Let K be a number field, and let E∕ K be an elliptic curve over K. The Mordell–Weil theorem
asserts that the K-rational points E (K) of E form a finitely generated abelian group. In this …

For a positive integer N, let C (N) be the subgroup of J 0 (N) generated by the equivalence
classes of cuspidal divisors of degree 0 and C (N)(Q):= C (N)∩ J 0 (N)(Q) be its Q-rational …

For a totally real field $ F $, a finite extension $\mathbf {F} $ of $\mathbf {F} _p $, and a
Galois character $\chi: G_F\to\mathbf {F}^{\times} $ unramified away from a finite set of …

We consider the generalised Jacobian\(J_ {0}(N) _ {{\textbf {m}}}\) of the modular curve\(X_
{0}(N)\) of level N, with respect to the modulus\({\textbf {m}}\) consisting of all cusps on the …

Let p be an odd prime and ka non-negative integer. Let N be a positive integer such that
p\not ∣ N p∤ N and λ λ a Dirichlet character modulo N. We obtain quantitative lower bounds …

For a positive integer N, let X 0 (N) X_0(N) be the modular curve over QQ and J 0 (N) J_0(N)
its Jacobian variety. We prove that the rational cuspidal subgroup of J 0 (N) J_0(N) is equal …

Let p be a prime and let J1 (pn) denote the Jacobian of the modular curve X1 (pn). The
Jacobian J1 (pn) contains a ℚ‐rational torsion subgroup generated by the cuspidal divisor …

Let $ F $ be a number field. Let $ p $ be a prime number. Washington proved the $\ell $-part
of the class numbers in cyclotomic $\mathbb {Z} _p $ extension of $ F $ is bounded when $ F …