This article explores the connection between radical isogenies and modular curves. Radical isogenies are formulas designed for the computation of chains of isogenies of fixed small degree $N$, introduced by Castryck, Decru, and Vercauteren at Asiacrypt 2020. One significant advantage of radical isogeny formulas over other formulas with a similar purpose is that they eliminate the need to generate a point of order $N$ that generates the kernel of the isogeny. While radical isogeny formulas were originally developed using elliptic curves in Tate normal form, Onuki and Moriya have proposed radical isogeny formulas of degrees $3$ and $4$ on Montgomery curves and attempted to obtain a simpler form of radical isogenies using enhanced elliptic and modular curves. In this article, we translate the original setup of radical isogenies in Tate normal form into the language of modular curves. Additionally, we solve an open problem introduced by Onuki and Moriya regarding radical isogeny formulas on $X_0(N).$
We provide explicit bounds on the difference of heights of the $j$-invariants of isogenous elliptic curves defined over $\overline{\mathbb{Q}}$. The first one is reminiscent of a classical estimate for the Faltings height of isogenous abelian varieties, which is indeed used in the proof. We also use an explicit version of Silverman's inequality and isogeny estimates by Gaudron and Rémond. We give applications to the study of Vélu's formulas and of modular polynomials.
Schoof's classic algorithm allows point-counting for elliptic curves over finite fields in polynomial time. This algorithm was subsequently improved by Atkin, using factorizations of modular polynomials, and by Elkies, using a theory of explicit isogenies. Moving to Jacobians of genus-2 curves, the current state of the art for point counting is a generalization of Schoof's algorithm. While we are currently missing the tools we need to generalize Elkies' methods to genus 2, recently Martindale and Milio have computed analogues of modular polynomials for genus-2 curves whose Jacobians have real …
The SEA algorithm for computing the cardinality of elliptic curves over finite fields in many characteristic uses modular polynomials. These polynomials come into different flavors, and methods to compute them flourished. Once equipped with some modular polynomials for prime $\ell$, algebraic formulas are used to compute a curve $\mathcal{E}^*/\mathbb{F}_q$ that is $\ell$-isogenous to the curve of interest $\mathcal{E}$. These formulas involve derivatives of the modular polynomial that may sometime vanish. One way to overcome this problem is to use alternative trivariate polynomials $U_\ell$, …
Let $\mathcal{O}$ be an order in the imaginary quadratic field $K$. For positive integers $M \mid N$, we determine the least degree of an $\mathcal{O}$-CM point on the modular curve $X(M,N)_{/K(ζ_M)}$ and also on the modular curve $X(M,N)_{/\mathbb{Q}(ζ_M)}$: that is, we treat both the case in which the complex multiplication is rationally defined and the case in which we do not assume that the complex multiplication is rationally defined. To prove these results we establish several new theorems on rational cyclic isogenies of CM elliptic curves. In particular, we extend a result of Kwon that …
Let $n$ be an integer such that the modular curve $X_0(n)$ is hyperelliptic of genus $\ge2$ and such that the Jacobian of $X_0(n)$ has rank $0$ over $\mathbb Q$. We determine all points of $X_0(n)$ defined over quadratic fields, and we give a moduli interpretation of these points. As a consequence, we show that up to $\overline{\mathbb Q}$-isomorphism, all but finitely many elliptic curves with $n$-isogenies over quadratic fields are in fact $\mathbb Q$-curves, and we list all exceptions. We also show that, again with finitely many exceptions up to $\overline{\mathbb Q}$-isomorphism, every $\m…
We provide examples of abelian surfaces over number fields $K$ whose reductions at almost all good primes possess an isogeny of prime degree $\ell$ rational over the residue field, but which themselves do not admit a $K$-rational $\ell$-isogeny. This builds on work of Cullinan and Sutherland. When $K=\mathbb{Q}$, we identify certain weight-$2$ newforms $f$ with quadratic Fourier coefficients whose associated modular abelian surfaces $A_f$ exhibit such a failure of a local-global principle for isogenies.
This is a sequel to the paper [F. Breuer, H.-G. Rück, Drinfeld modular polynomials in higher rank, J. Number Theory 129 (2009), 59-83.], in which we introduced Drinfeld modular polynomials of higher rank, using an analytic construction. These polynomials relate the isomorphism invariants of Drinfeld F_q[T]-modules of rank r\geq 2 linked by isogenies of a specified type. In the current paper, we give an algebraic construction of greater generality, and prove a generalization of the Kronecker congruences relations, which describe what happens when modular polynomials associated to P-isogenies ar…
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.
Elliptic $K$-curves are elliptic curves defined over some field extension $L/K$ that are isogenous to all of their Galois conjugates. We present a new result on $K$-curves $E$ that are given by a $K$-rational orbit $\{τ, -1/Nτ\}$ of the Fricke involution on $Y_0(N)$, giving a simple Diophantine condition on the extension $L/K$ that determines which twists of $E$ allow the isogeny between Galois conjugates to be defined over $L$. To support and illustrate this result, we also discuss parameterizations of cyclic $N$-isogenies corresponding to points on modular curves $X_0(N)$ of genus $0$. These…
We study the failure of a local-global principle for the existence of $l$-isogenies for elliptic curves over number fields $K$. Sutherland has shown that over $\mathbb{Q}$ there is just one failure, which occurs for $l=7$ and a unique $j$-invariant, and has given a classification of such failures when $K$ does not contain the quadratic subfield of the $l$'th cyclotomic field. In this paper we provide a classification of failures for number fields which do contain this quadratic field, and we find a new `exceptional' source of such failures arising from the exceptional subgroups of $\mbox{PGL}_…
In this article we introduce Hecke operators on the differential algebra of geometric quasi-modular forms. As an application for each natural number $d$ we construct a vector field in six dimensions which determines uniquely the polynomial relations between the Eisenstein series of weight 2,4 and 6 and their transformation under multiplication of the argument by $d$, and in particular, it determines uniquely the modular curve of degree $d$ isogenies between elliptic curves.
We present a new probabilistic algorithm to compute modular polynomials modulo a prime. Modular polynomials parameterize pairs of isogenous elliptic curves and are useful in many aspects of computational number theory and cryptography. Our algorithm has the distinguishing feature that it does not involve the computation of Fourier coefficients of modular forms. We avoid computing the exponentially large integral coefficients by working directly modulo a prime and computing isogenies between elliptic curves via Velu's formulas.
Let $p$ be an unramified prime in a totally real field $L$ such that $h^+(L)=1$. Our main result shows that Hilbert modular newforms of parallel weight two for $Γ_0(p)$ can be constructed naturally, via classical theta series, from modules of isogenies of superspecial abelian varieties with real multiplication on a Hilbert moduli space. This can be viewed as a geometric reinterpretation of the Eichler Basis Problem for Hilbert modular forms.
We present a new algorithm to compute the classical modular polynomial Phi_n in the rings Z[X,Y] and (Z/mZ)[X,Y], for a prime n and any positive integer m. Our approach uses the graph of n-isogenies to efficiently compute Phi_n mod p for many primes p of a suitable form, and then applies the Chinese Remainder Theorem (CRT). Under the Generalized Riemann Hypothesis (GRH), we achieve an expected running time of O(n^3 (log n)^3 log log n), and compute Phi_n mod m using O(n^2 (log n)^2 + n^2 log m) space. We have used the new algorithm to compute Phi_n with n over 5000, and Phi_n mod m with n over…
Let $p$ and $q$ be distinct primes, and let $X_{p,q}$ be the $(q+1)$-regular graph whose nodes are supersingular elliptic curves over $\overline{\mathbb{F}}_p$ and whose edges are $q$-isogenies. For fixed $p$, we compute the distribution of the $\ell$-Sylow subgroup of the sandpile group (i.e.\ Jacobian) of $X_{p,q}$ as $q \to \infty$. We find that the distribution disagrees with the Cohen-Lenstra heuristic in this context. Our proof is via Galois representations attached to modular curves. As a corollary of our result, we give an upper bound on the probability that the Jacobian is cyclic, whi…
Suppose that $G$ is a simple adjoint reductive group over $\mathbf{Q}$, with an exceptional Dynkin type, and with $G(\mathbf{R})$ quaternionic (in the sense of Gross-Wallach). Then there is a notion of modular forms for $G$, anchored on the so-called quaternionic discrete series representations of $G(\mathbf{R})$. The purpose of this paper is to give an explicit form of the Fourier expansion of modular forms on $G$, along the unipotent radical $N$ of the Heisenberg parabolic $P = MN$ of $G$.
Suppose that $G$ is a simple reductive group over $\mathbf{Q}$, with an exceptional Dynkin type, and with $G(\mathbf{R})$ quaternionic (in the sense of Gross-Wallach). In a previous paper, we gave an explicit form of the Fourier expansion of modular forms on $G$ along the unipotent radical of the Heisenberg parabolic. In this paper, we give the Fourier expansion of the minimal modular form $θ_{Gan}$ on quaternionic $E_8$, and some applications. The $Sym^{8}(V_2)$-valued automorphic function $θ_{Gan}$ is a weight four, level one modular form on $E_8$, which has been studied by Gan. The applicat…
