arXiv subjects

Ha T. N. Tran

Publications and source records attributed to Ha T. N. Tran.

8 recordsLinked to original sources

arXiv2026

On the Geometry and Shapes of Rank 2 Log Unit Lattices

Every number field canonically gives rise to two lattices: its ring of integers and its log unit lattice. While the shapes of the former have undergone extensive research, far less is known about the shapes of the latter, referred to as unit shapes. This paper presents an in-depth analysis of the unit shapes of number fields with unit rank 2. Our first main result characterizes, in many cases, the location of unit shapes within the fundamental domain of the space of rank 2 lattice shapes in terms of the Galois group of the field's Galois closure, and determines when these unit shapes are transcendental. Next, we establish that the unit shape uniquely determines the field up to isomorphism for totally imaginary $D_6$ non-CM sextic fields; this result fails in the CM case. Finally, for certain subfamilies of $D_6$ non-CM imaginary sextics, we offer a simple sufficient condition for their log unit lattices to be orthogonal and provide lower bounds on the proportion of fields with orthogonal log unit lattice.

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arXiv2026

Gross lattices of supersingular elliptic curves

Let $p$ be a prime, $E$ be a supersingular elliptic curve defined over $\bar{\mathbb{F}}_p$, and $\mathscr{O}$ be its (geometric) endomorphism ring. Earlier results of Chevyrev-Galbraith and Goren-Love have shown that the successive minima of the Gross lattice of $\mathscr{O}$ characterize the isomorphism class of $\mathscr{O}$. In this paper, we extend this work and show that the value of the third successive minimum $D_3$ of the Gross lattice gives necessary and sufficient conditions for the curve to have its $j$-invariant in the field $\mathbb{F}_p$ or in the set $\mathbb{F}_{p^2} \setminus \mathbb{F}_p$, as well as finer information about the endomorphism ring of $E$ when its $j$-invariant belongs to $\mathbb{F}_p$ and $p \equiv 3 \pmod{4}$. We end our article with an investigation of the geometry of Gross lattices of supersingular elliptic curves.

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arXiv2025

Principal Well-Rounded Ideals of real quadratic fields

It has been well known since Gauss that the principality of an ideal in a real quadratic field $K$ is equivalent to the solvability of a certain generalized Pell equations. In this paper, we combine this classical result with Srinivasan's conditions for the existence of well-rounded ideals in $K$ to obtain necessary and sufficient criteria for a real quadratic field to have principal well-rounded (PWR) ideals. Using these criteria, we prove that there are infinitely many real quadratic fields that have PWR ideals. Moreover, these ideals are pairwise non-similar. We then construct new algorithms that produce these PWR ideals, especially when the field discriminant is large. Our algorithms run in sub-exponential time theoretically; however, they are very fast in practice by employing some commonly used probabilistic algorithms for testing squarefreeness. Finally, we briefly consider criteria for the existence of prime PWR ideals and show that there are infinitely many real quadratic fields that have prime PWR ideals.

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arXiv2025

Well-Rounded Twists of the Ring of Integers in Cyclic Cubic Fields

Computing well-rounded twists of ideals in number fields has been done when the field degree is $2$. In this paper, we develop a new algorithm to detect whether a basis of an ideal $\mathfrak{I}$ in a cyclic cubic field $F$ yields a well-rounded twist of $\mathfrak{I}$. We then prove that under certain conditions on a given basis of the ring of integers $\mathcal{O}_F$, the existence of its well-rounded twist is equivalent to the existence of a principal well-rounded ideal in $K$. Applying the result and the algorithm, we explicitly compute well-rounded twists of the ring of integers for cyclic cubic fields in the families of Shanks, Washington, and Kishi. In addition, we show that infinitely many fields in Shanks's family have rings of integers that admit orthogonal well-rounded twists.

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arXiv2023

Well-Rounded ideal lattices of cyclic cubic and quartic fields

In this paper, we find criteria for when cyclic cubic and cyclic quartic fields have well-rounded ideal lattices. We show that every cyclic cubic field has at least one well-rounded ideal. We also prove that there exist families of cyclic quartic fields which have well-rounded ideals and explicitly construct their minimal bases. In addition, for a given prime number $p$, if a cyclic quartic field has a unique prime ideal above $p$, then we provide the necessary and sufficient conditions for that ideal to be well-rounded. Moreover, in cyclic quartic fields, we provide the prime decomposition of all odd prime numbers and construct an explicit integral basis for every prime ideal.

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arXiv2022

Orientations and cycles in supersingular isogeny graphs

The paper concerns several theoretical aspects of oriented supersingular $\ell$-isogeny volcanoes and their relationship to closed walks in the supersingular $\ell$-isogeny graph. Our main result is a bijection between the rims of the union of all oriented supersingular $\ell$-isogeny volcanoes over $\overline{\mathbb{F}}_p$ (up to conjugation of the orientations), and isogeny cycles (non-backtracking closed walks which are not powers of smaller walks) of the supersingular $\ell$-isogeny graph over $\overline{\mathbb{F}}_p$. The exact proof and statement of this bijection are made more intricate by special behaviours arising from extra automorphisms and the ramification of $p$ in certain quadratic orders. We use the bijection to count isogeny cycles of given length in the supersingular $\ell$-isogeny graph exactly as a sum of class numbers of these orders, and also give an explicit upper bound by estimating the class numbers.

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arXiv2022

Well-Rounded Twists of Ideal Lattices from Imaginary Quadratic Fields

In this paper, we investigate the properties of well-rounded twists of a given ideal lattice of an imaginary quadratic field $K$. We show that every ideal lattice $I$ of $K$ has at least one well-rounded twist lattice. Moreover, we provide an explicit algorithm to compute all well-rounded twists of $I$.

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arXiv2022

Orienteering with one endomorphism

In supersingular isogeny-based cryptography, the path-finding problem reduces to the endomorphism ring problem. Can path-finding be reduced to knowing just one endomorphism? It is known that a small endomorphism enables polynomial-time path-finding and endomorphism ring computation (Love-Boneh [36]). An endomorphism gives an explicit orientation of a supersingular elliptic curve. In this paper, we use the volcano structure of the oriented supersingular isogeny graph to take ascending/descending/horizontal steps on the graph and deduce path-finding algorithms to an initial curve. Each altitude of the volcano corresponds to a unique quadratic order, called the primitive order. We introduce a new hard problem of computing the primitive order given an arbitrary endomorphism on the curve, and we also provide a sub-exponential quantum algorithm for solving it. In concurrent work (Wesolowski [54]), it was shown that the endomorphism ring problem in the presence of one endomorphism with known primitive order reduces to a vectorization problem, implying path-finding algorithms. Our path-finding algorithms are more general in the sense that we don't assume the knowledge of the primitive order associated with the endomorphism.

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