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Bo-Hae Im

Publications and source records attributed to Bo-Hae Im.

At least 37 records · Page 2Linked to original sources
arXiv2022

On the common zeros of quasi-modular forms for $Γ_0^+(N)$ of level $N=1,2,3$

In this paper, we study common zeros of the iterated derivatives of the Eisenstein series for $Γ_0^+(N)$ of level $N=1,2$ and $3$, which are quasi-modular forms. More precisely, we investigate the common zeros of quasi-modular forms, and prove that all the zeros of the iterated derivatives of the Eisenstein series $\frac{d^m E_k^{(N)}(τ)}{dτ^m}$ of weight $k=2,4,6$ for $Γ_0^+(N)$ of level $N=2,3$ are simple by generalizaing the results of Meher \cite{MEH} and Gun and Oesterlé \cite{SJ20} for SL$_2(\mathbb{Z})$.

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arXiv2021

Rank gain of Jacobians over number field extensions with prescribed Galois groups

We investigate the rank gain of elliptic curves, and more generally, Jacobian varieties, over non-Galois extensions whose Galois closure has Galois group permutation-isomorphic to a prescribed group $G$ (in short, "$G$-extensions"). In particular, for alternating groups and (an infinite family of) projective linear groups $G$, we show that most elliptic curves over (e.g.) $\mathbb{Q}$ gain rank over infinitely many $G$-extensions, conditional only on the parity conjecture. More generally, we provide a theoretical criterion which allows to deduce that "many" elliptic curves gain rank over infinitely many $G$-extensions, conditional on the parity conjecture and on the existence of geometric Galois realizations with group $G$ and certain local properties.

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arXiv2020

Shimura Varieties, Kummer Varieties, and Rational Curves

For a very general product $A$ of seven or more elliptic curves, every rational curve on the Kummer variety of $A$ projects trivially onto the Kummer variety of at least one of its factors. As a consequence, a very general member of certain families of abelian varieties parametrized by connected Shimura varieties of unitary type has the property that its Kummer variety has no rational curves.

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arXiv2018

Waring's problem for rational functions in one variable

Let $f\in \mathbb{Q}(x)$ be a non-constant rational function. We consider "Waring's Problem for $f(x)$," i.e., whether every element of $\bbq$ can be written as a bounded sum of elements of $\{f(a)\mid a\in \mathbb{Q}\}$. For rational functions of degree $2$, we give necessary and sufficient conditions. For higher degrees, we prove that every polynomial of odd degree and every odd Laurent polynomial satisfies Waring's Problem. We also consider the "Easier Waring's Problem": whether every element of $\mathbb{Q}$ can be represented as a bounded sum of elements of $\{\pm f(a)\mid a\in \mathbb{Q}\}$.

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arXiv2017

Ranks of rational points of the Jacobian varieties of hyperelliptic curves

In this paper, we obtain bounds for the Mordell-Weil ranks over cyclotomic extensions of a wide range of abelian varieties defined over a number field $F$ whose primes above $p$ are totally ramified over $F/\mathbb{Q}$. We assume that the abelian varieties may have good non-ordinary reduction at those primes. Our work is a generalization of \cite{Kim}, in which the second author generalized Perrin-Riou's Iwasawa theory for elliptic curves over $\mathbb{Q}$ with supersingular reduction (\cite{Perrin-Riou}) to elliptic curves defined over the above-mentioned number field $F$. On top of non-ordinary reduction and the ramification of the field $F$, we deal with the additional difficulty that the dimensions of the abelian varieties can be any number bigger than 1 which causes a variety of issues. As a result, we obtain bounds for the ranks over cyclotomic extensions $\mathbb{Q}(μ_{p^{\max(M,N)+n}})$ of the Jacobian varieties of {\it ramified} hyperelliptic curves $y^{2p^M}=x^{3p^N}+ax^{p^N}+b$ among others.

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arXiv2016

Rank gain of Jacobians over finite Galois extensions

Let $\mathcal{X}$ be a Riemann surface of genus $g>0$ defined over a number field $K$ which is a degree $d$-covering of $\mathbb{P}^1_K$. In this paper we show the existence of infinitely many linearly disjoint degree $d$-extensions $L/K$ over which the Jacobian of $\mathcal{X}$ gains rank. In the case where 0, 1 and $\infty$ are the only branch points, and there is an automorphism $σ$ of $\mathcal{X}$ which cyclically permutes these branch points, we obtain the same result for the Jacobian of $\mathcal{X}/σ$. In particular if $\mathcal{X}$ is the Klein quartic, then the construction provides an elliptic curve which gains rank over infinitely many degree $7$-extensions of $\mathbb{Q}$. As an application, we show the existence of infinitely many elliptic curves that gain rank over infinitely many cyclic cubic extensions of $\mathbb{Q}$.

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arXiv2015

Notes on Weierstrass points of modular curves $X_0(N)$

We give conditions when the fixed points by the partial Atkin-Lehner involutions on $X_0(N)$ are Weierstrass points as an extension of the result by Lehner and Newman \cite{LN}. Furthermore, we complete their result by determining whether the fixed points by the full Atkin-Lehner involutions on $X_0(N)$ are Weierstrass points or not.

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arXiv2013

Rational curves on quotients of abelian varieties by finite groups

In [3], it is proved that the quotient of an abelian variety $A$ by a finite order automorphism $g$ is uniruled if and only if some power of $g$ satisfies a numerical condition $0<\age(g^k)<1$. In this paper, we show that $\age(g^k)=1$ is enough to guarantee that $A/\langle g\rangle$ has at least one rational curve.

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arXiv2012

Infinite rank of elliptic curves over $\mathbf{Q}^{\ab}$

If $E$ is an elliptic curve defined over a quadratic field $K$, and the $j$-invariant of $E$ is not 0 or 1728, then $E(\mathbf{Q}^{\ab})$ has infinite rank. If $E$ is an elliptic curve in Legendre form, $y^2 = x(x-1)(x-λ)$, where $\mathbf{Q}(λ)$ is a cubic field, then $E(K \mathbf{Q}^{\ab})$ has infinite rank. If $λ\in K$ has a minimal polynomial $P(x)$ of degree 4 and $v^2 = P(u)$ is an elliptic curve of positive rank over $\bbq$, we prove that $y^2 = x(x-1)(x-λ)$ has infinite rank over $K\bbq^{\ab}$.

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arXiv2012

Some Applications of the Hales-Jewett Theorem to Field Arithmetic

Let $K$ be a field whose absolute Galois group is finitely generated. If $K$ neither finite nor of characteristic 2, then every hyperelliptic curve over $K$ with all of its Weierstrass points defined over $K$ has infinitely many $K$-points. If, in addition, $K$ is not locally finite, then every elliptic curve over $K$ with all of its 2-torsion rational has infinite rank over $K$. These and similar results are deduced from the Hales-Jewett theorem.

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arXiv2010

Semi-direct Galois covers of the affine line

Let $k$ be an algebraically closed field of characteristic $p>0$. Let $G$ be $Z/\ell Z$ semi-direct product $Z/pZ$ where $\ell$ is a prime distinct from $p$. In this paper, we study Galois covers $ψ:Z \to P^1_k$ ramified only over $\infty$ with Galois group $G$. We find the minimal genus of a curve $Z$ that admits such a cover and show that it depends only on $\ell$, $p$, and the order $a$ of $\ell$ modulo $p$. We also prove that the number of curves $Z$ of this minimal genus which admit such a cover is at most $(p-1)/a$.

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arXiv2010

Realizing Rational Representations in Mordell-Weil Groups

Let G be a finite group and V a finite-dimensional rational G-representation. We ask whether there exists a finite Galois extension L/K of number fields with Galois group G, an elliptic curve E/K, and a G-submodule of E(L) tensor Q isomorphic to V.

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arXiv2008

A Weak Chevalley-Warning Theorem for Quasi-finite Fields

There exists a function f: N -> N such that for every positive integer d, every quasi-finite field K and every projective hypersurface X of degree d and dimension at least f(d), the set X(K) is non-empty. This is a special case of a more general result about intersections of hypersurfaces of fixed degree in projective spaces of sufficiently high dimension over fields with finitely generated Galois groups.

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arXiv2006

Abelian Varieties over Cyclic Fields

Let K be a field not of characteristic 2 such that every finite separable extension of K is cyclic. Let A be an abelian variety over K. If K is infinite, then A(K) is Zariski-dense in A. If K is not locally finite, the rank of A over K is infinite.

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arXiv2006

Heegner points and the rank of elliptic curves over large extensions of global fields

Let k be a global field, $\bar{k}$ a separable closure of k, and $G_k$ the absolute Galois group $\Gal(\bar{k}/k)$ of $\bar{k}$ over k. For every g in $G_k$, let $\bar{k}^g$ be the fixed subfield of $\bar{k}$ under g. Let E/k be an elliptic curve over k. We show that for each g in $G_k$, the Mordell-Weil group $E(\bar{k}^g)$ has infinite rank in the following two cases. Firstly when k is a global function field of odd characteristic and E is parametrized by a Drinfeld modular curve, and secondly when k is a totally real number field and E/k is parametrized by a Shimura curve. In both cases our approach uses the non-triviality of a sequence of Heegner points on E defined over ring class fields.

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