Modular Curves Research Articles

Exceptional isogenies between reductions of pairs of elliptic curves

Let E and E' be two elliptic curves over a number field. We prove that the reductions of E and E' at a finite place p are geometrically isogenous for infinitely many p, and draw consequences for the existence of supersingular primes. This result is an analogue for distributions of Frobenius traces of known results on the density of Noether-Lefschetz loci in Hodge theory. The proof relies on dynamical properties of the Hecke correspondences on the modular curve.

The Chevalley-Weil Formula for Orbifold Curves

In the 1930s Chevalley and Weil gave a formula for decomposing the canonical representation on the space of differential forms of the Galois group of a ramified Galois cover of Riemann surfaces. In this article we prove an analogous Chevalley-Weil formula for ramified Galois covers of orbifold curves. We then specialize the formula to the case when the base orbifold curve is the (reduced) modular orbifold. As an application of this latter formula we decompose the canonical representations of modular curves of full, prime level and of Fermat curves of arbitrary exponent.

The action of the Hecke operators on the component groups of modular Jacobian varieties

For a prime number $q\geq 5$ and a positive integer $N$ prime to $q$, Ribet proved the action of the Hecke algebra on the component group of the Jacobian variety of the modular curve of level $Nq$ at $q$ is "Eisenstein", which means the Hecke operator $T_\ell$ acts by $\ell+1$ when $\ell$ is a prime number not dividing the level. In this paper, we completely compute the action of the Hecke algebra on this component group by a careful study of supersingular points with extra automorphisms.

Double covers of Cartan modular curves

We present a strategy to obtain explicit equations for the modular double covers associated respectively to both a split and a non-split Cartan subgroup of GL2(Fp) with p prime. Then we apply it successfully to the level 13 case.

Algebraic de Rham theory for weakly holomorphic modular forms of level one

We establish an Eichler-Shimura isomorphism for weakly modular forms of level one. We do this by relating weakly modular forms with rational Fourier coefficients to the algebraic de Rham cohomology of the modular curve with twisted coefficients. This leads to formulae for the periods and quasi-periods of modular forms.

Modular curves and symmetries of Hecke type

We give a characterization of modular curves by a single symmetry of Hecke type. In the proof, we use the theorem of André, which characterizes modular curves in terms of special points.

On the $p$-adic periods of the modular curve $X(\Gamma _0(p) \cap \Gamma (2))$

On the $p$-adic periods of the modular curve $X(\Gamma _0(p) \cap \Gamma (2))$

Modularity of generating series of winding numbers

The Shimura correspondence connects modular forms of integral weights and half-integral weights. One of the directions is realized by the Shintani lift, where the inputs are holomorphic differentials and the outputs are holomorphic modular forms of half-integral weight. In this article, we generalize this lift to differentials of the third kind. As an application we obtain a modularity result concerning the generating series of winding numbers of closed geodesics on the modular curve.

Indecomposable vector-valued modular forms and periods of modular curves

We classify the three-dimensional representations of the modular group that are reducible but indecomposable, and their associated spaces of holomorphic vector-valued modular forms. We then demonstrate how such representations may be employed to compute periods of modular curves. This technique obviates the use of Hecke operators, and therefore provides a method for studying noncongruence modular curves as well as congruence.

Rational torsion subgroups of modular Jacobian varieties

In this article, we study the Q-rational torsion subgroups of the Jacobian varieties of modular curves. The main result is that, for any positive integer N, J0(N)(Q)tor[q∞]=0 if q is a prime not dividing 6⋅N⋅∏p|N(p2−1). To prove the result, we explicitly construct a collection of Eisenstein series with rational Fourier expansions, and then determine their constant terms to control the size of the rational torsion subgroups.

On divisors of modular forms

The denominator formula for the Monster Lie algebra is the product expansion for the modular function J(z)−J(τ) given in terms of the Hecke system of SL2(Z)-modular functions jn(τ). It is prominent in Zagier's seminal paper on traces of singular moduli, and in the Duncan–Frenkel work on Moonshine. The formula is equivalent to the description of the generating function for the jn(z) as a weight 2 modular form with a pole at z. Although these results rely on the fact that X0(1) has genus 0, here we obtain a generalization, framed in terms of polar harmonic Maass forms, for all of the X0(N) modular curves. We use these functions to study divisors of modular forms.

Eta-quotients and embeddings of $$X_0(N)$$ X 0 ( N ) in the projective plane

In this paper, we find projective plane models of the modular curves $$X_0(N)$$ by constructing maps from $$X_0(N)$$ to the projective plane using modular forms. We use eta-quotients of weight 12. We find those eta-quotients in $$M_{12}(\varGamma _0(N))$$ which have maximal order of vanishing at infinity.

Congruence formulas for Legendre modular polynomials

Let p≥5 be a prime number. We generalize the results of E. de Shalit [4] about supersingular j-invariants in characteristic p. We consider supersingular elliptic curves with a basis of 2-torsion over F‾p, or equivalently supersingular Legendre λ-invariants. Let Fp(X,Y)∈Z[X,Y] be the p-th modular polynomial for λ-invariants. A simple generalization of Kronecker's classical congruence shows that R(X):=Fp(X,Xp)p is in Z[X]. We give a formula for R(λ) if λ is supersingular. This formula is related to the Manin–Drinfeld pairing used in the p-adic uniformization of the modular curve X(Γ0(p)∩Γ(2)). This pairing was computed explicitly modulo principal units in a previous work of both authors. Furthermore, if λ is supersingular and is in Fp, then we also express R(λ) in terms of a CM lift (which is shown to exist) of the Legendre elliptic curve associated to λ.

Paramodular forms of level 8 and weights 10 and 12

We study degree 2 paramodular eigenforms of level 8 and weights 10 and 12, and determine all their local representations. We prove dimensions by the technique of Jacobi restriction. A level divisible by a cube permits a wide variety of local representations, but also complicates the Hecke theory by involving Fourier expansions at more than one zero-dimensional cusp. We overcome this difficulty by the technique of restriction to modular curves. An application of our determination of the local representations is that we obtain the Euler 2-factor of each newform.

On a simple model of $$X_0(N)$$ X 0 ( N )

We find plane models for all $$X_0(N)$$ , $$N\ge 2$$ . We observe a map from the modular curve $$X_0(N)$$ to the projective plane constructed using modular forms of weight 12 for the group $$\Gamma _0(N)$$ ; the Ramanujan function $$\Delta $$ , $$\Delta (N\cdot )$$ and the third power of Eisestein series of weight 4, $$E_4^3$$ , and prove that this map is birational equivalence for every $$N\ge 2$$ . The equation of the model is the minimal polynomial of $$\Delta (N\cdot )/\Delta $$ over $${\mathbb {C}}(j)$$ .

Reduction of dynatomic curves

In this paper, we make partial progress on a function field version of the dynamical uniform boundedness conjecture for certain one-dimensional families ${\mathcal{F}}$ of polynomial maps, such as the family $f_{c}(x)=x^{m}+c$, where $m\geq 2$. We do this by making use of the dynatomic modular curves $Y_{1}(n)$ (respectively $Y_{0}(n)$) which parametrize maps $f$ in ${\mathcal{F}}$ together with a point (respectively orbit) of period $n$ for $f$. The key point in our strategy is to study the set of primes $p$ for which the reduction of $Y_{1}(n)$ modulo $p$ fails to be smooth or irreducible. Morton gave an algorithm to construct, for each $n$, a discriminant $D_{n}$ whose list of prime factors contains all the primes of bad reduction for $Y_{1}(n)$. In this paper, we refine and strengthen Morton’s results. Specifically, we exhibit two criteria on a prime $p$ dividing $D_{n}$: one guarantees that $p$ is in fact a prime of bad reduction for $Y_{1}(n)$, yet this same criterion implies that $Y_{0}(n)$ is geometrically irreducible. The other guarantees that the reduction of $Y_{1}(n)$ modulo $p$ is actually smooth. As an application of the second criterion, we extend results of Morton, Flynn, Poonen, Schaefer, and Stoll by giving new examples of good reduction of $Y_{1}(n)$ for several primes dividing $D_{n}$ when $n=7,8,11$, and $f_{c}(x)=x^{2}+c$. The proofs involve a blend of arithmetic and complex dynamics, reduction theory for curves, ramification theory, and the combinatorics of the Mandelbrot set.

On Hecke’s decomposition of the regular differentials on the modular curve of prime level

In this paper, we review Hecke’s decomposition of the regular differentials on the modular curve of prime level p under the action of the group $${{\mathrm{SL}}}_2(p)/\langle {\pm 1}\rangle $$ SL 2 ( p ) / ⟨ ± 1 ⟩ . We show that his distinguished subspace corresponds to a factor of the Jacobian which decomposes as a product of conjugate, isogenous elliptic curves with complex multiplication.

On the essential minimum of Faltings’ height

We study the essential minimum of the (stable) Faltings height on the moduli space of elliptic curves. We prove that, in contrast to the Weil height on a projective space and the N{e}ron-Tate height of an abelian variety, Faltings' height takes at least two values that are smaller than its essential minimum. We also provide upper and lower bounds for this quantity that allow us to compute it up to five decimal places. In addition, we give numerical evidence that there are at least four isolated values before the essential minimum. One of the main ingredients in our analysis is a good approximation of the hyperbolic Green function associated to the cusp of the modular curve of level one. To establish this approximation, we make an intensive use of distortion theorems for univalent functions. Our results have been motivated and guided by numerical experiments that are described in detail in the companion files.

The pro-l outer Galois actions associated to modular curves of prime power level

The pro-l outer Galois actions associated to modular curves of prime power level

The Belyi Characterization of a Class of Modular Curves

The Belyi Characterization of a Class of Modular Curves