Six unlikely intersection problems in search of effectivity

We give a new algorithm for constructing Picard curves over a finite field with a given endomorphism ring. This has important applications in cryptography since curves of genus 3 allow for smaller key sizes than elliptic curves. For a sextic CM-field $K$ containing the cube roots of unity, we define and compute certain class polynomials modulo small primes and then use the Chinese Remainder Theorem to construct the class polynomials over the rationals. We also give some examples.

We prove several results on torsion points and Galois representations for complex multiplication (CM) elliptic curves over a number field containing the CM field. One result computes the degree in which such an elliptic curve has a rational point of order $N$, refining results of Silverberg. Another result bounds the size of the torsion subgroup of an elliptic curve with CM by a nonmaximal order in terms of the torsion subgroup of an elliptic curve with CM by the maximal order. Our techniques also yield a complete classification of both the possible torsion subgroups and the rational cyclic isogenies of a $K$-CM elliptic curve $E$ defined over $K(j(E))$.

We give a new characterisation of elliptic curves of Shimura type in terms commuting families of Frobenius lifts and also strengthen an old principal ideal theorem for ray class fields. These two results combined yield the existence of global minimal models for elliptic curves of Shimura type, generalising a result of Gross. Along the way we also prove a handful of small but new results regarding elliptic curves with complex multiplication.

Rational points on elliptic curves with complex multiplication by the ring of integers in [formula omitted](√−7)

In the p n -th cyclotomic field ℚ p n , p a prime number, n ∈ ℕ , the prime p is totally ramified and the only ideal above p is generated by ω n = ζ p n - 1 , with the primitive p n -th root of unity ζ p n = e 2 π i p n . Moreover these numbers represent a norm coherent set, i.e. N ℚ p n + 1 / ℚ p n ( ω n + 1 ) = ω n . It is the aim of this article to establish a similar result for the ray class field K 𝔭 n of conductor 𝔭 n over an imaginary quadratic number field K where 𝔭 n is the power of a prime ideal in K . Therefore the exponential function has to be replaced by a suitable elliptic function.

Jacobi forms with CM and applications

Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k"> <mml:semantics> <mml:mi>k</mml:mi> <mml:annotation encoding="application/x-tex">k</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a field, <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A"> <mml:semantics> <mml:mi>A</mml:mi> <mml:annotation encoding="application/x-tex">A</mml:annotation> </mml:semantics> </mml:math> </inline-formula> a finite-dimensional Frobenius <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k"> <mml:semantics> <mml:mi>k</mml:mi> <mml:annotation encoding="application/x-tex">k</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -algebra and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="rho colon upper A right-arrow upper A"> <mml:semantics> <mml:mrow> <mml:mi> ρ </mml:mi> <mml:mo> : </mml:mo> <mml:mi>A</mml:mi> <mml:mo stretchy="false"> → </mml:mo> <mml:mi>A</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\rho \colon A\to A</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , the Nakayama automorphism of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A"> <mml:semantics> <mml:mi>A</mml:mi> <mml:annotation encoding="application/x-tex">A</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with respect to a Frobenius homomorphism <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="phi colon upper A right-arrow k"> <mml:semantics> <mml:mrow> <mml:mi> φ </mml:mi> <mml:mo> : </mml:mo> <mml:mi>A</mml:mi> <mml:mo stretchy="false"> → </mml:mo> <mml:mi>k</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\varphi \colon A\to k</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . Assume that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="rho"> <mml:semantics> <mml:mi> ρ </mml:mi> <mml:annotation encoding="application/x-tex">\rho</mml:annotation> </mml:semantics> </mml:math> </inline-formula> has finite order <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="m"> <mml:semantics> <mml:mi>m</mml:mi> <mml:annotation encoding="application/x-tex">m</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k"> <mml:semantics> <mml:mi>k</mml:mi> <mml:annotation encoding="application/x-tex">k</mml:annotation> </mml:semantics> </mml:math> </inline-formula> has a primitive <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="m"> <mml:semantics> <mml:mi>m</mml:mi> <mml:annotation encoding="application/x-tex">m</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -th root of unity <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="w"> <mml:semantics> <mml:mi>w</mml:mi> <mml:annotation encoding="application/x-tex">w</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . Consider the decomposition <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A equals upper A 0 circled-plus midline-horizontal-ellipsis circled-plus upper A Subscript m minus 1"> <mml:semantics> <mml:mrow> <mml:mi>A</mml:mi> <mml:mo>=</mml:mo> <mml:msub> <mml:mi>A</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo> ⊕ </mml:mo> <mml:mo> ⋯ </mml:mo> <mml:mo> ⊕ </mml:mo> <mml:msub> <mml:mi>A</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>m</mml:mi> <mml:mo> − </mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">A = A_0\oplus \cdots \oplus A_{m-1}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A"> <mml:semantics> <mml:mi>A</mml:mi> <mml:annotation encoding="application/x-tex">A</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , obtained by defining <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A Subscript i Baseline equals StartSet a element-of upper A colon rho left-parenthesis a right-parenthesis equals w Superscript i Baseline a EndSet"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>A</mml:mi> <mml:mi>i</mml:mi> </mml:msub> <mml:mo>=</mml:mo> <mml:mo fence="false" stretchy="false">{</mml:mo> <mml:mi>a</mml:mi> <mml:mo> ∈ </mml:mo> <mml:mi>A</mml:mi> <mml:mo>:</mml:mo> <mml:mi>

Let $E$ be an elliptic curve defined over $\Q$ with complex multiplication by the ring of integers of an imaginary quadratic field $K$. We apply an elliptic unit version of Ichimura–Sumida criterion for the $\Z_p$-extension $F_\infty/F$ associated to $E$ and try to determine the characteristic polynomial of the maximal unramified abelian $p$-extension of $F_\infty$.

Let H be the Hilbert class field of an imaginary quadratic field K. An elliptic curve E over H with complex multiplication by K is called a Q-curve if E is isogenous over H to all its Galois conjugates. We classify Q-curves over H, relating them with the cohomology group H2(H/Q,±1). The structures of the abelian varieties over Q obtained from Q-curves by restriction of scalars are investigated.

This paper is a contribution to the verification of conjectures of Birch and Swinnerton-Dyer about elliptic curves (1). The evidence that they produce is largely derived from curves with complex multiplication byi. It is natural to consider other kinds of complex multiplications and here we shall make a start on the case when the ring of complex multiplications is isomorphic to the ring Z[σ], where σ2= − 2.

Some remarks on the two-variable main conjecture of Iwasawa theory for elliptic curves without complex multiplication

Following Ramanujan's work on modular equations and approximations of π, there are formulas for 1/π of the form Following Ramanujan's work on modular equations and approximations of π, there are formulas for 1/π of the form ∑ k = 0 ∞ ( 1 2 ) k ( 1 d ) k ( d - 1 d ) k k ! 3 ( a k + 1 ) ( λ d ) k = δ π for d=2,3,4,6, where łd are singular values that correspond to elliptic curves with complex multiplication, and a,δ are explicit algebraic numbers. In this paper we prove a p-adic version of this formula in terms of the so-called Ramanujan type congruence. In addition, we obtain a new supercongruence result for elliptic curves with complex multiplication.

The computation of the order of Frobenius action on the $\\ell$-torsion is a\npart of Schoof-Elkies-Atkin algorithm for point counting on an elliptic curve\n$E$ over a finite field $\\mathbb{F}_q$. The idea of Schoof's algorithm is to\ncompute the trace of Frobenius $t$ modulo primes $\\ell$ and restore it by the\nChinese remainder theorem. Atkin's improvement consists of computing the order\n$r$ of the Frobenius action on $E[\\ell]$ and of restricting the number $t\n\\pmod{\\ell}$ to enumerate by using the formula $t^2 \\equiv q (\\zeta +\n\\zeta^{-1})^2 \\pmod{\\ell}$. Here $\\zeta$ is a primitive $r$-th root of unity.\nIn this paper, we generalize Atkin's formula to the general case of abelian\nvariety of dimension $g$. Classically, finding of the order $r$ involves\nexpensive computation of modular polynomials. We study the distribution of the\nFrobenius orders in case of abelian surfaces and $q \\equiv 1 \\pmod{\\ell}$ in\norder to replace these expensive computations by probabilistic algorithms.\n

Let $k$ be a field of characteristic zero containing a primitive $n$-th root of unity. Let $C^0_n$ be a singular plane curve of degree $n$ over $k$ admitting an order $n$ automorphism, $n$ nodes as the singularities, and $C_n$ be its normalization. In this paper we study the factors of Prym variety $\mbox{Prym}(\widetilde{C}_n/C_n)$ associated to the double cover $\widetilde{C}_n$ of $C_n$ exactly ramified at the points obtained by the blow-up of the singularities. We provide explicit models of some algebraic curves related to the construction of $\mbox{Prym}(\widetilde{C}_n/C_n)$ as a Prym variety and determine the interesting simple factors other than elliptic curves or hyperelliptic curves with small genus which come up in $J_n$ so that the endomorphism rings contains the totally real field $\mathbb{Q}(\zeta_n+\zeta^{-1}_n)$.

Let F be the cyclotomic field of fifth roots of unity. We computationally investigate modularity of elliptic curves over F.