Let ℓ and n be positive integers with ℓ prime. The modular curves X_1(ℓ^n) and X_0(ℓ^n) are algebraic curves over ℚ whose non-cuspidal points parameterize elliptic curves with a distinguished point of order ℓ^n or a distinguished cyclic subgroup of order ℓ^n, respectively. We wish to understand isolated points on these curves, which are roughly those not belonging to an infinite parameterized family of points having the same degree. Our first main result is that there are precisely 15 j-invariants in ℚ which arise as the image of an isolated point x∈ X_1(ℓ^n) under the natural map j:X_1(ℓ^n) → X_1(1). This completes a prior partial classification of Ejder. We also identify the 19 rational j-invariants which correspond to isolated points on X_0(ℓ^n).

Given a number field k, and a quadratic rational function f(x) is an element of k(x),the associated arboreal representation of the absolute Galois group of k is a subgroup of the automorphism group of a regular rooted binary tree. Boston and Jones conjectured that the image of such a representation for f is an element of Z[x] contains a dense set of settled elements. An automorphism is settled if the number of its orbits on the nth level of the tree remains small as n goes to infinity. In this article, we exhibit many quadratic rational functions whose associated Arboreal Galois groups are not densely settled. These examples arise from quadratic rational functions whose critical points lie in a single periodic orbit. To prove our results, we present a detailed study of the iterated monodromy groups (IMG) of f, which also allows us to provide a negative answer to Jones and Levy's question regarding settled pairs. Furthermore, we study the iterated extension k(f-infinity(t)) generated by adjoining to k(t) all roots of fn(x) = t for n >= 1 for a parameter t. We call the intersection of k(f-infinity(t)) with k, the field of constants associated with f. When one of the two critical points of f is the image of the other, we show that the field of constants is contained in the cyclotomic extension of k generated by all 2-power roots of unity. In particular, we prove the conjecture of Ejder, Kara, and Ozman regarding the rational function 1 (x-1)2 . (c) 2025 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

Let f(x) ∈ K(x) be a quadratic polynomial where K is a field of characteristic not equal to 2. The associated arboreal Galois representation of the absolute Galois group of K acts on a regular rooted binary tree. Boston and Jones conjectured that, for f ∈ℤ[x], the image of this representation contains a dense set of settled elements. Roughly speaking, a cycle of an automorphism τ of the tree is called stable if its length strictly increases at each subsequent level, and τ is called settled if the proportion of vertices contained in stable cycles goes to 1 as the level goes to infinity. In this article, we prove that the arithmetic iterated monodromy groups of postcritically finite quadratic polynomials in K[x] with periodic postcritical orbits are densely settled. In the number field case, by a result of Benedetto–Ghioca–Juul–Tucker , it follows that for infinitely many a ∈ K, the associated arboreal Galois representations are densely settled. In particular, our results apply to the arithmetic IMG of the Basilica map f(x)=x^2-1.

We study the arithmetic and geometric iterated monodromy groups associated to the postcritically finite (PCF) quadratic rational function f(x)=2/(x-1)^2 defined over a number field k, whose critical points are both strictly pre-periodic. We give explicit recursive descriptions of the topological generators of the geometric iterated monodromy group of f and show that the arithmetic iterated monodromy group has Hausdorff dimension zero. We describe an explicit criterion to determine the values a∈ k for which the associated arboreal Galois group achieves its maximum possible size. In particular, we show that maximality of the arboreal Galois group can already be verified at level four, which is computationally accessible. Finally, we determine the intersection of the constant field of the arithmetic iterated monodromy group with k(μ_2^∞), providing the first full study of a PCF quadratic map with non-abelian constant field.

Let & ell; and n be positive integers with & ell; prime. The modular curves X1(& ell;n) and X0(& ell;n) are algebraic curves over Q whose non-cuspidal points parameterize elliptic curves with a distinguished point of order & ell;n or a distinguished cyclic subgroup of order & ell;n, respectively. We wish to understand isolated points on these curves, which are roughly those not belonging to an infinite parameterized family of points having the same degree. Our first main result is that there are precisely 15 j-invariants in Q which arise as the image of an isolated point x is an element of X1(& ell;n) under the natural map j:X1(& ell;n)-> X1(1). This completes a prior partial classification of Ejder. We also identify the 19 rational j-invariants which correspond to isolated points on X0(& ell;n).

We study the postcritically finite non-polynomial map $f(x)=\frac{1}{(x-1)^2}$ over a number field $k$ and prove various results about the geometric $G^{\text{geom}}(f)$ and arithmetic $G^{\text{arith}}(f)$ iterated monodromy groups of $f$. We show that the elements of $G^{\text{geom}}(f)$ are the ones in $G^{\text{arith}}(f)$ that are fixing the roots of unity by assuming a conjecture on the size of $G^{\text{geom}}_n(f)$. Furthermore, we describe exactly for which $a \in k$ the Arboreal Galois group $G_a(f)$ and $G^{\text{arith}}(f)$ are equal.

Let $$\ell $$ be a prime and let $$n\ge 1$$ . In this note we show that if there is a non-cuspidal, non-CM isolated point x with a rational j-invariant on the modular curve $$X_1(\ell ^n)$$ , then $$\ell =37$$ and the j-invariant of x is either $$7\cdot 11^3$$ or $$-7.137^3\cdot 2083^3$$ . The reverse implication holds for the first j-invariant but it is currently unknown whether or not it holds for the second.

Let $$\ell $$ ℓ be a prime and let $$n\ge 1$$ n ≥ 1 . In this note we show that if there is a non-cuspidal, non-CM isolated point x with a rational j -invariant on the modular curve $$X_1(\ell ^n)$$ X 1 ( ℓ n ) , then $$\ell =37$$ ℓ = 37 and the j -invariant of x is either $$7\cdot 11^3$$ 7 · 11 3 or $$-7.137^3\cdot 2083^3$$ - 7 . 137 3 · 2083 3 . The reverse implication holds for the first j-invariant but it is currently unknown whether or not it holds for the second.

We consider a large family of dynamical Belyi maps of arbitrary degree and study the arithmetic monodromy groups attached to the iterates of such maps. Building on the results of Bouw-Ejder-Karemaker on the geometric monodromy groups of these maps, we show that the quotient of the arithmetic monodromy group by the geometric monodromy group has order either 1 or 2. Prior to this article, a result of this kind was only known for quadratic maps (Pink) and a few examples in degree 3.

Let ℓ be a prime and let n≥ 1 . In this note we show that if there is a non-cuspidal, non-CM isolated point x with a rational j -invariant on the modular curve X_1(ℓ ^n) , then ℓ =37 and the j -invariant of x is either 7· 11^3 or -7.137^3· 2083^3 . The reverse implication holds for the first j-invariant but it is currently unknown whether or not it holds for the second.

Let l be a prime and let n >= 1. In this note we show that if there is a non-cuspidal, non-CM isolated point x with a rational j-invariant on the modular curve X-1(l(n)), then l = 37 and the j-invariant of x is either 7 center dot 11(3) or -7.137(3) center dot 2083(3). The reverse implication holds for the first j-invariant but it is currently unknown whether or not it holds for the second.

We consider a large class of so-called dynamical Belyi maps and study the Galois groups of iterates of such maps. From the combinatorial invariants of the maps, we construct a useful presentation of the geometric Galois groups as subgroups of automorphism groups of regular trees, in terms of iterated wreath products. Using results on the reduction of dynamical Belyi maps modulo certain primes, we obtain results on the corresponding arithmetic Galois groups of iterates. These lead to results on the behavior of the arithmetic Galois groups under specialization, with applications to dynamical sequences.

We say a closed point x on a curve C is sporadic if C has only finitely many closed points of degree at most deg⁡(x) and that x is isolated if it is not in a family of effective degree d divisors parametrized by P1 or a positive rank abelian variety (see Section 4 for more precise definitions and a proof that sporadic points are isolated). Motivated by well-known classification problems concerning rational torsion of elliptic curves, we study sporadic and isolated points on the modular curves X1(N). In particular, we show that any non-cuspidal non-CM sporadic, respectively isolated, point x∈X1(N) maps down to a sporadic, respectively isolated, point on a modular curve X1(d), where d is bounded by a constant depending only on j(x). Conditionally, we show that d is bounded by a constant depending only on the degree of Q(j(x)), so in particular there are only finitely many j-invariants of bounded degree that give rise to sporadic or isolated points.

Let F-n denote the Fermat curve given by x(n)+ y(n) = z(n) and let mu(n) denote the Galois module of nth roots of unity. It is known that the integral homology group H-1(F-n, Z) is a cyclic Z[mu(n) x mu(n)] module. In this paper, we prove this result using modular symbols and the modular description of Fermat curves; moreover we find a basis for the integral homology group H-1(F-n, Z). We also construct a family of Fermat curves using the Fermat surface and compute its monodromy.

We study the dynamical properties of a large class of rational maps with exactly three ramification points. By constructing families of such maps, we obtain infinitely many conservative maps of degree $d$; this answers a question of Silverman. Rather precise results on the reduction of these maps yield strong information on the rational dynamics.

Let K denote the quadratic field Q(d) where d=−1 or −3 and let E be an elliptic curve defined over K. In this paper, we analyze the torsion subgroups of E in the maximal elementary abelian 2-extension of K.

Let F_n denote the Fermat curve given by x^n+y^n=z^n and let μ_n denote the Galois module of nth roots of unity. It is known that the integral homology group H_1(F_n,) is a cyclic [μ_n×μ_n] module. In this paper, we prove this result using modular symbols and the modular description of Fermat curves; moreover we find a basis for the integral homology group H_1(F_n,). We also construct a family of Fermat curves using the Fermat surface and compute its monodromy.