Searcharxiv⌕ Search

arXiv subjects

Bo-Hae Im

Publications and source records attributed to Bo-Hae Im.

At least 19 recordsLinked to original sources
arXiv2026

The threshold for linear independence of multiple zeta values in positive characteristic

A fundamental conjecture formulated by Thakur in 2009, which has guided significant developments in function field arithmetic, asserts that multiple zeta values (MZV's) in positive characteristic of fixed weight are linearly independent over $\mathbb{F}_q$. In this paper we settle this conjecture by determining the precise threshold for this independence. We prove that linear independence holds for all weights up to 2q, while for weight 2q+1 we establish the existence of a unique and explicit $\mathbb{F}_q$-linear relation. This result provides the first counterexample to Thakur's conjecture. Our proof relies on a new connection between MZVs and Carlitz multiple polylogarithms over $\mathbb{F}_q$, generalizing a central result of [IKLNDP24]. We also introduce a modification of the algorithm from [ND21] that yields a weight-preserving operator acting on $\mathbb{F}_q$-linear relations, providing the algebraic framework for these results.

math.NT↗
arXiv2026

Analytic rank-one elliptic curves over function fields and their rank over certain ring class fields

Let $E/k$ be a non-isotrivial elliptic curve over a global function field $k$ of characteristic $p>3$, and $G\subset \mathrm{Gal}(k^{\mathrm{sep}}/k)$ be a topologically finitely generated subgroup. We prove that if $E/k$ has analytic rank $1$, then its rank over the fixed subfield $L^G$ is infinite, where $L$ is the infinite ring class extension of some finite separable extension $K/k$. If $E/k$ has analytic rank $0$, then we prove that the same holds provided there exists an imaginary quadratic extension $K/k$ such that $E/K$ has analytic rank $1$ and satisfies the Heegner hypothesis.

math.NT↗
arXiv2025

Stability of torsion subgroups of elliptic curves over non-Galois extensions of odd prime degree

Let $K$ be a field of characteristic $0$ and $E/K$ an elliptic curve over $K$. For a finite extension $L/K$ and a prime~$\ell$, we provide Galois-theoretic sufficient conditions on $L/K$ under which $E\left(L\right)\left[\ell^{\infty}\right] = E\left(K\right)\left[\ell^{\infty}\right]$. For a non-Galois extension $L/K$ of prime degree, we relate the growth of the $\ell^{\infty}$-torsion subgroup of $E$ under the base change $L/K$ to the image of the mod-$\ell$ cyclotomic character. In particular, In particular, we refine Gonz{\'a}lez-Jim{\'e}nez's result by ruling out certain torsion structures for quintic non-Galois extensions $L/\mathbb{Q}$.

math.NT↗
arXiv2025

Elliptic curves and finitely generated Galois groups

Let $K$ be an extension of $\mathbb{Q}$ and $A/K$ an elliptic curve. If $\mathrm{Gal}(\bar K/K)$ is finitely generated, then $A$ is of infinite rank over $K$. In particular, this implies the $g=1$ case of the Junker-Koenigsmann conjecture. This "anti-Mordellic'' result follows from a new "Mordellic'' theorem, which asserts that if $K_0$ is finitely generated over $\mathbb{Q}$, the points of an abelian variety $A_0/K_0$ over the compositum of all bounded-degree Galois extensions of $K_0$ form a virtually free abelian group. This, in turn, follows from a second Mordellic result, which asserts that the group of $A_0$ over the extension of $K_0$ defined by the torsion of $A_0(\bar K_0)$ is free modulo torsion.

math.NT↗
arXiv2025

The automorphism group of torsion points of an elliptic curve over a field of characteristic $\ge 5$

For a field $\mathbb{K}$ of characteristic $p\ge5$ containing $\mathbb{F}_{p}^{\operatorname{alg}}$ and the elliptic curve $E_{s,t}: y^{2} = x^{3} + sx + t$ defined over the function field $\mathbb{K}\left(s,t\right)$ of two variables $s$ and $t$, we prove that for a non-negative positive integer $e$ and a positive integer $N$ which is not divisible by $p$, the automorphism group of the normal extension $\mathbb{K}\left(s,t\right)\left(E_{s,t}\left[p^{e} N\right]\right)$ over $\mathbb{K}\left(s,t\right)$ is isomorphic to $\left(\mathbb{Z}/p^{e}\mathbb{Z}\right)^{\times} \times \operatorname{SL}_{2} \left(\mathbb{Z}/N\mathbb{Z}\right)$.

math.NT↗
arXiv2025

Larsen's conjecture for elliptic curves over $\mathbb{Q}$ with analytic rank at most $1$

We prove Larsen's conjecture for elliptic curves over $\mathbb{Q}$ with analytic rank at most $1$. Specifically, let $E/\mathbb{Q}$ be an elliptic curve over $\mathbb{Q}$. If $E/\mathbb{Q}$ has analytic rank at most $1$, then we prove that for any topologically finitely generated subgroup $G$ of $\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$, the rank of $E$ over the fixed subfield $\overline{\mathbb{Q}}^G$ of $\overline{\mathbb{Q}}$ under $G$ is infinite.

math.NT↗
arXiv2024

Rank growth of abelian varieties over certain finite Galois extensions

Let $A/K$ be an abelian variety over a number field $K$. We prove that a finite automorphism group $G \subseteq \mathrm{Aut}_K(X)$ of a smooth projective variety $X/K$ such that $X/G \cong \mathbb{P}_K^d$ can force the rank growth of $A$ over infinitely many mutually linearly disjoint $G$-extensions $L_i/K$. The proof is based on Hilbert irreducibility and N\'{e}ron specialization. We then combine the theorem with finite group representations to obtain explicit lower bounds for rank growth. As applications, we obtain rank growth results for Jacobian varieties and construct explicit examples. We further prove arbitrarily large rank growth of abelian varieties over symmetric extensions. Finally, we study the connection with infinite rank conjectures.

math.NT↗
arXiv2024

The automorphism group of the $p^{n}$-torsion points of an elliptic curve over a field of characteristic $p \ge 5$

For a field $K$ of characteristic $p\ge5$ and the elliptic curve $E_{s,t}: y^{2} = x^{3} + sx + t$ defined over the function field $K\left(s,t\right)$ of two variables $s$ and $t$, we prove that for a positive integer $n$, the automorphism group of the normal extension $K\left(s,t\right)\left(E_{s,t}\left[p^{n}\right]\right)/K\left(s,t\right)$ is isomorphic to $\left(\mathbb{Z}/p^{n}\mathbb{Z}\right)^{\times}$, and its inseparable degree is $p^{n}$.

math.NT↗
arXiv2024

Zagier-Hoffman's conjectures in positive characteristic

Multiples zeta values and alternating multiple zeta values in positive characteristic were introduced by Thakur and Harada as analogues of classical multiple zeta values of Euler and Euler sums. In this paper we determine all linear relations among alternating multiple zeta values and settle the main goals of these theories. As a consequence we completely establish Zagier-Hoffman's conjectures in positive characteristic formulated by Todd and Thakur which predict the dimension and an explicit basis of the span of multiple zeta values of Thakur of fixed weight.

math.NT↗
arXiv2024

Zagier-Hoffman's conjectures in positive characteristic II

Zagier-Hoffman's conjectures predict the dimension and a basis for the $\mathbb Q$-vector spaces spanned by $N$th cyclotomic multiple zeta values (MZV's) of fixed weight where $N$ is a natural number. For $N=1$ (MZV's case), half of these conjectures have been solved by the work of Terasoma, Deligne-Goncharov and Brown with the help of Zagier's identity. The other half are completely open. For $N=2$ (alternating MZV's case) and $N=3,4,8$, Deligne-Goncharov and Deligne solved the same half of these conjectures for $N$th-cyclotomic MZV's. For other values of $N$, no sharp upper bound on the dimension is known. In this paper we completely establish, for all $N$, Zagier-Hoffman's conjectures for $N$th cyclotomic multiple zeta values in positive characteristic. By working with the tower of all cyclotomic extensions, we present a proof that is uniform on $N$ and give an effective algorithm to express any cyclotomic multiple zeta value in the chosen basis. This generalizes all previous work on these conjectures for MZV's and alternating MZV's in positive characteristic.

math.NT↗
arXiv2024

On the real zeros of depth 1 quasimodular forms

We discuss the critical points of modular forms, or more generally the zeros of quasimodular forms of depth $1$ for $\mathrm{PSL}_2(\mathbb Z)$. In particular, we consider the derivatives of the unique weight $k$ modular forms $f_k$ with the maximal number of consecutive zero Fourier coefficients following the constant $1$. Our main results state that (1) every zero of a depth $1$ quasimodular form near the derivative of the Eisenstein series in the standard fundamental domain lies on the geodesic segment $\{z \in \mathbb H: \Re(z)=1/2\}$, and (2) more than half of zeros of $f_k$ in the standard fundamental domain lie on the geodesic segment $\{z \in \mathbb H: \Re(z)=1/2\}$ for large enough $k$ with $k\equiv 0 \pmod{12}$.

math.NT↗
arXiv2023

Growth of Torsion Groups of Elliptic Curves Over Number Fields without Rationally Defined CM

For a quadratic field $\mathcal{K}$ without rationally defined CM, we prove that there exists of a prime $p_{\mathcal{K}}$ depending only on $\mathcal{K}$ such that if $d$ is a positive integer whose minimal prime divisor is greater than $p_{\mathcal{K}}$, then for any extension $L/\mathcal{K}$ of degree d and any elliptic curve $E/\mathcal{K}$, we have $E\left(L\right)_{\operatorname{tors}} = E\left(\mathcal{K}\right)_{\operatorname{tors}}$. By not assuming the GRH, this is a generalization of the results by Genao, and Gonález-Jiménez and Najman.

math.NT↗
arXiv2023

Hopf algebras and alternating multiple zeta values in positive characteristic

In \cite{IKLNDP23} we presented a systematic study of algebra structures of multiple zeta values in positive characteristic introduced by Thakur as analogues of classical multiple zeta values of Euler. In this paper we construct algebra and Hopf algebra structures of alternating multiple zeta values introduced by Harada, extending our previous work. Our results could be considered as an analogue of those of Hoffman \cite{Hof00} and Racinet \cite{Rac02} in the classical setting. The proof is based on two new ingredients: the first one is a direct and explicit construction of the shuffle Hopf algebra structure, and the second one is the notion of horizontal maps.

math.NT↗
arXiv2023

Hopf algebras and multiple zeta values in positive characteristic

Multiples zeta values (MZV's for short) in positive characteristic were introduced by Thakur as analogues of classical multiple zeta values of Euler. In this paper we give a systematic study of algebraic structures of MZV's in positive characteristic. We construct both the stuffle algebra and the shuffle algebra of these MZV's and equip them with algebra and Hopf algebra structures. In particular, we completely solve a problem suggested by Deligne and Thakur \cite{Del17} in 2017 and establish Shi's conjectures \cite{Shi18}. The construction of the stuffle algebra is based on our recent work \cite{IKLNDP22}.

math.NT↗
arXiv2022

Density of Elliptic Curves over Number Fields with Prescribed Torsion Subgroups

Let $K$ be a number field. For positive integers $m$ and $n$ such that $m\mid n$, we let $\mathscr{S}_{m,n}$ be the set of elliptic curves $E/K$ defined over $K$ such that $E(K)_{\operatorname{tors}}\supseteq \mathscr{T}\cong \mathbb{Z}/m\mathbb{Z}\times \mathbb{Z}/n\mathbb{Z}$. We prove that if the genus of the modular curve $X_{1}(m,n)$ is $0$, then `almost all' $E\in \mathscr{S}_{m,n}$ satisfy that $E(K)_{\operatorname{tors}}= \mathscr{T}$, i.e., not larger than $\mathscr{T}$. In particular, if $m=n=1$, this result generalizes Duke's theorem over $\mathbb{Q}$ to arbitrary number fields $K$ for the trivial torsion subgroup.

math.NT↗
arXiv2022

Non-holomorphic Eisenstein series for certain Fuchsian groups and class numbers

We study certain types of Fuchsian groups of the first kind denoted by $R(N)$, which coincide with the Fricke groups or the arithmetic Hecke triangle groups of low levels. We find all elliptic points and cusps of $R(p)$ for a prime $p$, and prove that there is a one-to-one correspondence between the set of equivalence classes of elliptic points of $R(p)$ and the imaginary quadratic class group. We also find the explicit formula of the Fourier expansion of the non-holomorphic Eisenstein series for $R(N)$ and study their analytic properties. These non-holomorphic Eisenstein series together with cusp forms provide a basis for the space of polyharmonic Maass forms for $R(N)$.

math.NT↗
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$_2(\mathbb{Z})$.

math.NT↗