Helena A. Verrill

Modular Forms on Noncongruence Subgroups and Atkin–Swinnerton-Dyer Relations

We give new examples of noncongruence subgroups Γ ⊂ SL2(ℤ) whose space of weight-3 cusp forms S 3(Γ) admits a basis satisfying the Atkin–Swinnerton-Dyer congruence relations with respect to a weight-3 newform for a certain congruence subgroup.

CUSPIDAL MODULAR SYMBOLS ARE TRANSPORTABLE

Modular symbols of weight 2 for a congruence subgroup0 satisfy the identity f; ./ gDf ; ./g for all ; in the extended upper half plane and 2 0. The analogue of this identity is false for modular symbols of weight greater than 2. This paper provides a definition of transportable modular symbols, which are symbols for which an analogue of the above identity holds, and proves that every cuspidal symbol can be written as a transportable symbol. As a corollary, an algorithm is obtained for computing periods of cuspforms.

Symmetric Groups and Conjugacy Classes

Abstract Let S n be the symmetric group of degree n where n > 5. Given any non-trivial , we prove that the product of the conjugacy classes and is never a conjugacy class. Furthermore, if n is odd and not a multiple of three, then is the union of at least three distinct conjugacy classes. We also describe the elements in the case when is the union of exactly two distinct conjugacy classes.

Lifts of projective congruence groups

We show that noncongruence subgroups of SL2(Z) projectively equivalent to congruence subgroups are ubiquitous. More precisely, they al- ways exist if the congruence subgroup in question is a principal congruence subgroup ( N) of level N > 2, and they exist in many cases also for 0(N). The motivation for asking this question is related to modular forms: pro- jectively equivalent groups have the same spaces of cusp forms for all even weights whereas the spaces of cusp forms of odd weights are distinct in gen- eral. We make some initial observations on this phenomenon for weight 3 via geometric considerations of the attached elliptic modular surfaces. We also develop algorithms that construct all subgroups projectively equiv- alent to a given congruence subgroup and decide which of them are congruence. A crucial tool in this is the generalized level concept of Wohlfahrt.

On ℓ-adic representations for a space of noncongruence cuspforms

This paper is concerned with a compatible family of 4-dimensional l-adic representations ρl of GQ := Gal(Q/Q) attached to the space of weight-3 cuspforms S3(Γ) on a noncongruence subgroup Γ ⊂ SL2(Z). For this representation we prove that: 1. It is automorphic: the L-function L(s,ρl∨) agrees with the L-function for an automorphic form for GL4(AQ), where ρl∨ is the dual of ρl. 2. For each prime p≥5 there is a basis hp = {hp+, hp-} of S3(Γ) whose expansion coefficients satisfy 3-term Atkin and Swinnerton-Dyer (ASD) relations, relative to the q-expansion coefficients of a newform f of level 432. The structure of this basis depends on the class of p modulo 12. The key point is that the representation ρl admits a quaternion multiplication structure in the sense of Atkin, Li, Liu, and Long.

Computing with toric varieties

A computer algebra package (written by the second author) is described which deals with both affine and projective toric varieties in any number of dimensions (written in both MAGMA and GAP). Among other things, the package implements the desingularization procedure, constructs some error-correcting codes associated with toric varieties, and computes the Riemann-Roch space of a divisor on a toric variety.

Transportable Modular Symbols and the Intersection Pairing

Transportable modular symbols were originally introduced in order to compute periods of modular forms [18]. Here we use them to give an algorithm to compute the intersection pairing for modular symbols of weight k ? 2. This generalizes the algorithm given by Merel [13] for computing the intersection pairing for modular symbols of weight 2.We also define a certain subspace of the space of transportable modular symbols, and give numerical evidence to support a conjecture that this space should replace the usual space of cuspidal modular symbols.