Belyi's theorem asserts that a smooth projective curve $X$ defined over a number field can be realized as a cover of the projective line unramified outside three points. In this short paper we investigate the bejaviour of the minimal degree of such a cover. More precisely, we start by defining the absolute Belyi degree of X, which only depends on the $\bar{\bold Q}$-isomorphism class of $X$. We then give a lower bound of this invariant, only depending on the stable primes of bad reduction (as defined in the paper) and we show that this bound is sharp. In the second part of the paper, we introduce the relative Belyi degree of a curve X defined over a fixed number field $K$. We first prove that there exist finitely many $K$-isomorphism classes of curves of bounded (relative) Belyi degree and we then obtain a lower bound, only depending on the primes of bad reduction of the minimal regular model of $X$ over (the ring of integers of) $K$.
We show that every smooth projective curve over a finite field k admits a finite tame morphism to the projective line over k. Furthermore, we construct a curve with no such map when k is an infinite perfect field of characteristic two. Our work leads to a refinement of the tame Belyi theorem in positive characteristic, building on results of Saïdi, Sugiyama-Yasuda, and Anbar-Tutdere.
Motivated by a demand for explicit genus 1 Belyi maps from theoretical physics, we give an efficient method of explicitly computing genus one Belyi maps by (1) composing covering maps from elliptic curves to the Riemann sphere with simpler (univariate) genus zero Belyi maps, as well as by (2) composing further with isogenies between elliptic curves. This gives many new explicit dessins on the doubly periodic plane, including several which have been realized in the physics literature as so-called brane-tilings in the context of quiver gauge theories.
We give a Belyi-type characterisation of smooth complete intersections of general type over $\mathbb{C}$ which can be defined over $\bar{\mathbb{Q}}$. Our proof uses the higher-dimensional analogue of the Shafarevich boundedness conjecture for families of canonically polarized varieties, finiteness results for maps to varieties of general type, and rigidity theorems for Lefschetz pencils of complete intersections.
There are two types of Belyi's Theorem for curves defined over finite fields of characteristic p, namely the Wild and the Tame p-Belyi Theorems. In this paper, we discuss them in the language of function fields. We provide a self-contained proof for the Wild p-Belyi Theorem for any prime p and the Tame 2-Belyi Theorem.
We study the Betti map of a particular (but relevant) section of the family of Jacobians of hyperelliptic curves using the polynomial Pell equation $A^2-DB^2=1$, with $A,B,D\in \mathbb C[t]$ and certain ramified covers ${\mathbb P}^1\to {\mathbb P}^1$ arising from such equation and having heavy constrains on their ramification. In particular, we obtain a special case of a result of André, Covaja and Zannier on the submersivity of the Betti map by studying the locus of the polynomials $D$ that fit in a Pell equation inside the space of polynomials of fixed even degree. Moreover, Riemann Existen…
For a prime number p, we characterize the groups that may arise as torsion subgroups of an elliptic curve with complex multiplication defined over a number field of degree 2p. In particular, our work shows that a classification in the strongest sense is tied to determining whether there exist infinitely many Sophie Germain primes.
We provably compute the full set of rational points on 1403 Picard curves defined over $\mathbb{Q}$ with Jacobians of Mordell-Weil rank $1$ using the Chabauty-Coleman method. To carry out this computation, we extend Magma code of Balakrishnan and Tuitman for Coleman integration. The new code computes $p$-adic (Coleman) integrals on curves to points defined over number fields where the prime $p$ splits completely and implements effective Chabauty for curves whose Jacobians have infinite order points that are not the image of a rational point under the Abel-Jacobi map. We discuss several interes…
In this paper, we study the finiteness problem of torsion points on an elliptic curve whose coordinates satisfy some multiplicative dependence relations. In particular, we prove that on an elliptic curve defined over a number field there are only finitely many torsion points whose coordinates are multiplicatively dependent. Moreover, we produce an effective result when the elliptic curve is defined over the rational numbers or has complex multiplication.
Fix an abelian variety $A_0$ and a non-isotrivial abelian scheme over a smooth irreducible curve, both defined over the algebraic numbers. Consider the union of all images of translates of a fixed finite-rank subgroup of $A_0$, also defined over the algebraic numbers, by abelian subvarieties of $A_0$ of codimension at least $k$ under all isogenies between $A_0$ and some fiber of the abelian scheme. We characterize the curves inside the abelian scheme which are defined over the algebraic numbers, dominate the base curve and potentially intersect this set in infinitely many points. Our proof fol…
We prove that $\bar {\mathbb Q}_\ell$-local systems of bounded rank and ramification on a smooth variety $X$ defined over an algebraically closed field $k$ of characteristic $p\neq \ell$ are tamified outside of codimension $2$ by a finite separable cover of bounded degree. In rank one, there is a curve which preserves their monodromy. There is a curve defined over the algebraic closure of a purely transcendental extension of $k$ of finite degree which fulfills the Lefschetz theorem. Last version: minor typos corrected.
Let E be an elliptic curve defined over Q and let G = E(Q)_tors be the associated torsion subgroup. We study, for a given G, which possible groups G <= H could appear such that H=E(K)_tors, for [K:Q]=4 and H is one of the possible torsion structures that occur infinitely often as torsion structures of elliptic curves defined over quartic number fields.
For any number field, we prove that there exists an elliptic curve defined over this field such that its Shafarevich-Tate group has a nontrivial 2-torsion subgroup.
For any number field, we prove that there exists a stacky curve of genus one half defined over the ring of its integers violating the local-global principle for integral points.
I provide methods of constructing elliptic and hyperelliptic curves over global fields with interesting rational points over the given fields or over large field extensions. I also provide a elliptic curves defined over any given number field equipped with a rational point, (resp. with two rational points) of infinite order over the given number field, and elliptic curves over the rationals with two rational points over `simplest cubic fields.' I also provide hyperelliptic curves of genus exceeding any given number over any given number fields with points (over the given number field) which sp…
We characterize the possible reductions of $j$-invariants of elliptic curves which admit complex multiplication by an order $\mathcal{O}$ where the curve itself is defined over $\mathbb{Z}_p$. In particular, we show that the distribution of these $j$-invariants depends on which primes divide the discriminant and conductor of the order.
Associated to an open subgroup $G$ of $\GL_2(\Zhat)$ satisfying conditions $-I \in G$ and $\det(G) \subsetneq (\Zhat)^{\times}$ there is a modular curve $X_G$ which is a smooth compact curve defined over an extension of $\Q.$ In this article, we give a complete list of all such prime power level genus $0$ modular curves with a point.
We give some sufficient conditions on complex polynomials P and Q to assure that the algebraic plane curve P(x)-Q(y)=0 has no irreducible component of genus 0 or 1. Moreover, if deg (P)=deg (Q) and if both P, Q satisfy Hypothesis I introduced by H. Fujimoto, our sufficient conditions are necessary.
Let $[K:\mathbb{Q}]=p$ be a prime number and let $E/K$ be an elliptic curve with $j(E) \in \mathbb{Q}$. We determine the all possibilities for $E(K)_{tors}$. We obtain these results by studying Galois representations of $E$ and of it's quadratic twists.
Perfectoid versions of Abel Jacobi and Reimann Roch Theorem are proved, and perfectoid Elliptic Curve is constructed. A Perfectoid Tate Curve is defined and its cohomology computed via a Čech complex. Furthermore, perfectoid Theta function and Weierstraß series are also defined and suitably interpreted.
In "Proving that a genus 2 curve has complex multiplication", van Wamelen lists 19 curves of genus two over $\mathbf{Q}$ with complex multiplication (CM). For each of the 19 curves, the CM-field turns out to be cyclic Galois over $\mathbf{Q}$. The generic case of non-Galois quartic CM-fields did not feature in this list, as the field of definition in that case always contains a real quadratic field, known as the real quadratic subfield of the reflex field. We extend van Wamelen's list to include curves of genus two defined over this real quadratic field. Our list therefore contains the smalles…
The CM class number one problem for elliptic curves asked to find all elliptic curves defined over the rationals with non-trivial endomorphism ring. For genus-2 curves it is the problem of determining all CM curves of genus $2$ defined over the reflex field. We solve the problem by showing that the list given in Bouyer and Streng [LMS J. Comput. Math., 18(1):507{538, 2015] is complete.
We adopt a statistical point of view on the conjecture of Lang which predicts a lower bound for the canonical height of non-torsion rational points on elliptic curves defined over $\mathbb{Q}$. More specifically, we prove that among the family of all elliptic curves defined over $\mathbb{Q}$ and having positive rank, there is a density one subfamily of curves which satisfy a strong form of Lang's conjecture.
We study torsion subgroups of elliptic curves with complex multiplication (CM) defined over number fields which admit a real embedding. We give a complete classification of the groups which arise up to isomorphism as the torsion subgroup of a CM elliptic curve defined over a number field of odd degree: there are infinitely many. Restricting to the case of prime degree, we show that there are only finitely many isomorphism classes. More precisely, there are six "Olson groups" which arise as torsion subgroups of CM elliptic curves over number fields of every degree, and there are precisely 17 "n…
In this note we consider certain elliptic curves defined over real quadratic fields isogenous to their Galois conjugate. We give a construction of algebraic points on these curves defined over almost totally real number fields. The main ingredient is the system of Heegner points arising from Shimura curve uniformizations. In addition, we provide an explicit $p$-adic analytic formula which allows for the effective, algorithmic calculation of such points.
