arXiv subjects

Frederick Saia

Publications and source records attributed to Frederick Saia.

8 recordsLinked to original sources

arXiv2025

Effective surjectivity of Galois representations of products of elliptic curves over function fields

We prove an effective surjectivity result for Galois representations of products of non-isotrivial, non-isogenous elliptic curves over certain function fields of characteristic $0$. This is by way of an isogeny degree bound in this setting, generated from bounds for elliptic curves by Griffon--Pazuki and from the function field analogue of the Frey--Mazur conjecture, by employing techniques originating in work by Serre and Masser--W{\"{u}}stholz in the number field setting.

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arXiv2025

Shimura curve Atkin--Lehner quotients of genus at most two

We provide a complete enumeration of all quotients of genus $0, 1$ and $2$ of the Shimura curves $X_0^D(N)$ over $\mathbb{Q}$ by non-trivial subgroups of Atkin--Lehner involutions. For all $1270$ genus $1$ quotients $X$ with $N$ squarefree, we determine the isomorphism class of the Jacobian $X$. For $146$ non-elliptic genus $1$ curves $X$ and for $405$ curves genus $2$ quotients $X$, we provide a defining equation for $X$. A main tool for us is the theory of \v{C}erednik--Drinfeld uniformizations of the curves $X_0^D(N)$, which we implement in wider generality than has previously been done in the literature.

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arXiv2025

Replacement dynamics of binary quadratic forms

For an $S$-valued function $f$ of $m \geq 1$ variables we consider the dynamical process in which the output $f(\overline{v})$ replaces exactly one entry of the input $\overline{v} \in S^m$ at each step. This can be viewed as a special case of multivariate polynomial semigroup dynamics, and our study focuses on periodic vectors with respect to this process. We define a stratification of periodic vectors according to their type, and characterize types for which the determination of periodic vectors comes down to dynamics of univariate polynomials. We then restrict to the case of a diagonal binary quadratic form $f$ over $\mathbb{Q}$, and classify rational periodic vectors for all types of period up to $5$. This includes two types, of periods $4$ and $5$, which do not arise from the univariate case, and we prove that there are no periodic vectors over the rationals of the single non-univariate type of period $4$.

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arXiv2025

Point counts, automorphisms, and gonalities of Shimura curves

We implement an algorithm to compute the number of points over finite fields for the Shimura curves $X_0^D(N)$ over $\mathbb{Q}$ and their Atkin--Lehner quotients. Our computations identify $116$ such quotients over finite fields (out of $783514$ tested) that attain a number of rational points exceeding that of any previously known curve of the same genus over the same finite field. To illustrate the utility of our point counts algorithm in addressing arithmetic questions, we prove that all automorphisms are Atkin--Lehner for $9288$ of the $10609$ curves $X_0^D(N)$ of genus $g > 2$ with $D$ the discriminant of an indefinite quaternion algebra over $\mathbb{Q}$, $N$ a squarefree positive integer coprime to $D$, and $DN\leq 10000$, and we determine all tetragonal and geometrically tetragonal curves $X_0^D(N)$ up to a small number of possible exceptions.

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arXiv2024

Bielliptic Shimura curves $X_0^D(N)$ with nontrivial level

We work towards completely classifying all bielliptic Shimura curves $X_0^D(N)$ with nontrivial level $N$ coprime to $D$, extending a result of Rotger that provided such a classification for level one. Combined with prior work, this allows us to determine the list of all relatively prime pairs $(D,N)$ for which $X_0^D(N)$ has infinitely many degree $2$ points. As an application, we use these results to make progress on determining which curves $X_0^D(N)$ have sporadic points. Using tools similar to those that appear in this study, we also determine all of the geometrically trigonal Shimura curves $X_0^D(N)$ with $\gcd(D,N)=1$ (none of which are trigonal over $\mathbb{Q}$).

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arXiv2022

CM Elliptic Curves: Volcanoes, Reality and Applications, Part II

Let $M \mid N$ be positive integers, and let $Δ$ be the discriminant of an order in an imaginary quadratic field $K$. When $Δ_K < -4$, the first author determined the fiber of the morphism $X_0(M,N) \rightarrow X(1)$ over the closed point $J_Δ$ corresponding to $Δ$ and showed that all fibers of the map $X_1(M,N) \rightarrow X_0(M,N)$ over $J_Δ$ were connected. Here we complement this prior work by addressing the most difficult cases $Δ_K \in \{-3,-4\}$. These works provide all the information needed to compute, for each positive integer $d$, all subgroups of $E(F)[\operatorname{tors}]$, where $F$ is a number field of degree $d$ and $E_{/F}$ is an elliptic curve with complex multiplication.

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arXiv2022

CM points on Shimura curves via QM-equivariant isogeny volcanoes

We study CM points on the Shimura curves $X_0^D(N)_{/\mathbb{Q}}$ and $X_1^D(N)_{/\mathbb{Q}}$, parametrizing abelian surfaces with quaternionic multiplication and extra level structure. A description of the locus of points with CM by a specified order is obtained for general level, via an isogeny-volcano approach in analogy to work of Clark and Clark--Saia in the $D=1$ case of modular curves. This allows for a count of all points with CM by a specified order on such a curve, and a determination of all primitive residue fields and primitive degrees of such points on $X_0^D(N)_{/\mathbb{Q}}$. We leverage computations of least degrees towards the existence of sporadic CM points on $X_0^D(N)_{/\mathbb{Q}}$.

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arXiv2016

Classically Integral Quadratic Forms Excepting at Most Two Values

Let $S \subseteq \mathbb{N}$ be finite. Is there a positive definite quadratic form that fails to represent only those elements in $S$? For $S = \emptyset$, this was solved (for classically integral forms) by the $15$-Theorem of Conway-Schneeberger in the early 1990s and (for all integral forms) by the $290$-Theorem of Bhargava-Hanke in the mid-2000s. In 1938 Halmos attempted to list all weighted sums of four squares that failed to represent $S=\{m\}$; of his $88$ candidates, he could provide complete justifications for all but one. In the same spirit, we ask, "for which $S = \{m, n\}$ does there exist a quadratic form excepting only the elements of $S$?" Extending the techniques of Bhargava and Hanke, we answer this question for quaternary forms. In the process, we prove what Halmos could not; namely, that $x^2+2y^2+7z^2+13w^2$ represents all positive integers except $5$. We develop new strategies to handle forms of higher dimensions, yielding an enumeration of and proofs for the $73$ possible pairs that a classically integral positive definite quadratic form may except.

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