Plectic Stark–Heegner points

Plectic points were introduced by Fornea and Gehrmann [Adv. Math. 414 (2023)] as certain tensor products of local points on elliptic curves over arbitrary number fields F F . They are constructed in rank r ≤ [ F : Q ] r\leq [F:\mathbb {Q}] -situations, and they conjecturally come from p p -adic regulators of basis of the Mordell-Weil group defined over dihedral extensions of F F . In this article we define anticyclotomic p p -adic L-functions with variables of type weight and level attached to a family of overconvergent modular symbols defined over totally real fields F F and a quadratic extension K / F K/F . Their restriction to the weight space provides Hida-Rankin p p -adic L-functions. If such a family passes through an overconvergent modular symbol attached to a modular elliptic curve E E , we obtain a p p -adic Gross-Zagier formula that computes higher derivatives of such Hida-Rankin p p -adic L-functions in terms of plectic points. This result generalizes that of Bertolini and Darmon [Ann. of Math. (2) 170 (2009), pp. 343–370], which has been key to the recent approach towards the algebraicity of Darmon points.

The article [Darmon 02] proposes a conjectural p-adic analytic construction of points on (modular) elliptic curves, points which are defined over ring class fields of real quadratic fields. These points are related to classical Heegner points in the same way as Stark units to circular or elliptic units. For this reason they are called “Stark-Heegner points,” following a terminology introduced in [Darmon 98]. If K is a real quadratic field, the Stark-Heegner points attached to K are conjectured to satisfy an analogue of the Shimura reciprocity law, so that they can in principle be used to find explicit generators for the ring class fields of K. It is also expected that their heights can be expressed in terms of derivatives of the Rankin L-series attached to E and K, in analogy with the Gross-Zagier formula . The main goal of this paper is to describe algorithms for calculating Stark-Heegner points and supply numerical evidence for the Shimura reciprocity and Gross-Zagier conjectures, focussing primarily on elliptic curves of prime conductor.

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

Let E be a modular elliptic curve over a totally real cubic field. We have a cubic-triple product motive over \mathbb{Q} constructed from E through multiplicative induction; it is of rank 8. We show that, under certain assumptions on E , the nonvanishing of the central critical value of the L -function attached to the motive implies that the dimension of the associated Bloch-Kato Selmer group is 0.

Let Γ \Gamma be a congruence subgroup of type ( N 1 , N 2 ) (N_1,N_2) and of level N N . We study congruences between weight 2 normalized newforms f f and Eisenstein series E E on Γ \Gamma modulo a prime p \mathfrak {p} above a rational prime p p . Assume that p ∤ 6 N p\nmid 6N , E E is a common eigenfunction for all Hecke operators and f f is ordinary at p \mathfrak {p} . We show that the abelian variety associated to f f and the cuspidal subgroup associated to E E intersect non-trivially in their p p -torsion points. Let A A be a modular elliptic curve over Q \mathbb {Q} with good ordinary reduction at p p . We apply the above result to show that an isogeny of degree divisible by p p from the optimal curve A 1 A_1 in the Q \mathbb {Q} -isogeny class of elliptic curves containing A A to A A extends to an étale morphism of Néron models over Z p \mathbb {Z}_p if p > 7 p>7 . We use this to show that p p -adic distributions associated to the p p -adic L L -functions of A A are Z p \mathbb {Z}_p -valued.

For a modular elliptic curve E/, we show a number of links between the primes ℓ for which the mod ℓ representation of E/ has projective dihedral image and congruence primes for the newform associated to E/.

We prove the finiteness of the subgroup of the Jacobian generated by the cusps of a Drinfeld modular curve. This is an analogue in positive characteristic of a classical result of Drinfeld and Manin about elliptic modular curves, and generalizes earlier work on special cases.

We discuss the computational application of Heegner points to the study of elliptic curves over Q, concentrating on the curves E d : Dy2 = x3 − x arising in the “congruent number” problem. We begin by briefly reviewing the cyclotomic construction of units in real quadratic number fields, which is analogous in many ways to the Heegner-point approach to the arithmetic of elliptic curves, and allows us to introduce some of the key ideas in a more familiar setting. We then quote the key results and conjectures that we shall need concerning elliptic curves and Heegner points, and show how they yield practical algorithms for finding rational points on E d and other properties of such curves. We conclude with a report on more recent work along similar lines on the elliptic curves x3 + y3 = A.

Let p be an odd prime and K an imaginary quadratic field where p splits. Under appropriate hypotheses, Bertolini showed that the Selmer group of a p-ordinary elliptic curve over the anticyclotomic \({\mathbb {Z}}_p\)-extension of K does not admit any proper \(\Lambda \)-submodule of finite index, where \(\Lambda \) is a suitable Iwasawa algebra. We generalize this result to the plus and minus Selmer groups (in the sense of Kobayashi) of p-supersingular elliptic curves. In particular, in our setting the plus/minus Selmer groups have \(\Lambda \)-corank one, so they are not \(\Lambda \)-cotorsion. As an application of our main theorem, we prove results in the vein of Greenberg–Vatsal on Iwasawa invariants of p-congruent elliptic curves, extending to the supersingular case results for p-ordinary elliptic curves due to Hatley–Lei.

The concept of public key cryptography was first introduced by Diffie and Hellman in 1976 using discrete logarithm problem as base of difficulty. In 1985, T. ElGamal proposed public key cryptosystem scheme based on discrete logarithm problem. Elliptic curve cryptosystems were first proposed in 1985 independently by Neil Koblitz and Victor Miller. Elliptic curve cryptosystems are unique in using elliptic curve groups for arithmetic. This cryptosystem is based on discrete logarithm problem in the group of points of an elliptic curve defined over a finite field. The discrete logarithm problem in an elliptic curve group appears to be much harder than the discrete logarithm problem in other groups. Hence elliptic curves cryptosystem can match the security of other cryptosystems while using smaller key. In this paper we will discuss a VLSl implementation of elliptic curves cryptosystem for ElGamal encryption scheme.

Let K be an arbitrary number field, and let ρ : Gal($\math{\bar{K}}$/K) → GL2(E) be a nearly ordinary irreducible geometric Galois representation. In this paper, we study the nearly ordinary deformations of ρ. When K is totally real and ρ is modular, results of Hida imply that the nearly ordinary deformation space associated to ρ contains a Zariski dense set of points corresponding to ‘automorphic’ Galois representations. We conjecture that if K is not totally real, then this is never the case, except in three exceptional cases, corresponding to: (1) ‘base change’, (2) ‘CM’ forms, and (3) ‘even’ representations. The latter case conjecturally can only occur if the image of ρ is finite. Our results come in two flavours. First, we prove a general result for Artin representations, conditional on a strengthening of the Leopoldt Conjecture. Second, when K is an imaginary quadratic field, we prove an unconditional result that implies the existence of ‘many’ positive-dimensional components (of certain deformation spaces) that do not contain infinitely many classical points. Also included are some speculative remarks about ‘p-adic functoriality’, as well as some remarks on how our methods should apply to n-dimensional representations of Gal($\math{\bar{\QQ}}$/ℚ) when n > 2.

Fermat's Last Theorem is a well-known classical theorem in mathematics. Andrew Willes has proven this theorem using the modular elliptic curve. However, the proposed proof is difficult for mathematicians and researchers to understand. For this reason, in this study, we provide evidence of several properties of Fermat's Last Theorem with a simple concept. We use Newton's Binomial Theorem, well-known in Fermat's time. In this study, we prove Fermat's Last Theorem for case . We also use the Newton’s Binomial theorem to verify several cases .

In this paper, we consider normalized newforms $f\in S_k(\Gamma_0(N),\varepsilon_f)$ whose non-constant term Fourier coefficients are congruent to those of an Eisenstein series modulo some prime ideal above a rational prime $p$. In this situation, we establish a congruence between the anticyclotomic $p$-adic $L$-function of Bertolini-Darmon-Prasanna and the Katz two-variable $p$-adic $L$-function. From this, we derive congruences between images under the $p$-adic Abel-Jacobi map of certain generalized Heegner cycles attached to $f$ and special values of the Katz $p$-adic $L$-function. In particular, our results apply to newforms associated with elliptic curves $E/\mathbb{Q}$ whose mod $p$ Galois representations $E[p]$ are reducible at a good prime $p$. As a consequence, we show the following: if $K$ is an imaginary quadratic field satisfying the Heegner hypothesis with respect to $E$ and in which $p$ splits, and if the bad primes of $E$ satisfy certain congruence conditions mod $p$ and $p$ does not divide certain Bernoulli numbers, then the Heegner point $P_{E}(K)$ is non-torsion, in particular implying that $\text{rank}_{\mathbb{Z}}E(K) = 1$. From this, we show that when $E$ is semistable with reducible mod $3$ Galois representation, then a positive proportion of real quadratic twists of $E$ have rank 1 and a positive proportion of imaginary quadratic twists of $E$ have rank 0.

A promising direction for constructing cryptographically stable pseudo-random sequence generators is an application of transformations in a group of points of elliptic and hypereliptic curves. This will allow building evidence-stable crypto algorithms, the problem of finding the private key in which is associated with solving a theoretically complex elliptic curve discrete logarithm problem. This paper proposes a method for generating pseudo-random sequences of the maximal period using transformations on elliptic curves. This method consists in the application of recurrent transformations with sequential formation of elements of points group of elliptic curves. This allows providing the maximum period of pseudo-random sequences with the reduction of the problem of finding the private key to the solution of the theoretically complex elliptic curve discrete logarithm problem. The block diagram of the device for generating pseudo-random sequences and the scheme for generating the internal states of the generator are given. We also present the results of statistical testing of some generators, which show that the generated sequences are indistinguishable in a statistical sense from truly random ones.KeywordsPseudo-random sequencesElliptic curvesMaximum period

When D is positive, let D > 1 denote the fundamental unit of O D .It has norm -1 when the wide and narrow class numbers agree, and norm 1 otherwise.The quantity log | D | is called the regulator of O D .