Case Of Elliptic Curves Research Articles

Elliptic curves with a rational 2-torsion point ordered by conductor and the boundedness of average rank

In this paper we refine recent work due to A. Shankar, A. N. Shankar, and X. Wang on counting elliptic curves by conductor to the case of elliptic curves with a rational 2-torsion point. This family is a small family, as opposed to the large families considered by the aforementioned authors. We prove the analogous counting theorem for elliptic curves with so-called square-free index as well as for curves with suitably bounded Szpiro ratios. We note that the assumptions we impose on the size of the Szpiro ratios are less stringent than would be expected by the naive generalization of their approach; this requires an application of a theorem of Browning and Heath-Brown. Our proof also relies on linear programming techniques.

Lie invariant Frobenius lifts

We begin with the observation that the p-adic completion of any affine elliptic curve with ordinary reduction possesses Frobenius lifts ϕ that are “Lie invariant mod p” in the sense that the “normalized” action of ϕ on 1-forms preserves mod p the space of invariant 1-forms. Our main result is that, after removing the 2-torsion sections, the above situation can be “infinitesimally deformed” in the sense that the above mod p result has a mod p2 analogue. We end by showing that, in contrast with the case of elliptic curves, the following holds: if G is a linear algebraic group over a number field and if G is not a torus then for all but finitely many primes p the p-adic completion of G does not possess a Frobenius lift that is Lie invariant mod p.

Isogeny graphs on superspecial abelian varieties: eigenvalues and connection to Bruhat–Tits buildings

Abstract We study for each fixed integer $g \ge 2$ , for all primes $\ell $ and p with $\ell \neq p$ , finite regular directed graphs associated with the set of equivalence classes of $\ell $ -marked principally polarized superspecial abelian varieties of dimension g in characteristic p, and show that the adjacency matrices have real eigenvalues with spectral gaps independent of p. This implies a rapid mixing property of natural random walks on the family of isogeny graphs beyond the elliptic curve case and suggests a potential construction of the Charles–Goren–Lauter-type cryptographic hash functions for abelian varieties. We give explicit lower bounds for the gaps in terms of the Kazhdan constant for the symplectic group when $g \ge 2$ . As a byproduct, we also show that the finite regular directed graphs constructed by Jordan and Zaytman also has the same property.

Counting the number of the twists of certain polarized abelian varieties

We count the number of isomorphism classes of degree d-twists of some polarized abelian varieties of dimension g over finite fields where either g is an odd prime or else g=2. This can be seen as a higher dimensional analogue of the counting problem for the case of elliptic curves.

Invariants for trees of non-archimedean polynomials and skeleta of superelliptic curves

AbstractIn this paper we generalize the j-invariant criterion for the semistable reduction type of an elliptic curve to superelliptic curves X given by $$y^{n}=f(x)$$ y n = f ( x ) . We first define a set of tropical invariants for f(x) using symmetrized Plücker coordinates and we show that these invariants determine the tree associated to f(x). This tree then completely determines the reduction type of X for n that are not divisible by the residue characteristic. The conditions on the tropical invariants that distinguish between the different types are given by half-spaces as in the elliptic curve case. These half-spaces arise naturally as the moduli spaces of certain Newton polygon configurations. We give a procedure to write down their equations and we illustrate this by giving the half-spaces for polynomials of degree $$d\le {5}$$ d ≤ 5 .

Elements of prime order in Tate–Shafarevich groups of abelian varieties over

Abstract For each prime p, we show that there exist geometrically simple abelian varieties A over ${\mathbb Q}$ with . Specifically, for any prime $N\equiv 1 \ \pmod p$ , let $A_f$ be an optimal quotient of $J_0(N)$ with a rational point P of order p, and let $B = A_f/\langle P \rangle $ . Then the number of positive integers $d \leq X$ with is $ \gg X/\log X$ , where $\widehat B_d$ is the dual of the dth quadratic twist of B. We prove this more generally for abelian varieties of $\operatorname {\mathrm {GL}}_2$ -type with a p-isogeny satisfying a mild technical condition. In the special case of elliptic curves, we give stronger results, including many examples where for an explicit positive proportion of integers d.

Yield criteria representable by elliptic curves and Weierstrass form

Elliptic curve terminology comes from the close association with elliptic functions, and not because of any physical resemblance to an ellipse. The curves investigated here represent various yield loci in the plane having cubic algebraic relationships between the second and third invariants of the deviatoric stress tensor. A well-known yield condition attributed to Drucker falls into this classification. In addition, the more commonly used Tresca yield condition represents a limiting case of elliptic curves. All yield criteria based on elliptic curves, including the Tresca, can be parameterized in terms of the Weierstrass elliptic &-function. The properties of elliptic curves as they pertain to the formulation of various plastic yield criteria of materials are the topic of this investigation. Various perfectly plastic solutions of mode I crack problems are discussed.

Comparing anticyclotomic Selmer groups of positive coranks for congruent modular forms – Part II

We study the Selmer group associated to a p-ordinary newform f∈S2r(Γ0(N)) over the anticyclotomic Zp-extension of an imaginary quadratic field K/Q. Under certain assumptions, we prove that this Selmer group has no proper Λ-submodules of finite index. This generalizes work of Bertolini in the elliptic curve case. We also offer both a correction and an improvement to an earlier result on Iwasawa invariants of congruent modular forms by the present authors. Along the way, we prove a general result on the vanishing of several anticyclotomic μ-invariants attached to f.

Torsion growth over cubic fields of rational elliptic curves with complex multiplication

This article is a contribution to the project of classifying the torsion growth of elliptic curve upon base-change. In this article we treat the case of elliptic curve defined over the rationals with complex multiplication. For this particular case, we give a description of the possible torsion growth over cubic fields and a completely explicit description of this growth in terms of some invariants attached to a given elliptic curve.

Euler characteristics and their congruences in the positive rank setting

AbstractThe notion of the truncated Euler characteristic for Iwasawa modules is an extension of the notion of the usual Euler characteristic to the case when the homology groups are not finite. This article explores congruence relations between the truncated Euler characteristics for dual Selmer groups of elliptic curves with isomorphic residual representations, over admissible p-adic Lie extensions. Our results extend earlier congruence results from the case of elliptic curves with rank zero to the case of higher rank elliptic curves. The results provide evidence for the p-adic Birch and Swinnerton-Dyer formula without assuming the main conjecture.

Rank and bias in families of hyperelliptic curves via Nagao's conjecture

Let X:y2=f(x) be a hyperelliptic curve over Q(T) of genus g≥1. Assume that the jacobian of X over Q(T) has no subvariety defined over Q. Denote by Xt the specialization of X to an integer T=t, let aXt(p) be its trace of Frobenius, and let AX,r(p)=1p∑t=1paXt(p)r be its r-th moment. The first moment is related to the rank of the jacobian JX(Q(T)) by a generalization of a conjecture of Nagao:limX→∞⁡1X∑p≤X−AX,1(p)log⁡p=rankJX(Q(T)). Generalizing a result of S. Arms, Á. Lozano-Robledo, and S.J. Miller, we compute first moments for various families resulting in infinitely many hyperelliptic curves over Q(T) having jacobian of moderately large rank 4g+2, where g is the genus; by Silverman's specialization theorem, this yields hyperelliptic curves over Q with large rank jacobian. Note that Shioda has the better record in this direction: he constructed hyperelliptic curves of genus g with jacobian of rank 4g+7. In the case when X is an elliptic curve, Michel proved p⋅AX,2=p2+O(p3/2). For the families studied, we observe the same second moment expansion. Furthermore, we observe the largest lower order term that does not average to zero is on average negative, a bias first noted by S.J. Miller in the elliptic curve case. We prove this bias for a number of families of hyperelliptic curves.

On the p-adic Birch and Swinnerton-Dyer conjecture for elliptic curves over number fields

We formulate a multivariable p-adic Birch and Swinnerton-Dyer conjecture for p-ordinary elliptic curves A over number fields K. It generalizes the one-variable conjecture of Mazur, Tate, and Teitelbaum, who studied the case K=Q and the phenomenon of exceptional zeros. We discuss old and new theoretical evidence toward our conjecture and in particular we fully prove it, under mild conditions, in the following situation: K is imaginary quadratic, A=EK is the base change to K of an elliptic curve over the rationals, and the rank of A is either 0 or 1. The proof is naturally divided into a few cases. Some of them are deduced from the purely cyclotomic case of elliptic curves over Q, which we obtain from a refinement of recent work of Venerucci alongside the results of Greenberg, Stevens, Perrin-Riou, and the author. The only genuinely multivariable case (rank 1, two exceptional zeros, three partial derivatives) is newly established here. Its proof generalizes to show that the “almost-anticyclotomic” case of our conjecture is a consequence of conjectures of Bertolini and Darmon on families of Heegner points, and of (partly conjectural) p-adic Gross–Zagier and Waldspurger formulas in families.

Explicit root numbers of abelian varieties

The Birch and Swinnerton-Dyer conjecture predicts that the parity of the algebraic rank of an abelian variety over a global field should be controlled by the expected sign of the functional equation of its L L -function, known as the global root number. In this paper, we give explicit formulae for the local root numbers as a product of Jacobi symbols. This enables one to compute the global root number, generalising work of Rohrlich, who studies the case of elliptic curves. We provide similar formulae for the root numbers after twisting the abelian variety by a self-dual Artin representation. As an application, we find a rational genus two hyperelliptic curve with a simple Jacobian whose root number is invariant under quadratic twist.

Hash functions from superspecial genus-2 curves using Richelot isogenies

AbstractIn 2018 Takashima proposed a version of Charles, Goren and Lauter’s hash function using Richelot isogenies, starting from a genus-2 curve that allows for all subsequent arithmetic to be performed over a quadratic finite field 𝔽p2. In 2019 Flynn and Ti pointed out that Takashima’s hash function is insecure due to the existence of small isogeny cycles. We revisit the construction and show that it can be repaired by imposing a simple restriction, which moreover clarifies the security analysis. The runtime of the resulting hash function is dominated by the extraction of 3 square roots for every block of 3 bits of the message, as compared to one square root per bit in the elliptic curve case; however in our setting the extractions can be parallelized and are done in a finite field whose bit size is reduced by a factor 3. Along the way we argue that the full supersingular isogeny graph is the wrong context in which to study higher-dimensional analogues of Charles, Goren and Lauter’s hash function, and advocate the use of the superspecial subgraph, which is the natural framework in which to view Takashima’s 𝔽p2-friendly starting curve.

Computing the endomorphism ring of an ordinary abelian surface over a finite field

We present a new algorithm for computing the endomorphism ring of an ordinary abelian surface over a finite field which is subexponential and generalizes an algorithm of Bisson and Sutherland for elliptic curves. The correctness of this algorithm only requires the heuristic assumptions required by the algorithm of Biasse and Fieker [2] which computes the class group of an order in a number field in subexponential time. Thus we avoid the multiple heuristic assumptions on isogeny graphs and polarized class groups which were previously required. The output of the algorithm is an ideal in the maximal totally real subfield of the endomorphism algebra, generalizing the elliptic curve case.

Higher distances for constant dimensions codes: the case of osculating spaces to a Veronese variety

In linear network coding the information is encoded in terms of a basis of a vector space and it is received as a basis of a possible altered vector space. In the constant dimension case Koetter and Kschischang introduced a metric on the Grassmannian and proved efficient and correct decoding in terms of this metric. Here we introduce a second order invariant of the code: the minimum dimension of the linear span of 3 different linear subspaces belonging to the code. This is the case \(s=3\) of a family \(d_s\) and \(d'_s\), \(s\ge 3\), of invariants of network codes. We study these invariants in a case recently proposed by Hansen (the set of all osculating spaces of a Veronese embedding of a finite projective space) and for a related case (the set of osculating spaces to curves of positive genus) with a complete description of the case of elliptic curves and the ones related to the Hermitian curve.

Moduli of elliptic curves via twisted stable maps

Abramovich, Corti and Vistoli have studied modular compactifications of stacks of curves equipped with abelian level structures arising as substacks of the stack of twisted stable maps into the classifying stack of a finite group, provided the order of the group is invertible on the base scheme. Recently Abramovich, Olsson and Vistoli extended the notion of twisted stable maps to allow arbitrary base schemes, where the target is a tame stack, not necessarily Deligne-Mumford. We use this to extend the results of Abramovich, Corti and Vistoli to the case of elliptic curves with level structures over arbitrary base schemes; we prove that we recover the compactified Katz-Mazur regular models, with a natural moduli interpretation in terms of level structures on Picard schemes of twisted curves. Additionally, we study the interactions of the different such moduli stacks contained in a stack of twisted stable maps in characteristics dividing the level.

The complex AGM, periods of elliptic curves over C and complex elliptic logarithms

We give an account of the complex Arithmetic–Geometric Mean (AGM), as first studied by Gauss, together with details of its relationship with the theory of elliptic curves over C, their period lattices and complex parametrisation. As an application, we present efficient methods for computing bases for the period lattices and elliptic logarithms of points, for arbitrary elliptic curves defined over C. Earlier authors have only treated the case of elliptic curves defined over the real numbers; here, the multi-valued nature of the complex AGM plays an important role. Our method, which we have implemented in both MAGMA and Sage, is illustrated with several examples using elliptic curves defined over number fields with real and complex embeddings.

Powers of 2 in modular degrees of modular abelian varieties

An analogue, for modular abelian varieties A , of a conjecture of Watkins on elliptic curves over Q , would say that 2 R divides the modular degree, where R is the rank of the Mordell–Weil group A ( Q ) . We exhibit some numerical evidence for this. We examine various sources of factors of 2 in the modular degree, and the extent to which they are independent. Assuming that a certain 2-adic Hecke ring is a local complete intersection, and is isomorphic to a Galois deformation ring (a 2-adic “ R ≃ T ” theorem), we show how the analogue of Watkinsʼs conjecture follows, under certain conditions on A , extending and correcting earlier work on the elliptic curve case.

On uniform large Galois images for modular abelian varieties

We formulate a question regarding uniform versions of "large Galois image properties" for modular abelian varieties of higher dimension, generalizing the well-known case of elliptic curves. We then answer our question affirmatively in the exceptional image case, and provide lower estimates for uniform bounds in the remaining cases.