Kamal Khuri-Makdisi

Asymptotically fast group operations on Jacobians of general curves

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On the Iwahori-Hecke algebra of a p-adic group

with the subcategories indexed by pairs (M,σ) with M a Levi subgroup, and σ an irreducible supercuspidal representation of M , under a certain equivalence relation (“association”) to be explained below. In [BR], based on unpublished work of Bernstein, the authors construct a projective generator (to be defined below) for each subcategory Si(G). For a given subcategory, the generators are not unique, and in fact the method in general gives a way to produce a finite number of non-isomorphic generators. Nevertheless, there is, up to isomorphism, a canonical choice of generator, Πi, for each Si(G), and this leads naturally to the equivalence of categories ModEnd S(G)(Πi) ≡ Si(G), where ModR is the category of right modules over a ring R. This paper concerns analyzing the algebra End S(G)(Πi) in the case where Si(G) is the familiar unramified principal series. We note, however, that the results in [BR] are not needed for this special case. Our study works once the structure theory of the Iwahori-Hecke algebra in section 3 along with the classical “Borel result” on the unramified principal series are known. The latter says that this category is equivalent to the category of representations of the Iwahori-Hecke algebra (this was done first for admissible representations in [Bo] and then generally for smooth representations by Matsumoto [Ma]).

On the Fourier coefficients of nonholomorphic Hilbert modular forms of half-integral weight

Let f(z) = ∑ n≥1 ane 2πinz be a Hecke eigenform of half-integral weight m+1/2, and let g(z) = ∑ n≥1 bne 2πinz be the corresponding even-weight form, in the sense of [Sh 73]. In particular, g has weight 2m, and belongs to the same eigenvalues of Hecke operators as f . If n = qr with squarefree r, then an is expressible in terms of ar and the {bj}. At the end of [Sh 77], Shimura suggested that ar should be related to special values of Dirichlet series associated to g. This was borne out in [Wa 81], where Waldspurger proved the striking relation that for squarefree r, ar is essentially proportional to ∑ n≥1 φr(n)bnn −s ∣∣ s=m . Here we have twisted the standard Dirichlet series for g by a character φr obtained from the character of f and the quadratic character ( r · ) . The purpose of this paper is to derive generalizations of Waldspurger’s relation, with an explicit proportionality constant, in the case where f and g are nonholomorphic Hilbert modular forms over a totally real number field F . If F = Q, such forms are also called Maass forms. The method of proof, which ought to generalize to arbitrary number fields, follows, with some simplifications, that in [Sh 93a], which treats the case of holomorphic Hilbert modular forms. Previous investigations into this topic have included work by Kohnen and Zagier ([Ko-Za 81] and [Ko 85]) in the holomorphic case, and Katok and Sarnak ([Ka-Sa 93]) in the nonholomorphic case. Both of these treatments deal only with forms on the upper half-plane (i.e. F = Q), with some additional restrictions. Recent (not yet published) work of M. Furusawa suggests that the method in [Sh 93a] and in this paper should generalize to yield a similar formula, in the case of the correspondence between automorphic forms on Sp(n) and on O(2n+ 1). Extending Shimura’s work in [Sh 93a] to the nonholomorphic case involves two main difficulties. First, as the Fourier expansions of Maass forms involve Whittaker functions instead of exponentials, Mellin transforms and Rankin-Selberg convolutions produce more complicated “Gamma-factors” than usual; these factors must be explicitly evaluated, in order to yield precise versions of Waldspurger’s relation. Second, whereas the Fourier expansions of holomorphic forms are indexed only by totally positive elements of the field F , the expansions of Maass forms are indexed by field elements of arbitrary signature; this makes the calculations rather more delicate. Section 3 of this paper explains in explicit detail how one overcomes both of these problems in constructing a Dirichlet series from Hilbert modular forms of arbitrary integral weight. The results in section 3 are in principle known from [Ma 53], [J-L], and [W], but are not found in this form in the literature (see the discussion at the beginning of section 3). This applies particularly to the appendix to section 3, where we explicitly compute the Mellin transforms of all possible Whittaker functions at the archimedean places. Sections 1 through 4 contain nothing new, but rather set up precise definitions and normalizations of all automorphic forms, special functions, and parameters

MODULI INTERPRETATION OF EISENSTEIN SERIES

Let L >= 3. Using the moduli interpretation, we define certain elliptic modular forms of level Gamma(L) over any field k where 6L is invertible and k contains the Lth roots of unity. These forms generate a graded algebra R_L, which, over C, is generated by the Eisenstein series of weight 1 on Gamma(L). The main result of this article is that, when k=C, the ring R_L contains all modular forms on Gamma(L) in weights >= 2. The proof combines algebraic and analytic techniques, including the action of Hecke operators and nonvanishing of L-functions. Our results give a systematic method to produce models for the modular curve X(L) defined over the Lth cyclotomic field, using only exact arithmetic in the L-torsion field of a single Q-rational elliptic curve E^0.

Fast Jacobian group operations for C_{3,4} curves over a large finite field

Let C be an arbitrary smooth algebraic curve of genus g over a large finite field K. We revisit fast addition algorithms in the Jacobian of C due to Khuri-Makdisi (math.NT/0409209, to appear in Math. Comp.). The algorithms, which reduce to linear algebra in vector spaces of dimension O(g) once |K| >> g, and which asymptotically require O(g^{2.376}) field operations using fast linear algebra, are shown to perform efficiently even for certain low genus curves. Specifically, we provide explicit formulae for performing the group law on Jacobians of C_{3,4} curves of genus 3. We show that, typically, the addition of two distinct elements in the Jacobian of a C_{3,4} curve requires 117 multiplications and 2 inversions in K, and an element can be doubled using 129 multiplications and 2 inversions in K. This represents an improvement of approximately 20% over previous methods.

GENERATING FUNCTIONS FOR HECKE OPERATORS

Fix a prime N, and consider the action of the Hecke operator TN on the space of modular forms of full level and varying weight κ. The coefficients of the matrix of TN with respect to the basis {E4i E6j | 4i + 6j = κ} for can be combined for varying κ into a generating function FN. We show that this generating function is a rational function for all N, and present a systematic method for computing FN. We carry out the computations for N = 2, 3, 5, and indicate and discuss generalizations to spaces of modular forms of arbitrary level.

An Exact Sequence in the Representation Theory of SL(2)

Abstract Let Vbe the standard two-dimensional representation of the algebraic group G = SL(2, C), and write V n = Sym n Vfor the irreducible (n + 1)-dimensional representation of Gon the nth symmetric tensor power of V. Also consider the (2 n )-dimensional space W n = V ⊗n , obtained as the nth tensor power of V. It is known that each V n can be written in terms of W 0,…, W n as V n = W n − W n−2 + W n−4 −…, where we view V n and the W i as virtual representations of G. We explain this phenomenon by writing down an exact sequence that gives a “resolution” of V n in terms of W 0,…, W n . Dedicated in memory of my colleague Ahmad Shamsuddin.

On Jacobian group arithmetic for typical divisors on curves

In a previous joint article with Abu Salem, we gave efficient algorithms for Jacobian group arithmetic of “typical” divisor classes on

On Inverting the Koszul Complex

C_{3,4}