Modular Elliptic Curves Research Articles

Modular elliptic curves over the field of twelfth roots of unity

In this paper we perform an extensive study of the spaces of automorphic forms for $\text{GL}_{2}$ of weight $2$ and level $\mathfrak{n}$, for $\mathfrak{n}$ an ideal in the ring of integers of the quartic CM field $\mathbb{Q}({\it\zeta}_{12})$ of twelfth roots of unity. This study is conducted through the computation of the Hecke module $H^{\ast }({\rm\Gamma}_{0}(\mathfrak{n}),\mathbb{C})$, and the corresponding Hecke action. Combining this Hecke data with the Faltings–Serre method for proving equivalence of Galois representations, we are able to provide the first known examples of modular elliptic curves over this field.Supplementary materials are available with this article.

$p$-adic L-functions of automorphic forms and exceptional zeros

We construct p -adic L-functions for automorphic representations of \mathrm{GL} _2 of a number field F , and show that the corresponding p -adic L-function of a modular elliptic curve E over F has an extra zero at the central point for each prime above p at which E has split multiplicative reduction, a part of the exceptional zero conjecture.

Weierstrass mock modular forms and elliptic curves

Mock modular forms, which give the theoretical framework for Ramanujan’s enigmatic mock theta functions, play many roles in mathematics. We study their role in the context of modular parameterizations of elliptic curves \(E/\mathbb {Q}\). We show that mock modular forms which arise from Weierstrass ζ-functions encode the central L-values and L-derivatives which occur in the Birch and Swinnerton-Dyer Conjecture. By defining a theta lift using a kernel recently studied by Hovel, we obtain canonical weight 1/2 harmonic Maass forms whose Fourier coefficients encode the vanishing of these values for the quadratic twists of E. We employ results of Bruinier and the third author, which builds on seminal work of Gross, Kohnen, Shimura, Waldspurger, and Zagier. We also obtain p-adic formulas for the corresponding weight 2 newform using the action of the Hecke algebra on the Weierstrass mock modular form.

Uniformization of modular elliptic curves via p-adic periods

The Langlands Programme predicts that a weight 2 newform f over a number field K with integer Hecke eigenvalues generally should have an associated elliptic curve Ef over K. In [19], we associated, building on works of Darmon [8] and Greenberg [20], a p-adic lattice Λ to f, under certain hypothesis, and implicitly conjectured that Λ is commensurable with the p-adic Tate lattice of Ef. In this paper, we present this conjecture in detail and discuss how it can be used to compute, directly from f, a Weierstrass equation for the conjectural Ef. We develop algorithms to this end and implement them in order to carry out extensive systematic computations in which we compute Weierstrass equations of hundreds of elliptic curves, some with huge heights, over dozens of number fields. The data we obtain give extensive support for the conjecture and furthermore demonstrate that the conjecture provides an efficient tool to building databases of elliptic curves over number fields.

Computation of framed deformation functors

In this work we compute the framed deformation functor associated to a reducible representation given as a direct sum of 2-dimensional representations associated to elliptic curves with appropriate local conditions. Such conditions arise in the works of Schoof and correspond to reduction properties of modular elliptic curves.

Quadratic twists of elliptic curves

The paper generalizes, for a wide class of elliptic curves defined over Q, the celebrated classical lemma of Birch and Heegner about quadratic twists with prime discriminants, to quadratic twists by discriminants having any prescribed number of prime factors. In addition, it proves stronger results for the family of quadratic twists of the modular elliptic curve X 0 ( 49 ) , including showing that there is a large class of explicit quadratic twists whose complex L-series does not vanish at s = 1 , and for which the full Birch–Swinnerton-Dyer conjecture is valid.

A Flexible Architecture for Modular Arithmetic Hardware Accelerators based on RNS

Modular arithmetic is a building block for a variety of applications potentially supported on embedded systems. An approach to turn modular arithmetic more efficient is to identify algorithmic modifications that would enhance the parallelization of the target arithmetic in order to exploit the properties of parallel devices and platforms. The Residue Number System (RNS) introduces data-level parallelism, enabling the parallelization even for algorithms based on modular arithmetic with several data dependencies. However, the mapping of generic algorithms to full RNS-based implementations can be complex and the utilization of suitable hardware architectures that are scalable and adaptable to different demands is required. This paper proposes and discusses an architecture with scalability features for the parallel implementation of algorithms relying on modular arithmetic fully supported by the Residue Number System (RNS). The systematic mapping of a generic modular arithmetic algorithm to the architecture is presented. It can be applied as a high level synthesis step for an Application Specific Integrated Circuit (ASIC) or Field Programmable Gate Array (FPGA) design flow targeting modular arithmetic algorithms. An implementation with the Xilinx Virtex 4 and Altera Stratix II Field Programmable Gate Array (FPGA) technologies of the modular exponentiation and Elliptic Curve (EC) point multiplication, used in the Rivest-Shamir-Adleman (RSA) and (EC) cryptographic algorithms, suggests latency results in the same order of magnitude of the fastest hardware implementations of these operations known to date.

On special zeros of p-adic L-functions of Hilbert modular forms

Let E be a modular elliptic curve over a totally real number field F. We prove the weak exceptional zero conjecture which links a (higher) derivative of the p-adic L-function attached to E to certain p-adic periods attached to the corresponding Hilbert modular form at the places above p where E has split multiplicative reduction. Under some mild restrictions on p and the conductor of E we deduce the exceptional zero conjecture in the strong form (i.e. where the automorphic p-adic periods are replaced by the \(\mathcal {L}\)-invariants of E defined in terms of Tate periods) from a special case proved earlier by Mok. Crucial for our method is a new construction of the p-adic L-function of E in terms of local data.

Computation of ATR Darmon Points on Nongeometrically Modular Elliptic Curves

ATR points were introduced by Darmon as a conjectural construction of algebraic points on certain elliptic curves for which the Heegner-point method is not in general available. So far, the only numerical evidence, provided by Darmon–Logan and Gärtner, concerned curves arising as quotients of Shimura curves. In those special cases, the ATR points can be obtained from the already existing Heegner points, thanks to results from Zhang and Darmon–Rotger–Zhao. In this paper, we compute for the first time an algebraic ATR point on a curve that is not uniformizable by any Shimura curve, thus providing the first piece of numerical evidence that Darmon's construction works beyond geometric modularity. To this purpose, we improve the method proposed by Darmon and Logan by removing the requirement that the real quadratic base field be norm-Euclidean and accelerating the numerical integration of Hilbert modular forms.

Modular abelian varieties of odd modular degree

In this paper, we will study modular Abelian varieties with odd congruence numbers by examining the cuspidal subgroup of $J_0(N)$. We will show that the conductor of such Abelian varieties must be of a special type. For example, if $N$ is the conductor of an absolutely simple modular Abelian variety with an odd congruence number, then $N$ has at most two prime divisors, and if $N$ is odd, then $N=p^\alpha$ or $N=pq$ for some prime $p$ and $q$. In the second half of this paper, we will focus on modular elliptic curves with odd modular degree. Our results, combined with the work of Agashe, Ribet, and Stein, finds necessary condition for elliptic curves to have odd modular degree. In the process we prove Watkins's conjecture for elliptic curves with odd modular degree and a nontrivial rational torsion point.

On Modular Ball-Quotient Surfaces of Kodaira Dimension One

Let be a lattice which is not co-ompact, of finite covolume with respect to the Bergman metric and acting freely on the open unit ball . Then the toroidal compactification is a projective smooth surface with elliptic compactification divisor . In this short note we discover a new class of unramifed ball quotients . We consider ball quotients with kod and . We prove that each minimal surface with finite Mordell-Weil group in the class described admits an étale covering which is a pull-back of . Here denotes the elliptic modular surface parametrizing elliptic curves with 6-torsion points which generate [6].

Plus/minus p-adic L-functions for Hilbert modular forms

R. Pollack constructed in Pollack (2003) [13] plus/minus p-adic L-functions for elliptic modular forms, which are p-adically bounded, when the Hecke eigenvalues at p are zero (the most supersingular case). The goal of this work is to generalize his construction to Hilbert modular forms. We find a suitable condition for Hilbert modular forms corresponding to the vanishing of p-th Hecke eigenvalue in elliptic modular form case, which guarantees the existence of plus/minus p-adic L-functions which are p-adically bounded. As an application, we construct cyclotomic plus/minus p-adic L-functions for modular elliptic curves over a totally real field F under the assumption that a p ( E ) = 0 for each prime p dividing p. We formulate a cyclotomic plus/minus Iwasawa main conjecture for such elliptic curves.

Towers of function fields over finite fields corresponding to elliptic modular curves

In this paper, we find several equations of recursive towers of function fields over finite fields corresponding to sequences of elliptic modular curves. This is a continuation of the work of Noam D. Elkies [8,9] on modular equations of higher degrees.

Modular elliptic directions with complex multiplication (with an application to Gross’s elliptic curves)

Let A_f be the abelian variety attached by Shimura to a normalized newform f\in S_2(\Gamma_1(N)) and assume that A_f has elliptic quotients. The paper deals with the determination of the one dimensional subspaces (elliptic directions) in S_2(\Gamma_1(N)) corresponding to the pullbacks of the regular differentials of all elliptic quotients of A_f . For modular elliptic curves over number fields without complex multiplication (CM), the directions were studied by the authors in [8]. The main goal of the present paper is to characterize the directions corresponding to elliptic curves with CM. Then we apply the results obtained to the case N=p^2 , for primes p>3 and p\equiv 3 mod 4 . For this case we prove that if f has CM, then all optimal elliptic quotients of A_f are also optimal in the sense that its endomorphism ring is the maximal order of \mathbb{Q}(\sqrt{-p}) . Moreover, if f has trivial Nebentypus then all optimal quotients are Gross’s elliptic curve A(p) and its Galois conjugates. Among all modular parametrizations J_0(p^2)\to A(p) , we describe a canonical one and discuss some of its properties.

On the Birch and Swinnerton-Dyer conjecture

In this paper we prove that if the Birch and Swinnerton-Dyer conjecture holds for products of a finite number of modular elliptic curves defined over totally real number fields, then the Birch and SwinnertonDyer conjecture holds for products of a finite number of elliptic curves defined over totally real number fields.

ON ASYMPTOTICALLY OPTIMAL TOWERS OVER QUADRATIC FIELDS RELATED TO GAUSS HYPERGEOMETRIC FUNCTIONS

We define two asymptotically optimal towers over quadratic fields, and give the explicit descriptions of the ramification loci and the sets of places splitting completely, which relate to the elliptic modular curves X0(4n) and X0(3n), respectively. Moreover, in the last section, we determine completely the modularity of a tower given by Maharaj and Wulftange in [18].

Increasing the number of fibered faces of arithmetic hyperbolic 3-manifolds

We exhibit a closed hyperbolic 3-manifold which satisfies a very strong form of Thurston's Virtual Fibration Conjecture. In particular, this manifold has finite covers which fiber over the circle in arbitrarily many ways. More precisely, it has a tower of finite covers where the number of fibered faces of the Thurston norm ball goes to infinity, in fact faster than any power of the logarithm of the degree of the cover, and we give a more precise quantitative lower bound. The example manifold $M$ is arithmetic, and the proof uses detailed number-theoretic information, at the level of the Hecke eigenvalues, to drive a geometric argument based on Fried's dynamical characterization of the fibered faces. The origin of the basic fibration $M\to S^1$ is the modular elliptic curve $E=X_0(49)$, which admits multiplication by the ring of integers of ${\Bbb Q}[\sqrt{-7}]$. We first base change the holomorphic differential on $E$ to a cusp form on ${{\rm GL}(2)}$ over $K={\Bbb Q}[\sqrt{-3}]$, and then transfer over to a quaternion algebra $D/K$ ramified only at the primes above $7$; the fundamental group of $M$ is a quotient of the principal congruence subgroup of ${\cal O}_D^\ast$ of level $7$. To analyze the topological properties of $M$, we use a new practical method for computing the Thurston norm, which is of independent interest. We also give a noncompact finite-volume hyperbolic 3-manifold with the same properties by using a direct topological argument.

The rationality of Stark-Heegner points over genus fields of real quadratic fields

We study the algebraicity of Stark-Heegner points on a modular elliptic curve E. These objects are /?-adic points on E given by the values of certain /?-adic integrals, but they are conjecturally defined over ring class fields of a real quadratic field K. The present article gives some evidence for this algebraicity conjecture by showing that linear combinations of Stark-Heegner points weighted by certain genus characters of K are defined over the predicted quadratic extensions of K. The non- vanishing of these combinations is also related to the appropriate twisted Hasse-Weil L-series of E over K, in the spirit of the Gross-Zagier formula for classical Heegner points.

On the modularity of supersingular elliptic curves over certain totally real number fields

We study generalisations to totally real fields of the methods originating with Wiles and Taylor and Wiles [A. Wiles, Modular elliptic curves and Fermat's Last Theorem, Ann. of Math. 141 (1995) 443–551; R. Taylor, A. Wiles, Ring-theoretic properties of certain Hecke algebras, Ann. of Math. 141 (1995) 553–572]. In view of the results of Skinner and Wiles [C. Skinner, A. Wiles, Nearly ordinary deformations of irreducible residual representations, Ann. Fac. Sci. Toulouse Math. (6) 10 (2001) 185–215] on elliptic curves with ordinary reduction, we focus here on the case of supersingular reduction. Combining these, we then obtain some partial results on the modularity problem for semistable elliptic curves, and end by giving some applications of our results, for example proving the modularity of all semistable elliptic curves over Q ( 2 ) .

CM points on products of Drinfeld modular curves

Let X X be a product of Drinfeld modular curves over a general base ring A A of odd characteristic. We classify those subvarieties of X X which contain a Zariski-dense subset of CM points. This is an analogue of the André-Oort conjecture. As an application, we construct non-trivial families of higher Heegner points on modular elliptic curves over global function fields.