Cuspidal Divisor Class Research Articles

On the rational cuspidal divisor class groups of Drinfeld modular curves X0(𝔭r)

Let [Formula: see text] be the rational cuspidal divisor class group of the Drinfeld modular curve [Formula: see text] for a prime power level [Formula: see text]. We relate the rational cuspidal divisors of degree 0 on [Formula: see text] with [Formula: see text]-quotients, where [Formula: see text] is the Drinfeld discriminant function. As a result, we are able to determine explicitly the structure of [Formula: see text] for arbitrary prime [Formula: see text] and [Formula: see text].

The rational cuspidal subgroup of J0(p2M)$J_0(p^2M)$ with M squarefree

AbstractFor a positive integer N, let be the modular curve over and its Jacobian variety. We prove that the rational cuspidal subgroup of is equal to the rational cuspidal divisor class group of when for any prime p and any squarefree integer M. To achieve this, we show that all modular units on can be written as products of certain functions , which are constructed from generalized Dedekind eta functions. Also, we determine the necessary and sufficient conditions for such products to be modular units on under a mild assumption.

The rational torsion subgroup of J0(N)

Let N be a positive integer and let J0(N) be the Jacobian variety of the modular curve X0(N). For any prime p≥5 whose square does not divide N, we prove that the p-primary subgroup of the rational torsion subgroup of J0(N) is equal to that of the rational cuspidal divisor class group of X0(N), which is explicitly computed in [33]. Also, we prove the same assertion holds for p=3 under the extra assumption that either N is not divisible by 3 or there is a prime divisor of N congruent to −1 modulo 3.

The rational cuspidal divisor class group of X0(N)

For any positive integer N, we completely determine the structure of the rational cuspidal divisor class group of X0(N), which is conjecturally equal to the rational torsion subgroup of J0(N). More specifically, for a given prime ℓ, we construct a rational cuspidal divisor Zℓ(d) for any non-trivial divisor d of N. Also, we compute the order of the linear equivalence class of Zℓ(d) and show that the ℓ-primary subgroup of the rational cuspidal divisor class group of X0(N) is isomorphic to the direct sum of the cyclic subgroups generated by the linear equivalence classes of Zℓ(d).

Cuspidal divisor class groups of non-split Cartan modular curves

I find an explicit description of modular units in terms of Siegel functions for the modular curves $X^+_{ns}(p^k)$ associated to the normalizer of a non-split Cartan subgroup of level $p^k$ where $p\not=2,3$ is a prime. The Cuspidal Divisor Class Group $\mathfrak{C}^+_{ns}(p^k)$ on $X^+_{ns}(p^k)$ is explicitly described as a module over the group ring $R = \mathbb{Z}[(\mathbb{Z}/p^k\mathbb{Z})^*/\{\pm 1\}]$. In this paper I give a formula involving generalized Bernoulli numbers $B_{2,\chi}$ for $|\mathfrak{C}^+_{ns}(p^k)|$.

The rational torsion subgroups of Drinfeld modular Jacobians and Eisenstein pseudo-harmonic cochains

Let \(\mathfrak n\) be a square-free ideal of \(\mathbb {F}_q[T]\). We study the rational torsion subgroup of the Jacobian variety \(J_0(\mathfrak n)\) of the Drinfeld modular curve \(X_0(\mathfrak n)\). We prove that for any prime number \(\ell \) not dividing \(q(q-1)\), the \(\ell \)-primary part of this group coincides with that of the cuspidal divisor class group. We further determine the structure of the \(\ell \)-primary part of the cuspidal divisor class group for any prime \(\ell \) not dividing \(q-1\).

Some non-congruence subgroups and the associated modular curves

In this paper, we study two families of normal subgroups of Γ0(2) with abelian quotients and their associated modular curves. They are similar to Fermat groups and Fermat curves in some aspects but very different in other aspects. Almost all of them are non-congruence subgroups. These modular curves are either projective lines or hyperelliptic curves. There are few modular forms of weight 1 for these groups. We also determine their cuspidal divisor class groups and show that these groups are finite.

Eisenstein Ideals and the Rational Torsion Subgroups of Modular Jacobian Varieties II

We study the rational torsion subgroup of the modular Jacobian variety $J_0(N)$ when $N$ is square-free. We prove that the $p$-primary part of this group coincides with that of the cuspidal divisor class group for $p\geq 3$ when $3 \nmid N$, and for $p\geq 5$ when $3 \mid N$. We further determine the structure of each eigenspace of such $p$-primary part with respect to the Atkin-Lehner involutions. This is based on our study of the Eisenstein ideals in the Hecke algebras.

Cuspidal class number of the tower of modular curves X 1(Np n )

We consider a cuspidal class number, which is the order of a subgroup of the full cuspidal divisor class group of X 1(Np n ) with $${p\nmid N}$$ and n ≥ 1. By studying the second generalized Bernoulli numbers, we obtain results similar to ones (Ferrero and Washington in Ann Math (2) 109(2):377–395, 1979; Washington in Invent Math 49:87–97, 1978) about the relative class numbers of cyclotomic $${\mathbb{Z}_p}$$ -extension of an abelian number field.

Class invariants and cyclotomic unit groups from special values of modular units

In this article we obtain class invariants and cyclotomic unit groups by considering specializations of modular units. We construct these modular units from functional solutions to higher order q-recurrence equations given by Selberg in his work generalizing the Rogers-Ramanujan identities. As a corollary, we provide a new proof of a result of Zagier and Gupta, originally considered by Gauss, regarding the Gauss periods. These results comprise part of the author’s 2006 Ph.D. thesis [6] in which the structure of these modular unit groups and their associated cuspidal divisor class groups are also characterized, and a cuspidal divisor class number formula is given in terms of products of L-functions and compared to the classical relative class number formula within the cyclotomic fields [6, 7].

A note on the finiteness of certain cuspidal divisor class groups

We prove the finiteness of the subgroup of the Jacobian generated by the cusps of a Drinfeld modular curve. This is an analogue in positive characteristic of a classical result of Drinfeld and Manin about elliptic modular curves, and generalizes earlier work on special cases.

On the Drinfeld discriminant function

The discriminant function Δ is a certain rigid analytic modularform defined on Drinfeld’s upper half-plane Ο. Its absolutevalue ❘Δ❘ may be considered as a function on theassociated Bruhat–Tits tree T. We compare log ❘Δ❘ with the conditionally convergent complex-valued Eisenstein series Edefined on T and thereby obtain results about the growth of ❘Δ❘ and of some related modular forms. We further determine to what extent roots may be extracted of Δ(z)/Δ(nz),regarded as a holomorphic function on Ο. In some cases, this enables us to calculate cuspidal divisor class groups of modular curves.

On the cuspidal divisor class group of a Drinfeld modular curve

The theory of theta functions for arithmetic groups \Gamma that act on the Drinfeld upper half-plane is extended to allow degenerate parameters. This is used to investigate the cuspidal divisor class groups of Drinfeld modular curves. These groups are finite for congruence subgroups \Gamma and may be described through the corresponding quotients of the Bruhat-Tits tree by \Gamma . The description given is fairly explicit, notably in the most important special case of Hecke congruence subgroups \Gamma over a polynomial ring.

The Manin—Drinfeld theorem and Ramanujan sums

The Manin—Drinfeld theorem asserts the finiteness of the cuspidal divisor class group of a modular curve corresponding to a congruence subgroup. The purpose of the note is to draw attention to the connection between this theorem and Ramanujan sums, and to the question of what happens for non-congruence subgroups.

The cuspidal group and special values of 𝐿-functions

The structure of the cuspidal divisor class group is investigated by relating this structure to arithmetic properties of special values of L L -functions of weight two Eisenstein series. A new proof of a theorem of Kubert (Proposition 3.1) concerning the group of modular units is derived as a consequence of the method. The key lemma is a nonvanishing result (Theorem 2.1) for values of the “ L L -function” attached to a one-dimensional cohomology class over the relevant-congruence subgroup. Proposition 4.7 provides data regarding Eisenstein series and associated subgroups of the cuspidal divisor class group which the author hopes will simplify future calculations in the cuspidal group.