Modular curves are the backstage pass to a vast, intricate concert of elliptic curves. Each curve tells a story about how these geometric objects can sit inside families, weave through isogenies, and reveal echoes of number theory in the geometry itself. A recent study by Maarten Derickx and Petar Orlić, based at the University of Zagreb, takes a close look at X0(p), the modular curve that classifies elliptic curves together with a cyclic p-isogeny. Their question is deceptively simple: when you try to map X0(p) to another curve of genus at least 2, what can go on? The answer, it turns out, is a mix of elegant rigidity and stubborn arithmetic that only falls into a handful of familiar channels for primes p below 3000.
Behind that question lies a bigger one: how do low-degree morphisms from a fixed modular curve to higher-genus curves shape the landscape of rational points? The De Franchis–Severi theorem tells us there are only finitely many maps from a fixed curve X to a fixed Y of genus at least 2. But when you fix X0(p) and let Y range over all curves of genus at least 2, a clever blending of geometry, arithmetic, and computation is required to enumerate what actually exists. Derickx and Orlić rise to the challenge with a mix of theory and heavy computation, and they don’t just settle the question for a single p. Their work lays out a classification for all p < 3000 and sets the stage for understanding how many degree six points X0(p) can harbor over Q, a question that sits at the crossroads of geometry and number theory.
Since the authors are part of the University of Zagreb, the study also shines a light on the way modern arithmetic geometry blends local computations with global structure. The lead researchers—Maarten Derickx and Petar Orlić—lead a team that leans on detailed Jacobian decompositions of J0(p), a careful use of Atkin–Lehner involutions, and a suite of computer algebra tools to navigate a landscape where numbers and shapes cohabitate in surprising ways. The outcome is not merely a catalog of which p’s admit exotic maps, but a deeper picture of how the arithmetic of modular curves constrains their geometry in a way that matters for how many rational points they can hold at low degrees.
A Map-Only Landscape: X0(p) and the Genus≥2 Constraint
What makes X0(p) a natural testing ground is simple to state: X0(p) is the moduli space of elliptic curves equipped with a cyclic p-isogeny. Algebraically, it sits as a smooth curve whose genus grows with p, a measure of its geometric complexity. If you ask for every possible non-constant map f from X0(p) to some other curve Y of genus at least 2, a powerful structural constraint appears: such maps are not arbitrary. They must respect the deep arithmetic encoded in the Jacobian J0(p), the space that remembers how all the degree-zero divisors on X0(p) interact with each other under addition, and carry the fingerprints of modular forms.
Derickx and Orlić prove a striking rigidity for p < 3000. If Y has genus at least 2 and f is a non-constant map defined over Q, then f must be isomorphic to the quotient map X0(p) → X0+(p), where X0+(p) is the quotient of X0(p) by the Atkin–Lehner involution wp. In plain terms: for these primes, the only nontrivial way to map X0(p) onto a higher-genus curve is through the canonical “half-turn” symmetry that identifies each point with its wp-partner, collapsing X0(p) onto its Atkin–Lehner quotient X0+(p). There aren’t hidden, exotic routes that land in another high-genus town—the road map is remarkably narrowed down.
Why does this happen? The proof leans on the decomposition of J0(p) into simple abelian varieties that come from newforms, which, for prime p, occur with multiplicity one. The authors enumerate these abelian subvarieties and examine whether any could come from a map to a genus ≥2 curve. A central tool is a corollary derived from a polarization argument: if the image of J(Y) inside J0(p) has too large a polarization kernel relative to the target genus, then such a map cannot exist. In effect, many putative targets Y are ruled out by a geometry that lives inside the Jacobian itself. When that line of attack leaves a few potential cases, they invoke known automorphisms of X0(p) and Riemann–Hurwitz-type constraints to show the only viable nontrivial map is the quotient by wp.
All of this wouldn’t be credible without numbers. The authors don’t stop at a conceptual proof; they run explicit computations of the Jacobian factors for primes up to 3000. One of the hardest test cases—the level p = 2089—reveals a Jacobian with a dozen simple factors. The endurance of these computations, carried out with Magma and Sage, is a reminder that modern number theory often wears two hats: deep theory and meticulous calculation. And it’s not just for a niche curiosity. The conclusion that 77636 primes below 1,000,000 behave almost identically in this respect—barring the odd exceptional display—speaks to a robust structural phenomenon in modular curves of prime level.
When Degree 6 Points Show Up: Theorems and Tangles
Beyond the geometry of maps between curves, the paper tackles a more arithmetic question: for which p do X0(p) curves have infinitely many points of degree six over Q? A degree-d point is simply a point whose coordinates lie in a field extension of Q of degree d. The question matters because Faltings’ theorem tells us that for a curve of genus at least 2, having infinitely many rational points is an exceptional property tied to the curve’s gonality and its diophantine geometry. In this setting, the authors leverage a network of ideas that connect maps to genus ≥2 curves, gonality, and density of points of bounded degree.
The backbone of their analysis is a blend of geometric and arithmetic density results. A key input is Frey’s observation: if a curve C over a number field K has infinitely many points of degree ≤ d, then its gonality over K is at most 2d. That links the abundance of low-degree points to a geometric complexity bound. Yet X0(p) is often too
