Modular Elliptic Curves Research Articles

A lower bound on the proportion of modular elliptic curves over Galois CM fields

We calculate an explicit lower bound on the proportion of elliptic curves that are modular over any Galois CM field not containing [Formula: see text]. Applied to imaginary quadratic fields, this proportion is at least [Formula: see text]. Applied to cyclotomic fields [Formula: see text] with [Formula: see text], this proportion is at least [Formula: see text] with only finitely many exceptions of [Formula: see text], for any choice of [Formula: see text].

The Recursive Structures of Manin Symbols over Q, Cusps and Elliptic Points on X0 (N)

Firstly, we present a more explicit formulation of the complete system D(N) of representatives of Manin’s symbols over Q, which was initially given by Shimura. Then, we establish a bijection between D(M)×D(N) and D(MN) for (M,N)=1, which reveals a recursive structure between Manin’s symbols of different levels. Based on Manin’s complete system Π(N) of representatives of cusps on X0(N) and Cremona’s characterization of the equivalence between cusps, we establish a bijection between a subset C(N) of D(N) and Π(N), and then establish a bijection between C(M)×C(N) and C(MN) for (M,N)=1. We also provide a recursive structure for elliptical points on X0(N). Based on these recursive structures, we obtain recursive algorithms for constructing Manin symbols over Q, cusps, and elliptical points on X0(N). This may give rise to more efficient algorithms for modular elliptic curves. As direct corollaries of these recursive structures, we present a recursive version of the genus formula and prove constructively formulas of the numbers of D(N), cusps, and elliptic points on X0(N).

On the 2-part of the Birch–Swinnerton-Dyer conjecture for elliptic curves with complex multiplication by the ring of integers of ℚ(−7)

For a modular elliptic curve [Formula: see text] and its quadratic twists [Formula: see text], we give equivalent conditions such that the [Formula: see text]-Selmer group [Formula: see text] is minimal, namely, it is of order [Formula: see text]. One of these conditions is described by the [Formula: see text]-value [Formula: see text]. The other conditions are described by quadratic and biquadratic residue symbol, so explicit and computable (and one can compute the density of [Formula: see text]). Also we prove the full Birch–Swinnerton-Dyer conjecture when the equivalent conditions are satisfied. This generalizes a result by J. Coates, Y. Li, Y. Tian and S. Zhai.

Plectic Stark–Heegner points

We propose a conjectural construction of determinants of global points on modular elliptic curves over arbitrary number fields, generalizing both the p-adic construction of Heegner points via Čerednik–Drinfeld uniformization and the definition of classical Stark–Heegner points. In alignment with Nekovář and Scholl's plectic conjectures, we expect the non-triviality of these plectic Stark–Heegner points to control the Mordell–Weil group of higher rank elliptic curves. We provide some indirect evidence for our conjectures by showing that higher order derivatives of anticyclotomic p-adic L-functions compute plectic invariants.

Proving The Fermat Last Theorem for Case q≤n

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 .

Plectic p-adic invariants

For modular elliptic curves over number fields of narrow class number one, and with multiplicative reduction at a collection of p-adic primes, we define new p-adic invariants. Inspired by Nekovář and Scholl's plectic conjectures, we believe these invariants control the Mordell–Weil group of higher rank elliptic curves and we support our expectations with numerical experiments.

A short note on inadmissible coefficients of weight 2 and 2k+1 newforms

Let f(z)=q+sum _{nge 2}a(n)q^n be a weight k normalized newform with integer coefficients and trivial residual mod 2 Galois representation. We extend the results of Amir and Hong in Amir and Hong (On L-functions of modular elliptic curves and certain K3 surfaces, Ramanujan J, 2021) for k=2 by ruling out or locating all odd prime values |ell |<100 of their Fourier coefficients a(n) when n satisfies some congruences. We also study the case of odd weights kge 1 newforms where the nebentypus is given by a quadratic Dirichlet character.

On L-functions of modular elliptic curves and certain K3 surfaces

Inspired by Lehmer’s conjecture on the non-vanishing of the Ramanujan tau -function, one may ask whether an odd integer alpha can be equal to tau (n) or any coefficient of a newform f(z). Balakrishnan, Craig, Ono and Tsai used the theory of Lucas sequences and Diophantine analysis to characterize non-admissible values of newforms of even weight kge 4. We use these methods for weight 2 and 3 newforms and apply our results to L-functions of modular elliptic curves and certain K3 surfaces with Picard number ge 19. In particular, for the complete list of weight 3 newforms f_lambda (z)=sum a_lambda (n)q^n that are eta -products, and for N_lambda the conductor of some elliptic curve E_lambda , we show that if |a_lambda (n)|<100 is odd with n>1 and (n,2N_lambda )=1, then aλ(n)∈{-5,9,±11,25,±41,±43,-45,±47,49,±53,55,±59,±61,±67,-69,±71,±73,75,±79,±81,±83,±89,±93±97,99}.\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} a_\\lambda (n) \\in&\\{-5,9,\\pm 11,25, \\pm 41, \\pm 43, -45,\\pm 47,49, \\pm 53,55, \\pm 59, \\pm 61,\\\\&\\pm 67, -69,\\pm 71,\\pm 73,75, \\pm 79,\\pm 81, \\pm 83, \\pm 89,\\pm 93 \\pm 97, 99\\}. \\end{aligned}$$\\end{document}Assuming the Generalized Riemann Hypothesis, we can rule out a few more possibilities leaving aλ(n)∈{-5,9,±11,25,-45,49,55,-69,75,±81,±93,99}.\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} a_\\lambda (n) \\in \\{-5,9,\\pm 11,25,-45,49,55,-69,75,\\pm 81,\\pm 93, 99\\}. \\end{aligned}$$\\end{document}

Quadratic twists of X0(14)

In the present paper, we prove the 2-part of Birch and Swinnerton-Dyer conjecture for an explicit infinite family of rank 0 quadratic twists of the modular elliptic curve X0(14), using an explicit form of the Waldspurger formula. We also give an explicit infinite family of rank 1 quadratic twists of X0(14) whose Tate–Shafarevich group is of odd cardinality.

From Pythagoras Theorem to Fermat’s Last Theorem and the Relationship between the Equation of Degree &amp;lt;i&amp;gt;n&amp;lt;/i&amp;gt; with One Unknown

The most interesting and famous problem that puzzled the mathematicians all around the world is much likely to be the Fermat’s Last Theorem. However, since the Theorem was proposed, people can’t find a way to solve the problem until Andrew Wiles proved the Fermat’s Last Theorem through a very difficult method called Modular elliptic curves in 1995. In this paper, I firstly constructed a geometric method to prove Fermat’s Last Theorem, and in this way we can easily get the conclusion below: If a and b are integer and a = b, n ∈ Q and n > 1, the value of c satisfies the function an + bn = cn that can never be integer; if a, b and c are integer and a ≠ b, n is integer and n > 2, the function an + bn = cn cannot be established.

Anticyclotomic 𝑝-adic 𝐿-functions and the exceptional zero phenomenon

Let A A be a modular elliptic curve over a totally real field F F , and let K / F K/F be a totally imaginary quadratic extension. In the event of exceptional zero phenomenon, we prove a formula for the derivative of the multivariable anticyclotomic p p -adic L L -function attached to ( A , K ) (A,K) , in terms of the Hasse-Weil L L -function and certain p p -adic periods attached to the respective automorphic forms. Our methods are based on a new construction of the anticyclotomic p p -adic L L -function by means of the corresponding automorphic representation.

Bounding cubic-triple product Selmer groups of elliptic curves

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.

Growth of the analytic rank of modular elliptic curves over quintic extensions

Given $F$ a totally real field and $E_{/F}$ a modular elliptic curve, we denote by $G_5(E_{/F};X)$ the number of quintic extensions $K$ of $F$ such that the norm of the relative discriminant is at most $X$ and the analytic rank of $E$ grows over $K$, i.e., $r_\mathrm{an}(E/K)>r_\mathrm{an}(E/F)$. We show that $G_5(E_{/F};X)\asymp_{+\infty} X$ when the elliptic curve $E_{/F}$ has odd conductor and at least one prime of multiplicative reduction. As Bhargava, Shankar and Wang \cite{BSW} showed that the number of quintic extensions of $F$ with norm of the relative discriminant at most $X$ is asymptotic to $c_{5,F} X$ for some positive constant $c_{5,F}$, our result exposes the growth of the analytic rank as a very common circumstance over quintic extensions.

Functoriality of Automorphic $\mathrm{L}$-Invariants and Applications

We study the behaviour of automorphic \mathrm{L} -invariants associated to cuspidal representations of \mathrm{GL}(2) of cohomological weight 0 under abelian base change and Jacquet-Langlands lifts to totally definite quaternion algebras. Under a standard non-vanishing hypothesis on automorphic \mathrm{L} -functions and some technical restrictions on the automorphic representation and the base field we get a simple proof of the equality of automorphic and arithmetic \mathrm{L} -invariants. This together with Spieß' results on p -adic \mathrm{L} -functions yields a new proof of the exceptional zero conjecture for modular elliptic curves – at least, up to sign.

On some new arithmetic functions involving prime divisors and perfect powers

Integer division and perfect powers play a central role in numerous mathematical results, especially in number theory. Classical examples involve perfect squares like in Pythagora's theorem, or higher perfect powers as the conjectures of Fermat (solved in 1994 by A. Wiles [Wiles, A.J., Modular elliptic curves and Fermat's Last Theorem, Annals of Mathematics, 141 (1995), 443–551.]) or Catalan (solved in 2002 by P. Mihăilescu [Mihăilescu, P., Primary cyclotomic units and a proof of Catalan's conjecture, J. Reine Angew. Math., 572 (2004), 167–195.]). The purpose of this paper is two-fold. First, we present some new integer sequences a(n), counting the positive integers smaller than n, having a maximal prime factor. We introduce an arithmetic function counting the number of perfect powers ij obtained for 1≤i,j≤n. Along with some properties of this function, we present the sequence A303748, which was recently added to the Online Encyclopedia of Integer Sequences (OEIS) [The On-Line Encyclopedia of Integer Sequences, http://oeis.org, OEIS Foundation Inc. 2011.]. Finally, we discuss some other novel integer sequences.

Iwasawa theory of quadratic twists of X 0(49)

The field \(K = \mathbb{Q}\left( {\sqrt { - 7} } \right)\) is the only imaginary quadratic field with class number 1, in which the prime 2 splits, and we fix one of the primes p of K lying above 2. The modular elliptic curve X0(49) has complex multiplication by the maximal order O of K, and we let E be the twist of X0(49) by the quadratic extension \(KK(\sqrt M )/K\), where M is any square free element of O with M ≡ 1 mod 4 and (M,7) = 1. In the present note, we use surprisingly simple algebraic arguments to prove a sharp estimate for the rank of the Mordell-Weil group modulo torsion of E over the field F∞ = K(Ep∞), where Ep∞ denotes the group of p∞-division points on E. Moreover, writing B for the twist of X0(49) by \(K(\sqrt[4]{{ - 7}})/K\), our Iwasawa-theoretic arguments also show that the weak form of the conjecture of Birch and Swinnerton-Dyer implies the non-vanishing at s = 1 of the complex L-series of B over every finite layer of the unique Z2-extension of K unramified outside p. We hope to give a proof of this last non-vanishing assertion in a subsequent paper.

Automorphicity and mean-periodicity

If $C$ is a smooth projective curve over a number field $k$, then, under fair hypotheses, its $L$-function admits meromorphic continuation and satisfies the anticipated functional equation if and only if a related function is $\mathfrak{X}$-mean-periodic for some appropriate functional space $\mathfrak{X}$. Building on the work of Masatoshi Suzuki for modular elliptic curves, we will explore the dual relationship of this result to the widely believed conjecture that such $L$-functions should be automorphic. More precisely, we will directly show the orthogonality of the matrix coefficients of $GL_{2g}$-automorphic representations to the vector spaces $\mathcal{T}(h(\mathcal{S},\{k_i\},s))$, which are constructed from the Mellin transforms $f(\mathcal{S},\{k_i\},s)$ of certain products of arithmetic zeta functions $\zeta(\mathcal{S},2s)\prod_{i}\zeta(k_i,s)$, where $\mathcal{S}\rightarrow {\rm Spec}(\mathcal{O}_k)$ is any proper regular model of $C$ and $\{k_i\}$ is a finite set of finite extensions of $k$. To compare automorphicity and mean-periodicity, we use a technique emulating the Rankin–Selberg method, in which the function $h(\mathcal{S},\{k_i\},s))$ plays the role of an Eisenstein series, exploiting the spectral interpretation of the zeros of automorphic $L$-functions.

Integrals on $p$-adic upper half planes and Hida families over totally real fields

Bertolini--Darmon and Mok proved a formula of the second derivative of the two-variable $p$-adic $L$-function of a modular elliptic curve over a totally real field along the Hida family in terms of the image of a global point by some $p$-adic logarithm map. The theory of $p$-adic indefinite integrals and $p$-adic multiplicative integrals on $p$-adic upper half planes plays an important role in their work. In this paper, we generalize these integrals for $p$-adic measures which are not necessarily $\mathbb{Z}$-valued, and prove a formula of the second derivative of the two-variable $p$-adic $L$-function of an abelian variety of $\mathrm{GL}(2)$-type associated to a Hilbert modular form of weight 2.

Modular elliptic curves over real abelian fields and the generalized Fermat equation x2ℓ+ y2m= zp

Using a combination of several powerful modularity theorems and class field theory we derive a new modularity theorem for semistable elliptic curves over certain real abelian fields. We deduce that if $K$ is a real abelian field of conductor $n<100$, with $5 \nmid n$ and $n \ne 29$, $87$, $89$, then every semistable elliptic curve $E$ over $K$ is modular. Let $\ell$, $m$, $p$ be prime, with $\ell$, $m \ge 5$ and $p \ge 3$.To a putative non-trivial primitive solution of the generalized Fermat $x^{2\ell}+y^{2m}=z^p$ we associate a Frey elliptic curve defined over $\mathbb{Q}(\zeta_p)^+$, and study its mod $\ell$ representation with the help of level lowering and our modularity result. We deduce the non-existence of non-trivial primitive solutions if $p \le 11$, or if $p=13$ and $\ell$, $m \ne 7$.

Modular Forms and Weierstrass Mock Modular Forms

Alfes, Griffin, Ono, and Rolen have shown that the harmonic Maass forms arising from Weierstrass ζ-functions associated to modular elliptic curves “encode” the vanishing and nonvanishing for central values and derivatives of twisted Hasse-Weil L-functions for elliptic curves. Previously, Martin and Ono proved that there are exactly five weight 2 newforms with complex multiplication that are eta-quotients. In this paper, we construct a canonical harmonic Maass form for these five curves with complex multiplication. The holomorphic part of this harmonic Maass form arises from the Weierstrass ζ-function and is referred to as the Weierstrass mock modular form. We prove that the Weierstrass mock modular form for these five curves is itself an eta-quotient or a twist of one. Using this construction, we also obtain p-adic formulas for the corresponding weight 2 newform using Atkin’s U-operator.