Hwajong Yoo

Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script l greater-than-or-equal-to 5"> <mml:semantics> <mml:mrow> <mml:mi> ℓ </mml:mi> <mml:mo> ≥ </mml:mo> <mml:mn>5</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">\ell \geq 5</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a prime and let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext

Abstract Following the method of Seifert surfaces in knot theory, we define arithmetic linking numbers and height pairings of ideals using arithmetic duality theorems, and compute them in terms of $n$-th power residue symbols. This formalism leads to a precise arithmetic analogue of a “path-integral formula” for linking numbers.

Let $\ell \geq 5$ be a prime and let $N$ be a square-free integer prime to $\ell$. For each prime $p$ dividing $N$, let $a_p$ be either $1$ or $-1$. We give sufficient criteria for the existence of a newform $f$ of weight 2 for $Γ_0(N)$ such that the mod $\ell$ Galois representation attached to $f$ is reducible and $U_p f = a_p f$ for primes $p$ dividing $N$. The main techniques used are level raising methods based on an exact sequence due to Ribet.

Let N be a non-squarefree positive integer and let l be an odd prime such that l(2) does not divide N. Consider the Hecke ring T(N) of weight 2 for Gamma(0)(N) and its rational Eisenstein primes of T(N) containing l. If m is such a rational Eisenstein prime, then we prove that m is of the form (l, I-M,N(D)), where we also define the ideal I-M,N(D) of T(N). Furthermore, we prove that C(N)[m] not equal 0, where C(N) is the rational cuspidal group of J(0)(N). To do this, we compute the precise orde

Mazur's fundamental work on Eisenstein ideals of prime level has a variety of arithmetic applications. In this article, we generalize some of his work to square-free level. More specifically, we attempt to compute the index of an Eisenstein ideal and the dimension of the m-torsion of the modular Jacobian variety, where m is an Eisenstein maximal ideal. In many cases, the dimension of the m-torsion is 2, in other words, a multiplicity one theorem holds.

Let $\mathcal{C}_N$ be the cuspidal subgroup of the Jacobian $J_0(N)$ for a square-free integer $N&gt;6$. For any Eisenstein maximal ideal $\mathfrak{m}$ of the Hecke ring of level $N$, we show that $\mathcal{C}_N[\mathfrak{m}]\neq 0$. To prove this, we calculate the index of an Eisenstein ideal $\mathcal{I}$ contained in $\mathfrak{m}$ by computing the order of a cuspidal divisor annihilated by $\mathcal{I}$.

Let $p$ be a prime greater than 3. Consider the modular curve $X_0(3p)$ over $\mathbb Q$ and its Jacobian variety $J_0(3p)$ over $\mathbb Q$. Let $\mathcal T(3p)$ and $\mathcal C(3p)$ be the group of rational torsion points on $J_0(3p)$ and the cuspidal g

The purpose of this thesis is to explain modularity of residually reducible Galois representations. More precisely, for a given reducible mod l representation, we want to classify the set of newforms whose associated mod l representations are isomorphic to it.We describe the partial result of the above classification.

Let $\ell \geq 5$ be a prime and let $N$ be a non-squarefree integer not divisible by $\ell$. For a rational Eisenstein prime $\mathfrak{m}$ of the Hecke ring $\mathbb{T}(N)$ of level $N$ acting on $J_0(N)$, we precisely compute the dimension of the kernel $J_0(N)[\mathfrak{m}]$ under a mild assumption. In the case of level $qr^2$ which violates our mild assumption, we propose a conjecture based on Sage computations. Assuming this conjecture, we complete our computation in all the remaining case

Let $E$ be an elliptic curve over a number field $K$ defined by a monic irreducible cubic polynomial $F(x)$. When $E$ is \textit{nice} at all finite primes of $K$, we bound its $2$-Selmer rank in terms of the $2$-rank of a modified ideal class group of the field $L=K[x]/{(F(x))}$, which we call the \textit{semi-narrow class group} of $L$. We then provide several sufficient conditions for $E$ being nice at a finite prime. As an application, when $K$ is a real quadratic field, $E/K$ is semistable

Let $E$ be an elliptic curve over a number field $K$ defined by a monic irreducible cubic polynomial $F(x)$. When $E$ is \textit{nice} at all finite primes of $K$, we bound its $2$-Selmer rank in terms of the $2$-rank of a modified ideal class group of the field $L=K[x]/{(F(x))}$, which we call the \textit{semi-narrow class group} of $L$. We then provide several sufficient conditions for $E$ being nice at a finite prime. As an application, when $K$ is a real quadratic field, $E/K$ is semistable