Modular Curves Research Articles

Concentration of closed geodesics in the homology of modular curves

Abstract We prove that the homology classes of closed geodesics associated to subgroups of narrow class groups of real quadratic fields concentrate around the Eisenstein line. This fits into the framework of Duke’s Theorem and can be seen as a real quadratic analogue of results of Michel and Liu–Masri–Young on supersingular reduction of CM-elliptic curves. We also study the level aspect, as well as a homological version of the sup norm problem. Finally, we present applications to group theory and modular forms

Drinfeld discriminant function and Fourier expansion of harmonic cochains

Let $$F_{\infty }={{\mathbb {F}}_q}\left( \!\left( {1/T}\right) \!\right) $$ be the completion of $${\mathbb {F}}_q(T)$$ at 1/T. We develop a theory of Fourier expansions for harmonic cochains on the edges of the Bruhat–Tits building of $${{\,\textrm{PGL}\,}}_r(F_{\infty })$$ , $$r\ge 2$$ , generalizing an earlier construction of Gekeler for $$r=2$$ . We then apply this theory to study modular units on the Drinfeld symmetric space $$\Omega ^r$$ over $$F_{\infty }$$ , and the cuspidal divisor groups of Satake compactifications of certain Drinfeld modular varieties. In particular, we obtain a higher dimensional analogue of a result of Ogg for classical modular curves $$X_0(p)$$ of prime level.

Proving The Fermat Last Theorem for Case q≤n

Fermat's Last Theorem is a well-known classical theorem in mathematics. Andrew Willes has proven this theorem using the modular elliptic curve. However, the proposed proof is difficult for mathematicians and researchers to understand. For this reason, in this study, we provide evidence of several properties of Fermat's Last Theorem with a simple concept. We use Newton's Binomial Theorem, well-known in Fermat's time. In this study, we prove Fermat's Last Theorem for case . We also use the Newton’s Binomial theorem to verify several cases .

$p$-adic equidistribution of CM points

Let X be a modular curve and consider a sequence of Galois orbits of CM points in X , whose p -conductors tend to infinity. Its equidistribution properties in X(\mathbf{C}) and in the reductions of X modulo primes different from p are well understood. We study the equidistribution problem in the Berkovich analytification X_{p}^{\operatorname{an}} of X_{\operatorname{Q}_{p}} . We partition the set of CM points of sufficiently high conductor in X_{\operatorname{Q}_{p}} into finitely many explicit basins B_{V} , indexed by the irreducible components V of the \bmod\text{-}p reduction of the canonical model of X . We prove that a sequence z_{n} of local Galois orbits of CM points with p -conductor going to infinity has a limit in X_{p}^{\operatorname{an}} if and only if it is eventually supported in a single basin B_{V} . If so, the limit is the unique point of X_{p}^{\operatorname{an}} whose mod- p reduction is the generic point of V . The result is proved in the more general setting of Shimura curves over totally real fields. The proof combines Gross's theory of quasi-canonical liftings with a new formula for the intersection numbers of CM curves and vertical components in a Lubin–Tate space.

Triangular modular curves of small genus

Triangular modular curves are a generalization of modular curves that arise from quotients of the upper half-plane by congruence subgroups of hyperbolic triangle groups. These curves also arise naturally as a source of Belyi maps with monodromy text {PGL}_2(mathbb {F}_q) or text {PSL}_2(mathbb {F}_q). We present a computational approach to enumerate Borel-type triangular modular curves of low genus, and we carry out this enumeration for prime level and small genus.

On the statistics of pairs of logarithms of integers

We study the correlations of pairs of logarithms of positive integers at various scalings, either with trivial weigths or with weights given by the Euler function, proving the existence of pair correlation functions. We prove that at the linear scaling, the pair correlations exhibit level repulsion, as it sometimes occurs in statistical physics. We prove total loss of mass phenomena at superlinear scalings, and Poissonian behaviour at sublinear scalings. The case of Euler weights has applications to the pair correlation of the lengths of common perpendicular geodesic arcs from the maximal Margulis cusp neighborhood to itself in the modular curve $\operatorname{PSL}_2(\mathbb Z)\backslash\mathbb H^2_{\mathbb R}$.

Bielliptic quotient modular curves of 𝑋₀(𝑁)

Let N ≥ 1 N\geq 1 be a non-square free integer and let W N W_N be a nontrivial subgroup of the group of the Atkin-Lehner involutions of X 0 ( N ) X_0(N) such that the modular curve X 0 ( N ) / W N X_0(N)/W_N has genus at least two. We determine all pairs ( N , W N ) (N,W_N) such that X 0 ( N ) / W N X_0(N)/W_N is a bielliptic curve and the pairs ( N , W N ) (N,W_N) such that X 0 ( N ) / W N X_0(N)/W_N has an infinite number of quadratic points over Q \mathbb {Q} .

Modular curves over number fields and ECM

We construct families of elliptic curves defined over number fields and containing torsion groups $$\mathbb {Z}/{M_1}\mathbb {Z}\times \mathbb {Z}/{M_2}\mathbb {Z}$$ where $$(M_1, M_2)$$ belongs to $$\{(1, 11)$$ , (1, 14), (1, 15), (2, 10), (2, 12), (3, 9), (4, 8), $$(6, 6)\}$$ (i.e., when the corresponding modular curve $$X_1(M_1, M_2)$$ has genus 1). We provide formulae for the curves and give examples of number fields for which the corresponding elliptic curves have non-zero ranks, giving explicit generators using D. Simon’s program whenever possible. The reductions of these curves can be used to speed up ECM for factoring numbers with special properties, a typical example being (factors of) Cunningham numbers $$b^n-1$$ such that $$M_1 \mid n$$ . We explain how to find points of potentially large orders on the reduction, if we accept to use quadratic twists.

Revisiting the Iconic Spitzer Phase Curve of 55 Cancri e: Hotter Dayside, Cooler Nightside, and Smaller Phase Offset

Thermal phase curves of short-period exoplanets provide the best constraints on the atmospheric dynamics and heat transport in their atmospheres. The published Spitzer Space Telescope phase curve of 55 Cancri e, an ultra-short-period super-Earth, exhibits a large phase offset suggesting significant eastward heat recirculation, unexpected on such a hot planet. We present our rereduction and analysis of these iconic observations using the open source and modular Spitzer Phase Curve Analysis pipeline. In particular, we attempt to reproduce the published analysis using the same instrument detrending scheme as the original authors. We retrieve the dayside temperature ( K), nightside temperature ( K at 2σ), and longitudinal offset of the planet's hot spot, and quantify how they depend on the reduction and detrending. Our reanalysis suggests that 55 Cancri e has a negligible hot spot offset of degrees east. The small phase offset and cool nightside are consistent with the poor heat transport expected on ultra-short-period planets. The high dayside 4.5 μm brightness temperature is qualitatively consistent with SiO emission from an inverted rock vapor atmosphere.

A group theoretic perspective on entanglements of division fields

In this paper, we initiate a systematic study of entanglements of division fields from a group theoretic perspective. For a positive integer n n and a subgroup G ⊆ G L 2 ( Z / n Z ) G\subseteq GL_2( \mathbb {Z}/n\mathbb {Z}) with surjective determinant, we provide a definition for G G to represent an ( a , b ) (a,b) -entanglement and give additional criteria for G G to represent an explained or unexplained ( a , b ) (a,b) -entanglement. Using these new definitions, we determine the tuples ( ( p , q ) , T ) ((p,q),T) , with p > q ∈ Z p>q\in \mathbb {Z} distinct primes and T T a finite group, such that there are infinitely many non- Q ¯ \overline {\mathbb {Q}} -isomorphic elliptic curves over Q \mathbb {Q} with an unexplained ( p , q ) (p,q) -entanglement of type T T . Furthermore, for each possible combination of entanglement level ( p , q ) (p,q) and type T T , we completely classify the elliptic curves defined over Q \mathbb {Q} with that combination by constructing the corresponding modular curve and j j -map.

Rational points on hyperelliptic Atkin-Lehner quotients of modular curves and their coverings

We complete the computation of all mathbb {Q}-rational points on all the 64 maximal Atkin-Lehner quotients X_0(N)^* such that the quotient is hyperelliptic. To achieve this, we use a combination of various methods, namely the classical Chabauty–Coleman, elliptic curve Chabauty, quadratic Chabauty, and the bielliptic quadratic Chabauty method (from a forthcoming preprint of the fourth-named author) combined with the Mordell-Weil sieve. Additionally, for square-free levels N, we classify all mathbb {Q}-rational points as cusps, CM points (including their CM field and j-invariants) and exceptional ones. We further indicate how to use this to compute the mathbb {Q}-rational points on all of their modular coverings.

The modularity of elliptic curves over all but finitely many totally real fields of degree 5

We study the finiteness of low degree points on certain modular curves and their Atkin--Lehner quotients, and, as an application, prove the modularity of elliptic curves over all but finitely many totally real fields of degree $5$. On the way, we prove a criterion for the finiteness of rational points of degree $5$ on a curve of large genus over a number field using the results of Abramovich--Harris and Faltings on subvarieties of Jacobians.

Automorphisms of Cartan modular curves of prime and composite level

We study the automorphisms of modular curves associated to Cartan subgroups of $\mathrm{GL}_2(\mathbb Z/n\mathbb Z)$ and certain subgroups of their normalizers. We prove that if $n$ is large enough, all the automorphisms are induced by the ramified covering of the complex upper half-plane. We get new results for non-split curves of prime level $p\ge 13$: the curve $X_{\text{ns}}^+(p)$ has no non-trivial automorphisms, whereas the curve $X_{\text{ns}}(p)$ has exactly one non-trivial automorphism. Moreover, as an immediate consequence of our results we compute the automorphism group of $X_0^*(n):=X_0(n)/W$, where $W$ is the group generated by the Atkin-Lehner involutions of $X_0(n)$ and $n$ is a large enough square.

Higher Hida and Coleman theories on the modular curve

We construct Hida and Coleman theories for the degree 0 and 1 cohomology of automorphic line bundles on the modular curve and we define a p-adic duality pairing between the theories in degree 0 and 1.

Relations among Ramanujan-type congruences I

We prove that Ramanujan-type congruences for integral weight modular forms away from the level and the congruence prime are equivalent to specific congruences for Hecke eigenvalues. In particular, we show that Ramanujan-type congruences are preserved by the action of the shallow Hecke algebra. More generally, we show for weakly holomorphic modular forms of integral weight, that Ramanujan-type congruences naturally occur for shifts in the union of two square-classes as opposed to single square-classes that appear in the literature on the partition function. We also rule out the possibility of square-free periods, whose scarcity in the case of the partition function was investigated recently. We complement our obstructions on maximal Ramanujan-type congruences with several existence statements. Our results are based on a framework that leverages classical results on integral models of modular curves via modular representation theory, and applies to congruences of all weakly holomorphic modular forms. Steinberg representations govern all maximal Ramanujan-type congruences for integral weights. We discern the scope of our framework in the case of half-integral weights through example calculations.

Congruences of elliptic curves arising from nonsurjective mod 𝑁 Galois representations

We study N N -congruences between quadratic twists of elliptic curves. If N N has exactly two distinct prime factors we show that these are parametrised by double covers of certain modular curves. In many, but not all cases, the modular curves in question correspond to the normaliser of a Cartan subgroup of GL 2 ⁡ ( Z / N Z ) \operatorname {GL}_2(\mathbb {Z}/N\mathbb {Z}) . By computing explicit models for these double covers we find all pairs ( N , r ) (N, r) such that there exist infinitely many j j -invariants of elliptic curves E / Q E/\mathbb {Q} which are N N -congruent with power r r to a quadratic twist of E E . We also find an example of a 48 48 -congruence over Q \mathbb {Q} . We make a conjecture classifying nontrivial ( N , r ) (N,r) -congruences between quadratic twists of elliptic curves over Q \mathbb {Q} . Finally, we give a more detailed analysis of the level 15 15 case. We use elliptic Chabauty to determine the rational points on a modular curve of genus 2 2 whose Jacobian has rank 2 2 and which arises as a double cover of the modular curve X ( n s 3 + , n s 5 + ) X(\mathrm {ns\,} 3^+, \mathrm {ns\,} 5^+) . As a consequence we obtain a new proof of the class number 1 1 problem.

Torsion Subgroups of Modular Jacobian Varieties Over Splitting Fields of Cuspidal Subgroups

Abstract For any positive integer $N$, let $J_0(N)$ be the Jacobian variety of the modular curve $X_0(N)$ over ${\mathbb {Q}}$ and ${\mathcal {C}}_N$ its cuspidal subgroup. Let $F_N$ denote the splitting field of ${\mathcal {C}}_N$, which is the smallest number field whose absolute Galois group acts trivially on ${\mathcal {C}}_N$. Let ${\mathcal {J}}_N=J_0(N)(F_{N})_{\textrm {tor}}$ be the torsion subgroup of the group of $F_N$-rational points on $J_0(N)$. We prove that ${\mathcal {J}}_N$ coincides with ${\mathcal {C}}_N$ outside $6N[F_N:{\mathbb {Q}}]$.

Moduli-friendly Eisenstein series over the p-adics and the computation of modular Galois representations

We show how our p-adic method to compute Galois representations occurring in the torsion of Jacobians of algebraic curves can be adapted to modular curves. The main ingredient is the use of “moduli-friendly” Eisenstein series introduced by Makdisi, which allow us to evaluate modular forms at p-adic points of modular curves and dispenses us of the need for equations of modular curves and for q-expansion computations in the construction of models of modular Jacobians. The resulting algorithm compares very favourably to our complex-analytic method.

On a probabilistic local-global principle for torsion on elliptic curves

Let $m$ be a positive integer and let $E$ be an elliptic curve over $\mathbb{Q}$ with the property that $m\mid\#E(\mathbb{F}_p)$ for a density $1$ set of primes $p$. Building upon work of Katz and Harron-Snowden, we study the probability that $m$ divides the the order of the torsion subgroup of $E(\mathbb{Q})$: we find it is nonzero for all $m \in \{ 1, 2, \dots, 10, 12, 16\}$ and we compute it exactly when $m \in \{ 1,2,3,4,5,7 \}$. As a supplement, we give an asymptotic count of elliptic curves with extra level structure when the parametrizing modular curve is torsion free of genus zero.

Plectic p-adic invariants

For modular elliptic curves over number fields of narrow class number one, and with multiplicative reduction at a collection of p-adic primes, we define new p-adic invariants. Inspired by Nekovář and Scholl's plectic conjectures, we believe these invariants control the Mordell–Weil group of higher rank elliptic curves and we support our expectations with numerical experiments.