Modular Curves Research Articles

Modular curves of prime-power level with infinitely many rational points

For each open subgroup $G$ of ${\rm GL}_2(\hat{\mathbb{Z}})$ containing $-I$ with full determinant, let $X_G/\mathbb{Q}$ denote the modular curve that loosely parametrizes elliptic curves whose Galois representation, which arises from the Galois action on its torsion points, has image contained in $G$. Up to conjugacy, we determine a complete list of the $248$ such groups $G$ of prime power level for which $X_G(\mathbb{Q})$ is infinite. For each $G$, we also construct explicit maps from each $X_G$ to the $j$-line. This list consists of $220$ modular curves of genus $0$ and $28$ modular curves of genus $1$. For each prime $\ell$, these results provide an explicit classification of the possible images of the $\ell$-adic Galois representations arising from elliptic curves over $\mathbb{Q}$ that is complete except for a finite set of exceptional $j$-invariants.

Torsion points on elliptic curves over number fields of small degree

Barry Mazur famously classified the finitely many groups that can occur as a torsion subgroup of an elliptic curve over the rationals. This thesis deals with generalizations of this to higher degree number fields. Merel proved that for all integers d one has that the number of isomorphsim classes of torsion groups of elliptic curves over number fields of degree d is finite. This thesis consists of 4 chapters, the first is introductory and the other tree are research articles. Chapter two deals with the computation of gonalities of modular curves, and the application of these computations to the question which cyclic subgroups can occur as the torsion subgroup of infinitely many non-isomorphic elliptic curves over number fields of degree <7. In the second chapter a general theory for finding rational points on symmetric powers of curves is developed that is similar to symmetric power Chabauty. Application of this theory to symmetric powers of modular curves allows us to determine which primes can divide the order of the torsion subgroup of an elliptic curve over a number field of degree <7. The last chapter studies elliptic curve with a point of order 17 over a number field of degree 4.

Certification of modular Galois representations

We show how the output of the algorithm to compute modular Galois representations described in our previous article can be certified. We have used this process to compute certified tables of such Galois representations obtained thanks to an improved version of this algorithm, including representations modulo primes up to 31 and representations attached to a newform with non-rational (but of course algebraic) coefficients, which had never been done before. These computations take place in the Jacobian of modular curves of genus up to 26. The resulting data are available on the author's webpage.

A novel image encryption scheme based on logistic map and dynatomic modular curve

A new image encryption and decryption algorithm based on chaotic map and dynatomic modular curve is proposed in this paper. Firstly, the definition of dynatomic modular curve and its periodic points are introduced, and a property of the dynatomic modular curve is proved. Secondly, the relationship between the Logistic map and the dynatomic modular curve is discussed. Finally, the encryption algorithm which is composed of permutation of pixels and substitution is given. In order to eliminate sufficiently the relation between adjacent pixels in the image, pixel values of the original image are sorted as index function, which derives from Logistic map and dynatomic modular curve. And XOR operation is performed between the scrambled pixel sequence and projective transformation sequence. Simulation experiments and nonparametric hypothesis test demonstrate that the proposed algorithm is secure to resist different types of attacks and it can be applied to real-time encryption.

Curves on surfaces with trivial canonical bundle

We survey some results concerning Severi varieties and variation in moduli of curves lying on K3 surfaces or on abelian surfaces. A number of open problems is listed and some work in progress is mentioned.

Families of elliptic curves with the same mod 8 representations

AbstractWe compute certain twists of the classical modular curve X(8). Searching for rational points on these twists enables us to find non-trivial pairs non-isogenous elliptic curves over ℚ whose 8-torsion subgroups are isomorphic as Galois modules. We also show that there are infinitely many examples over ℚ.

Towers of GL($r$)-type of modular curves

Abstract We construct Galois covers X r , k ⁢ ( N ) {X^{r,k}(N)} over ℙ 1 / 𝔽 q ⁢ ( T ) {{\mathbb{P}}^{1}/{\mathbb{F}}_{q}(T)} with Galois groups close to GL ⁢ ( r , 𝔽 q ⁢ [ T ] / ( N ) ) {{\rm GL}(r,{\mathbb{F}}_{q}[T]/(N))} ( r ≥ 3 {r\geq 3} ) and rationality and ramification properties similar to those of classical modular curves X ⁢ ( N ) {X(N)} over ℙ 1 / ℚ {{\mathbb{P}}^{1}/{\mathbb{Q}}} . As application we find plenty of good towers (with lim sup ⁡ number of rational points genus &gt; 0 \limsup{\frac{\text{number~{}of~{}rational~{}points}}{{\rm genus}}&gt;0} ) of curves over the field 𝔽 q r {{\mathbb{F}}_{q^{r}}} with q r {q^{r}} elements.

On periodicity of geodesic continued fractions

In this paper, we present some generalizations of Lagrange's theorem in the classical theory of continued fractions motivated by the geometric interpretation of the classical theory in terms of closed geodesics on the modular curve. As a result, for an extension F/F′ of number fields with rank one relative unit group, we construct a geodesic multi-dimensional continued fraction algorithm to “expand” any basis of F over Q, and prove its periodicity. Furthermore, we show that the periods describe the relative unit group. By developing the above argument adelically, we also obtain a p-adelic continued fraction algorithm and its periodicity for imaginary quadratic irrationals.

NORMAL BASES FOR MODULAR FUNCTION FIELDS

We provide a concrete example of a normal basis for a finite Galois extension which is not abelian. More precisely, let$\mathbb{C}(X(N))$be the field of meromorphic functions on the modular curve$X(N)$of level$N$. We construct a completely free element in the extension$\mathbb{C}(X(N))/\mathbb{C}(X(1))$by means of Siegel functions.

On a modular Fermat equation

We study some diophantine problems suggested by the analogy between multiplicative groups and powers of the modular curve in problems of “unlikely intersections”. We prove a special case of the Zilber–Pink conjecture for curves.

8. Overconvergent Modular Forms and Perfectoid Shimura Curves

We give a new construction of overconvergent modular forms of arbitrary weights, defining them in terms of functions on certain affinoid subsets of Scholze's infinite-level modular curve. These affinoid subsets, and a certain canonical coordinate on them, play a role in our construction which is strongly analogous with the role of the upper half-plane and its coordinate `$z$' in the classical analytic theory of modular forms. As one application of these ideas, we define and study an overconvergent Eichler-Shimura map in the context of compact Shimura curves over $\mb{Q}$, proving stronger analogues of results of Andreatta-Iovita-Stevens.

Iwasawa theory of quadratic twists of X 0(49)

The field \(K = \mathbb{Q}\left( {\sqrt { - 7} } \right)\) is the only imaginary quadratic field with class number 1, in which the prime 2 splits, and we fix one of the primes p of K lying above 2. The modular elliptic curve X0(49) has complex multiplication by the maximal order O of K, and we let E be the twist of X0(49) by the quadratic extension \(KK(\sqrt M )/K\), where M is any square free element of O with M ≡ 1 mod 4 and (M,7) = 1. In the present note, we use surprisingly simple algebraic arguments to prove a sharp estimate for the rank of the Mordell-Weil group modulo torsion of E over the field F∞ = K(Ep∞), where Ep∞ denotes the group of p∞-division points on E. Moreover, writing B for the twist of X0(49) by \(K(\sqrt[4]{{ - 7}})/K\), our Iwasawa-theoretic arguments also show that the weak form of the conjecture of Birch and Swinnerton-Dyer implies the non-vanishing at s = 1 of the complex L-series of B over every finite layer of the unique Z2-extension of K unramified outside p. We hope to give a proof of this last non-vanishing assertion in a subsequent paper.

Triple Product Formula and Mass Equidistribution on Modular Curves of Level N

It was shown in previous works that the measure associated to holomorphic newforms of weight $k$ and level $q$ will tend weakly to the Haar measure on modular curve of level 1, as $qk\rightarrow \infty$. In this paper we proved that this phenomenon is also true on modular curves of general level N.

Fields of definition of elliptic curves with prescribed torsion

We prove that all elliptic curves over quadratic fields with a subgroup isomorphic to C16, as well as all elliptic curves over cubic fields with a subgroup isomorphic to C2 × C14, are base changes of elliptic curves defined over Q. We obtain these results by studying geometric properties of modular curves and maps between modular curves, and then obtaining a modular description of these curves and maps.

Automorphicity and mean-periodicity

If $C$ is a smooth projective curve over a number field $k$, then, under fair hypotheses, its $L$-function admits meromorphic continuation and satisfies the anticipated functional equation if and only if a related function is $\mathfrak{X}$-mean-periodic for some appropriate functional space $\mathfrak{X}$. Building on the work of Masatoshi Suzuki for modular elliptic curves, we will explore the dual relationship of this result to the widely believed conjecture that such $L$-functions should be automorphic. More precisely, we will directly show the orthogonality of the matrix coefficients of $GL_{2g}$-automorphic representations to the vector spaces $\mathcal{T}(h(\mathcal{S},\{k_i\},s))$, which are constructed from the Mellin transforms $f(\mathcal{S},\{k_i\},s)$ of certain products of arithmetic zeta functions $\zeta(\mathcal{S},2s)\prod_{i}\zeta(k_i,s)$, where $\mathcal{S}\rightarrow {\rm Spec}(\mathcal{O}_k)$ is any proper regular model of $C$ and $\{k_i\}$ is a finite set of finite extensions of $k$. To compare automorphicity and mean-periodicity, we use a technique emulating the Rankin–Selberg method, in which the function $h(\mathcal{S},\{k_i\},s))$ plays the role of an Eisenstein series, exploiting the spectral interpretation of the zeros of automorphic $L$-functions.

Constraints on the automorphism group of a curve

For a curve of genus > 1 defined over a finite field, we present a sufficient criterion for the non-existence of automorphisms of order a power of a rational prime. We show how this criterion can be used to determine the automorphism group of some modular curves of high genus.

On the automorphic atoms of length 2 of $\mathrm{GL}_2(\mathbb{Q}_p)$

Let p&gt;3 be a prime. The aim of this paper is to give a description of the invariant space, under principal and Iwahori congruence subgroups of arbitrary level, of extensions of generic principal series representations appearing in the p -modular local Langlands correspondence for \bold{GL}_2(\bold{Q}_p) . As an application we describe Hecke isotypical components of the mod p cohomology of the modular curve over \bold{Q} with deeply ramified level at p .

The rational torsion subgroups of Drinfeld modular Jacobians and Eisenstein pseudo-harmonic cochains

Let \(\mathfrak n\) be a square-free ideal of \(\mathbb {F}_q[T]\). We study the rational torsion subgroup of the Jacobian variety \(J_0(\mathfrak n)\) of the Drinfeld modular curve \(X_0(\mathfrak n)\). We prove that for any prime number \(\ell \) not dividing \(q(q-1)\), the \(\ell \)-primary part of this group coincides with that of the cuspidal divisor class group. We further determine the structure of the \(\ell \)-primary part of the cuspidal divisor class group for any prime \(\ell \) not dividing \(q-1\).

Optimal Quotients of Jacobians With Toric Reduction and Component Groups

AbstractLet J be a Jacobian variety with toric reduction over a local field K. Let J → E be an optimal quotient defined over K, where E is an elliptic curve. We give examples in which the functorially induced map on component groups of the Néron models is not surjective. This answers a question of Ribet and Takahashi. We also give various criteria under which is surjective and discuss when these criteria hold for the Jacobians of modular curves.

Genus of abstract modular curves with level-ℓ structures

Abstract We prove – in arbitrary characteristic – that the genus of abstract modular curves associated to bounded families of continuous geometrically perfect 𝔽 ℓ {\mathbb{F}_{\ell}} -linear representations of étale fundamental groups of curves goes to infinity with ℓ {\ell} . This applies to the variation of the Galois image on étale cohomology groups with coefficients in 𝔽 ℓ {\mathbb{F}_{\ell}} in 1-dimensional families of smooth proper schemes or, under certain assumptions, to specialization of first Galois cohomology groups.