants of its parameters (φ, ρ). An early example of this was the paper [17], which predicts branching laws for the restriction of irreducible representations of the group SOn(k) to the subgroup SOn−1(k), using the e-factor of a distinguished symplectic representation of the L-group of SOn × SOn−1. These conjectures have now been verified in several cases; see [19] and [20].

Group representations and lattices

This paper began as an attempt to understand the Euclidean lattices that were recently constructed (using the theory of elliptic curves) by Elkies [E] and Shioda [Shl, Sh2], from the point of view of group representations. The main idea appears in a note of Thompson [Th2]: if one makes strong irreducibility hypotheses on a rational representation V of a finite group G, then the G-stable Euclidean lattices in V are severely restricted. Unfortunately, these hypotheses are rarely satisfied when V is absolutely irreducible over Q. But there are many examples where the ring EndG(V) is an imaginary quadratic field or a definite quaternion algebra. These representations allow us to construct some of the Mordell-Weil lattices considered by Elkies and Shioda, as well as some interesting even unimodular lattices that do not seem to come from the theory of elliptic curves. In ? 1 we discuss lattices and Hermitian forms on Y7, and in ??2-4 the strong irreducibility hypotheses we wish to make. In ?5 we show how our hypotheses imply the existence of a finite number (up to isomorphism) of Euclidean Z[G]lattices L in V with EndG(L) a maximal order in EndG(V) . We give some examples with dim V < 8 in ?6, and in ??7-9 discuss the invariants of L, such as the dual lattice and theta function. The rest of the paper is devoted to examples: in most, G is a finite group of Lie type and V is obtained as an irreducible summand of the Weil representation of G. Some of the representation theoretic problems left open by this paper are: to find all examples of pairs (G, V) satisfying the strong irreducibility hypotheses of ??2-4, and to determine the invariants (shortest nonzero vector, theta function, Thompson series, ...) of the G-lattices L so effortlessly constructed inside V.

Algebraic modular forms

In this paper, we develop an algebraic theory of modular forms, for connected, reductive groupsG overQ with the property that every arithmetic subgroup Γ ofG(Q) is finite. This theory includes our previous work [15] on semi-simple groupsG withG(R) compact, as well as the theory of algebraic Hecke characters for Serre tori [20]. The theory of algebraic modular forms leads to a workable theory of modular forms (modp), which we hope can be used to parameterize odd modular Galois representations.The theory developed here was inspired by a letter of Serre to Tate in 1987, which has appeared recently [21]. I want to thank Serre for sending me a copy of this letter, and for many helpful discussions on the topic.

The Weyl character formula, the half-spin representations, and equal rank subgroups.

Let B be a reductive Lie subalgebra of a semi-simple Lie algebra F of the same rank both over the complex numbers. To each finite dimensional irreducible representation Vlambda of F we assign a multiplet of irreducible representations of B with m elements in each multiplet, where m is the index of the Weyl group of B in the Weyl group of F. We obtain a generalization of the Weyl character formula; our formula gives the character of Vlambda as a quotient whose numerator is an alternating sum of the characters in the multiplet associated to Vlambda and whose denominator is an alternating sum of the characters of the multiplet associated to the trivial representation of F.

Local Heights on Curves

In this paper we will review the theory of local heights on curves and describe its relationship to the global height pairing on the Jacobian. The local results are all special cases of Neron’s theory [9], [10]; the global pairing was discovered independently by Neron and Tate [5], We will also discuss extensions of the local pairing to divisors of arbitrary degree and to divisors which are not relatively prime. The first extension is due to Arakelov [1]; the second is implicit in Tate’s work on elliptic curves [12]. I have also included several sections of examples which illustrate the general theory.

A Distinguished Family of Unitary Representations for the Exceptional Groups of Real Rank = 4

In this note, we will construct three small unitary representations for each of the four simply-connected exceptional Lie groups G of real rank = 4. We will describe the restrictions of these representations to a maximal compact subgroup K of G, and will show they are multiplicity-free. The method of construction is by a continuation of the “quaternionic discrete series” for G. This works in more generality, and we will treat it fully in another paper, so we have only sketched the proofs here.

Some applications of Gelfand pairs to number theory

The classical theory of Gelfand pairs has found a wide range of applications, ranging from harmonic analysis on Riemannian symmetric spaces to coding theory. Here we discuss a generalization of this theory, due to Gelfand-Kazhdan, and Bernstein, which was developed to study the representation theory of p-adic groups. We also present some recent number-theoretic results, on local e-factors and on the central critical values of automorphic L-functions, which fit nicely into this framework.

Arithmetic invariant theory

Let k be a field, let G be a reductive algebraic group over k, and let V be a linear representation of G. Geometric invariant theory involves the study of the k-algebra of G-invariant polynomials on V, and the relation between these invariants and the G-orbits on V, usually under the hypothesis that the base field k is algebraically closed. In favorable cases, one can determine the geometric quotient \(V /\!/G = \mathrm{Spec}(\mathrm{Sym}^{{\ast}}(V ^{\vee })^{G})\) and can identify certain fibers of the morphism \(V \rightarrow V/\!/G\) with certain G-orbits on V. In this paper we study the analogous problem when k is not algebraically closed. The additional complexity that arises in the orbit picture in this scenario is what we refer to as arithmetic invariant theory. We illustrate some of the issues that arise by considering the regular semisimple orbits—i.e., the closed orbits whose stabilizers have minimal dimension—in three arithmetically rich representations of the split odd special orthogonal group \(G = \mathrm{SO}_{2n+1}\).

Motives with Galois group of type G2: An exceptional theta-correspondence

In this paper, we study an exceptional theta correspondence, obtained by restricting the minimal automorphic representation of the adjoint group of type E7 and rank 3 over Q to the dual pair GxPGSp6. Here G is the anisotropic form of G2 over Q; using the correspondence, we lift certain automorphic forms on G to holomorphic cusp forms on PGSp6. This lifting provides the first step in a project to construct motives of rank 7 and weight O over Q with Galois group of type G2.