A. J. Scholl

Motives for modular forms

Let M be a pure motive over a number field F of rank n with coefficients in T ⊂ C; M may be thought of as a direct factor of the cohomology of a smooth projective variety X over F cut out by an algebraic correspondence defined over T . It was Langlands who propounded that there should be a bijective correspondence between (resp. absoultey irreducible) motives M and algebraic (resp. cuspidal) automorphic representations Π of GLn(AF ), in such a way that their invariants, ‘L-functions in s ∈ C’, L(M, s) and L(Π, s) match up. Until twenty years ago, most of positive results had been about the direction Π 7→ M = MΠ inspired by Deligne’s insights in [4]; but there have been some spectacular development in the opposite ‘harder’ direction M 7→ Π = ΠM (commonly knowns as ‘modularity of Galois representations’) in Wiles/TaylorWiles on modularity of semi-stable elliptic curves over Q and Breuil-Conrad-Diamond-Taylor on the Shimura-Taniyama-Weil conjecture, culminating in Kisin and Emerton (independently) on the FontaineMazur conjecture for GL2 over Q. One can even formulate and prove some cases of mod p and local p-adic analogues of ‘Langlands correspondences’, while the research currently led by Calegari and Geraghty seems very promising to improve considerably our understanding of significantly many new cases of bijections between M ’s and Π’s. With that spectacular development aside, the goal of this seminar is to unravel the correspondence Π 7→M in the case n = 2 and F is a totally real field; more precisely, we shall read [4] and [8] for F = Q, and [1] for general F .

Modular forms on noncongruence subgroups

In this paper I will describe some results and open problems connected with noncongruence subgroups of PSL2(z). Most of these have their origins in the fundamental paper of Atkin and Swinnerton-Dyer [1]. Considering how much we know about congruence subgroups and the associated modular forms, it is remarkable how little we can say in the general case (to avoid cumbersome language I shall speak of “congruence modular forms” and “noncongruence modular forms”). The principal difficulty is the absence of a satisfactory theory of Hecke operators. For congruence subgroups, the Hecke operators not only provide a direct interpretation of Fourier coefficients of modular forms in terms of eigenvalues, but also furnish a link with arithmetic, essentially through the representation theory of adele groups. Moreover, using the action of Hecke operators one can calculate congruence modular forms with relative ease. For a noncongruence subgroup it is possible to define the Hecke algebra using double cosets (as in Ch. 3 of [9]), but it seems difficult to exploit (I take this opportunity to correct the erroneous assertion to the contrary at the beginning of [6]); and there seem to be no good alternative computational devices with which to calculate noncongruence modular forms. This problem is discussed in detail in [1].

Integral Elements in K-Theory and Products of Modular Curves

In the first part of this paper we use de Jong’s method of alterations to contruct unconditionally ‘integral’ subspaces of motivic cohomology (with rational coefficients) for Chow motives over local and global fields. In the second part, we investigate the integrality of the elements constructed by Beilinson in the motivic cohomology of the product of two modular curves, completing the discussion in section 6 of his paper [1].

On the Hecke Algebra of a Noncongruence Subgroup

We begin by recalling standard facts concerning Hecke algebras and modular forms, for details of which the reader is referred to Chapter 3 of Shimura’s book [11]. By definition H (in Shimura’s notation, R(Γ, GL2(Q) )⊗Q) is the Q-algebra spanned by double cosets [ΓγΓ], for γ ∈ GL2(Q) with det γ > 0. Write as usual Mk(Γ) for the complex vector space of holomorphic modular forms of weight k ≥ 0, and Sk(Γ) for the subspace of cusp forms. There is a natural action of H on Mk(Γ), which preserves Sk(Γ). (In fact there is more than one way to normalise this action; the choice is irrelevant for this paper.)

On modular units

Introduction. In [6], Kubert and Lang describe the group of integral modular units on Γ(n) (“units over Z” in their terminology), and in particular determine its rank. Their method is based on finding explicit generators for the group of all modular units, and then by calculating their q-expansions to determine which are integral. In §5 of , Beilinson suggests another approach, based on representation theory and the geometry of the moduli schemes Mn. In this note I shall carry out this programme and give a representation-theoretic description of the group of integral modular units tensored with Q. From this it will be a simple exercise to calculate the rank for any reasonable congruence subgroup; we give an example at the end of §2. The proof uses the adelic language, and exploits in an essential way the action of Hecke operators at primes dividing the level. This approach reduces the problem to showing that the modular units ∆(qz)/∆(z) (for a prime q) are not integral; we give a proof of this fact by “pure thought” at the very end of the paper.

Hypersurfaces and the Weil Conjectures

We give a proof that the Riemann hypothesis for hypersurfaces over finite fields implies the result for all smooth proper varieties, by a deformation argument which does not use the theory of Lefschetz pencils or the l-adic Fourier transform.

On the algebra of modular forms on a congruence subgroup

Let A be a subring of the complex numbers containing 1, and Γ a subgroup of the modular group of finite index. We say that a modular form on Γ is A-integral if the coefficients of its Fourier expansion at infinity lie in A. We denote by Mk(Γ,A) the A-module of holomorphic A-integral modular forms of weight k, and by M(Γ, A) the graded algebra of A-integral modular forms on Γ.