Cuspidal Group Research Articles

On rational Eisenstein primes and the rational cuspidal groups of modular Jacobian varieties

Let N N be a non-squarefree positive integer and let ℓ \ell be an odd prime such that ℓ 2 \ell ^2 does not divide N N . Consider the Hecke ring T ( N ) \mathbb {T}(N) of weight 2 2 for Γ 0 ( N ) \Gamma _0(N) and its rational Eisenstein primes of T ( N ) \mathbb {T}(N) containing ℓ \ell . If m \mathfrak {m} is such a rational Eisenstein prime, then we prove that m \mathfrak {m} is of the form ( ℓ , I M , N D ) (\ell , ~\mathcal {I}^D_{M, N}) , where we also define the ideal I M , N D \mathcal {I}^D_{M, N} of T ( N ) \mathbb {T}(N) . Furthermore, we prove that C ( N ) [ m ] ≠ 0 \mathcal {C}(N)[\mathfrak {m}] \neq 0 , where C ( N ) \mathcal {C}(N) is the rational cuspidal group of J 0 ( N ) J_0(N) . To do this, we compute the precise order of the cuspidal divisor C M , N D \mathcal {C}^D_{M, N} and the index of I M , N D \mathcal {I}^D_{M, N} in T ( N ) ⊗ Z ℓ \mathbb {T}(N)\otimes \mathbb {Z}_\ell .

On Eisenstein ideals and the cuspidal group of J 0(N)

Let CN be the cuspidal subgroup of the Jacobian J0(N) for a square-free integer N > 6. For any Eisenstein maximal ideal m of the Hecke ring of level N, we show that CN[m] ≠ 0. To prove this, we calculate the index of an Eisenstein ideal I contained in m by computing the order of the cuspidal divisor annihilated by I.

The $\mathbb{Q}$-rational cuspidal group of $J_{1}(2p)$

Let $p$ be a prime not equal to 2 or 3. In this paper we study the $\mathbb{Q}$-rational cuspidal group $\mathcal{C}_{\mathbb{Q}}$ of the jacobian $J_{1}(2p)$ of the modular curve $X_{1}(2p)$. We prove that the group $\mathcal{C}_{\mathbb{Q}}$ is generated by the $\mathbb{Q}$-rational cusps. We determine the order of $\mathcal{C}_{\mathbb{Q}}$, and give numerical tables for all $p\leq127$. These tables give also other cuspidal class numbers for the modular curves $X_{1}(2p)$ and $X_{1}(p)$. We give a basis of the group of the principal divisors supported on the $\mathbb{Q}$-rational cusps, and using this we determine the explicit structure of $\mathcal{C}_{\mathbb{Q}}$ for all $p\leq127$. We determine the structure of the Sylow $p$-subgroup of $\mathcal{C}_{\mathbb{Q}}$, and the explicit structure for all $p\leq4001$.

Monoids and Cuspidal Group Characters

Let M be a finite monoid with unit group G. By the work of Munn and Ponizovski, the irreducible complex representations of M are classified according to which J-class (apex) they come from. Consider the irreducible representations of M with apex \(\neq G\). These representations restrict to representations of G, whose components we view as coming from J-classes below G. The remaining irreducible representations (and their characters) of G are called cuspidal. We show that an irreducible character \(\chi\) of G is cuspidal if and only if \(\chi(\sum V(e)) =0\) for all idempotents \(e \neq 1\), where \(V(e) = \{x \in G \mid exe = e\}\).

Cuspidal groups, ordinary Eisenstein series, and Kubota–Leopoldt p-adic L-functions

Cuspidal groups, ordinary Eisenstein series, and Kubota–Leopoldt p-adic L-functions

A Note on the Shimura Subgroup of J0( p)

Let J 0( p) be the Jacobian of the modular curve X 0( p). The group of connected components of its Néron model is cyclic, and has two special generators. One comes from the cuspidal group, the other from the Shimura subgroup. We compute their ratio, answering a question raised by B. Mazur ( Inst. Hautes Etudes Sci. Publ. Math. 47, 1977, 33-186).

The Eisenstein ideal of a Fermat curve

The Eisenstein ideal for a Fermat curve, defined by Mazur to be the endomorphisms of the jacobian which annihilate the cuspidal group, is computed for the curve of degree 5.

The meromorphic continuation and functional equations of cuspidal Eisenstein series for maximal cuspidal groups

The meromorphic continuation and functional equations of cuspidal Eisenstein series for maximal cuspidal groups

Representation theory and the cuspidal group of X(p)

Representation theory and the cuspidal group of X(p)

The Cuspidal Group and Special Values of L-Functions

The Cuspidal Group and Special Values of L-Functions

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.