Modular Curves Research Articles

Measured Foliations and Hilbert 12th Problem

Yu. I. Manin conjectured that the maximal abelian extensions of the real quadratic number fields are generated by the pseudo-lattices with real multiplication. We prove this conjecture using theory of measured foliations on the modular curves.

An explicit bound for integral points on modular curves

In this paper, we give a constant $C$ in \cite[Theorem 1.2]{sha2014bounding} by using an explicit Baker's inequality, hence we have an explicit bound of the integral points on modular curves.

Real and Complex Multiplication on K3 Surfaces via Period Integration

We report on a new approach, as well as some related experiments, to construct families of K3 surfaces having real or complex multiplication. The approach is based on an explicit description of the transcendental part of the cohomology in a topological way, using topological tori. Fundamental ideas include considering the period space of marked K3 surfaces, determining the periods by numerical integration, as well as tracing the modular curve by a numerical continuation method.

Splitting of primes in number fields generated by points on some modular curves

We study the splitting of primes in number fields generated by points on modular curves. Momose (Nagoya Math J 96:139–165, 1984) was the first to notice that quadratic points on $$X_1(N)$$ generate quadratic fields over which certain primes split in a particular way and his results were later expanded upon by Krumm (Quadratic Points on Modular Curves. PhD thesis, University of Georgia, 2013). We prove results about the splitting behaviour of primes in quadratic fields generated by points on the modular curves $$X_0(N)$$ which are hyperelliptic (except for $$N=37$$ ) and in cubic fields generated by points on $$X_1(2,14)$$ .

Realization of modular Galois representations in the Jacobians of modular curves

In Tian (Acta Arith. 164:399–412, 2014), the author improved the algorithm proposed by Edixhoven and Couveignes for computing mod \(\ell \) Galois representations associated to eigenforms f for the cases that \(\ell \ge k-1\) and f has level one, where k is the weight of f. In this paper, we generalize the results of Tian (Acta Arith. 164:399–412, 2014) and present a method to find the Jacobians of modular curves of minimal dimensions to realize the modular Galois representations. Our method works for the cases that \(\ell \ge 5\) may be any prime without the assumption \(\ell \ge k-1\) and the eigenforms f have arbitrary levels prime to \(\ell \). Moreover, if \(k>2\), we give criteria for realizing the mod \(\ell \) Galois representations in the Jacobians \(J_0\) of \(X_0\).

Counting elliptic curves with prescribed level structures over number fields

Harron and Snowden counted the number of elliptic curves over $\mathbb{Q}$ up to height $X$ with torsion group $G$ for each possible torsion group $G$ over $\mathbb{Q}$. In this paper we generalize their result to all number fields and all level structures $G$ such that the corresponding modular curve $X_G$ is a weighted projective line $\mathbb{P}(w_0,w_1)$ and the morphism $X_G\to X(1)$ satisfies a certain condition. In particular, this includes all modular curves $X_1(m,n)$ with coarse moduli space of genus $0$. We prove our results by defining a size function on $\mathbb{P}(w_0,w_1)$ following unpublished work of Deng, and working out how to count the number of points on $\mathbb{P}(w_0,w_1)$ up to size $X$.

Retainment and retrenchment of employees in the education sector in India in comparison to FMCG sector during COVID 19

The COVID 19 pandemic has had an impact on every sector around the world. To combat this situation lockdown was imposed on March 25, 2020 which has adversely affected various job sectors though out the country. It is found that the education sector has been fighting to survive the crisis with a different approach and digitising the challenges to wash away the threat of pandemic. The FMCG sector has also suffered a great deal as labourers have moved to their native places due to coronavirus pandemic. Private sectors saw more modular curves in the process than the government sector jobs. The paper reviews and studies the impact of this pandemic on employees of education sector and FMCG sector. It highlights the comparison between these effects (i.e. the retainment and retrenchment of employees) of these two sectors. KEY WORDS: COVID 19, Employee retainment, Education sector, FMCG sector.

The $U$-plane of rank-one 4d $\mathcal{N}=2$ KK theories

The simplest non-trivial 5d superconformal field theories (SCFT) are the famous rank-one theories with E_nEn flavour symmetry. We study their UU-plane, which is the one-dimensional Coulomb branch of the theory on \mathbb{R}^4 \times S^1ℝ4×S1. The total space of the Seiberg-Witten (SW) geometry – the E_nEn SW curve fibered over the UU-plane – is described as a rational elliptic surface with a singular fiber of type I_{9-n}I9−n at infinity. A classification of all possible Coulomb branch configurations, for the E_nEn theories and their 4d descendants, is given by Persson’s classification of rational elliptic surfaces. We show that the global form of the flavour symmetry group is encoded in the Mordell-Weil group of the SW elliptic fibration. We study in detail many special points in parameters space, such as points where the flavour symmetry enhances, and/or where Argyres-Douglas and Minahan-Nemeschansky theories appear. In a number of important instances, including in the massless limit, the UU-plane is a modular curve, and we use modularity to investigate aspects of the low-energy physics, such as the spectrum of light particles at strong coupling and the associated BPS quivers. We also study the gravitational couplings on the UU-plane, matching the infrared expectation for the couplings A(U)A(U) and B(U)B(U) to the UV computation using the Nekrasov partition function.

Cubic and Quartic Points on Modular Curves Using Generalised Symmetric Chabauty

Abstract Answering a question of Zureick-Brown, we determine the cubic points on the modular curves $X_0(N)$ for $N \in \{53,57,61,65,67,73\}$ as well as the quartic points on $X_0(65)$. To do so, we develop a “partially relative” symmetric Chabauty method. Our results generalise current symmetric Chabauty theorems and improve upon them by lowering the involved prime bound. For our curves a number of novelties occur. We prove a “higher-order” Chabauty theorem to deal with these cases. Finally, to study the isolated quartic points on $X_0(65)$, we rigorously compute the full rational Mordell–Weil group of its Jacobian.

Elliptic curves over totally real quartic fields not containing √5 are modular

We prove that every elliptic curve defined over a totally real number field of degree 4 not containing 5 \sqrt {5} is modular. To this end, we study the quartic points on four modular curves.

The least degree of a CM point on a modular curve

For a modular curve X = X 0 ( N ) $X = X_0(N)$ , X 1 ( N ) $X_1(N)$ or X 1 ( M , N ) $X_1(M,N)$ defined over Q $\mathbb {Q}$ , we denote by d CM ( X ) $d_{\operatorname{CM}}(X)$ the least degree of a CM point on X $X$ . For each discriminant Δ < 0 $\Delta < 0$ , we determine the least degree of a point on X 0 ( N ) $X_0(N)$ with CM by the order of discriminant Δ $\Delta$ . This places us in a position to study d CM ( X ) $d_{\operatorname{CM}}(X)$ as an ‘arithmetic function’ and we do so, obtaining various upper bounds, lower bounds and typical bounds. We deduce that all but finitely many curves in each of the families have sporadic CM points. Finally, we supplement these results with a computational study, for example, computing d CM ( X 0 ( N ) ) $d_{\operatorname{CM}}(X_0(N))$ and d CM ( X 1 ( N ) ) $d_{\operatorname{CM}}(X_1(N))$ exactly for N ⩽ 10 6 $N \leqslant 10^6$ and determining whether X 0 ( N ) $X_0(N)$ (respectively, X 1 ( N ) $X_1(N)$ , X 1 ( M , N ) $X_1(M,N)$ ) has sporadic CM points for all but 106 values of N $N$ (respectively, 227 values of N $N$ , 146 pairs ( M , N ) $(M,N)$ with M ⩾ 2 $M \geqslant 2$ ).

Isolated points on $$X_1(\ell ^n)$$ with rational j-invariant

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.

Heegner points and Beilinson–Kato elements: A conjecture of Perrin-Riou

A conjecture of Perrin-Riou relating Heegner cycles to Beilinson–Kato elements is proved, by relating both objects to p-adic families of Beilinson–Flach elements in the higher Chow groups of products of two modular curves.

Modular curves, the Tate-Shafarevich group and Gopakumar-Vafa invariants with discrete charges

We show that the stringy Kähler moduli space of a generic genus one curve of degree N, for N ≤ 5, is the Γ1(N) modular curve X1(N). This implies a correspondence between the cusps of the modular curves and certain large volume limits in the stringy Kähler moduli spaces of genus one fibered Calabi-Yau manifolds with N-sections. Using Higgs transitions in M-theory and F-theory as well as modular properties of the topological string partition function, we identify these large volume limits with elements of the Tate-Shafarevich group of the genus one fibration. Singular elements appear in the form of non-commutative resolutions with a torsional B-field at the singularity. The topological string amplitudes that arise at the various large volume limits are related by modular transformations. In particular, we find that the topological string partition function of a smooth genus one fibered Calabi-Yau threefold is transformed into that of a non-commutative resolution of the Jacobian by a Fricke involution. In the case of Calabi-Yau threefolds, we propose an expansion of the partition functions of a singular fibration and its non-commutative resolutions in terms of Gopakumar-Vafa invariants that are associated to BPS states with discrete charges. For genus one fibrations with 5-sections, this provides an enumerative interpretation for the partition functions that arise at certain irrational points of maximally unipotent monodromy.

Torsion properties of modified diagonal classes on triple products of modular curves

AbstractConsider three normalized cuspidal eigenforms of weight $2$ and prime level p. Under the assumption that the global root number of the associated triple product L-function is $+1$ , we prove that the complex Abel–Jacobi image of the modified diagonal cycle of Gross–Kudla–Schoen on the triple product of the modular curve $X_0(p)$ is torsion in the corresponding Hecke isotypic component of the Griffiths intermediate Jacobian. The same result holds with the complex Abel–Jacobi map replaced by its étale counterpart. As an application, we deduce torsion properties of Chow–Heegner points associated with modified diagonal cycles on elliptic curves of prime conductor with split multiplicative reduction. The approach also works in the case of composite square-free level.

Modular parametrization as Polyakov path integral: cases with CM elliptic curves as target spaces

For an elliptic curve E over an abelian extension k/K with CM by K of Shimura type, the L-functions of its [k:K] Galois representations are Mellin transforms of Hecke theta functions; a modular parametrization (surjective map) from a modular curve to E pulls back the 1-forms on E to give the Hecke theta functions. This article refines the study of our earlier work and shows that certain class of chiral correlation functions in Type II string theory with [E]_C (E as real analytic manifold) as a target space yield the same Hecke theta functions as objects on the modular curve. The Kahler parameter of the target space [E]_C in string theory plays the role of the index (partially ordered) set in defining the projective/direct limit of modular curves.

Fermat’s Last Theorem and modular curves over real quadratic fields

In this paper we study the Fermat equation $x^n+y^n=z^n$ over quadratic fields $\mathbb{Q}(\sqrt{d})$ for squarefree $d$ with $26 \leq d \leq 97$. By studying quadratic points on the modular curves $X_0(N)$, $d$-regular primes, and working with Hecke operators on spaces of Hilbert newforms, we extend work of Freitas and Siksek to show that for most squarefree $d$ in this range there are no non-trivial solutions to this equation for $n \geq 4$.

-adic images of Galois for elliptic curves over (and an appendix with John Voight)

AbstractWe discuss the$\ell $-adic case of Mazur’s ‘Program B’ over$\mathbb {Q}$: the problem of classifying the possible images of$\ell $-adic Galois representations attached to elliptic curvesEover$\mathbb {Q}$, equivalently, classifying the rational points on the corresponding modular curves. The primes$\ell =2$and$\ell \ge 13$are addressed by prior work, so we focus on the remaining primes$\ell = 3, 5, 7, 11$. For each of these$\ell $, we compute the directed graph of arithmetically maximal$\ell $-power level modular curves$X_H$, compute explicit equations for all but three of them and classify the rational points on all of them except$X_{\mathrm {ns}}^{+}(N)$, for$N = 27, 25, 49, 121$and two-level$49$curves of genus$9$whose Jacobians have analytic rank$9$.Aside from the$\ell $-adic images that are known to arise for infinitely many${\overline {\mathbb {Q}}}$-isomorphism classes of elliptic curves$E/\mathbb {Q}$, we find only 22 exceptional images that arise for any prime$\ell $and any$E/\mathbb {Q}$without complex multiplication; these exceptional images are realised by 20 non-CM rationalj-invariants. We conjecture that this list of 22 exceptional images is complete and show that any counterexamples must arise from unexpected rational points on$X_{\mathrm {ns}}^+(\ell )$with$\ell \ge 19$, or one of the six modular curves noted above. This yields a very efficient algorithm to compute the$\ell $-adic images of Galois for any elliptic curve over$\mathbb {Q}$.In an appendix with John Voight, we generalise Ribet’s observation that simple abelian varieties attached to newforms on$\Gamma _1(N)$are of$\operatorname {GL}_2$-type; this extends Kolyvagin’s theorem that analytic rank zero implies algebraic rank zero to isogeny factors of the Jacobian of$X_H$.

Cusps and $q$-expansion principles for modular curves at infinite level

We develop an analytic theory of cusps for Scholze's p -adic modular curves at infinite level in terms of perfectoid parameter spaces for Tate curves. As an application, we describe a canonical tilting isomorphism between an anticanonical overconvergent neighbourhood of the ordinary locus of the modular curve at level \Gamma_1(p^\infty) and the analogous locus of an infinite level perfected Igusa variety. We also prove various q -expansion principles for functions on modular curves at infinite level, namely that the properties of extending to the cusps, vanishing, coming from finite level, and being bounded, can all be detected on q -expansions.

Cycle integrals of the Parson Poincaré series and intersection angles of geodesics on modular curves

We prove a geometric formula for the cycle integrals of Parson’s weight $2k$ modular integrals in terms of the intersection angles of geodesics on modular curves. Our result is an analog for modular integrals of a classical formula for the cycle integrals