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Register a new userModular elliptic directions with complex multiplication (with an application to Grosss elliptic curves)
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For every normalized newform f in S_2(Gamma_1(N)) with complex multiplication, we study the modular parametrizations of elliptic curves C from the abelian variety A_f. We apply the results obtained when C is Grosss elliptic curve A(p).
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We provide a framework for using elliptic curves with complex multiplication to determine the primality or compositeness of integers that lie in special sequences, in deterministic quasi-quadratic time. We use this to find large primes, including the largest prime currently known whose primality cannot feasibly be proved using classical methods.
Let G(A) be an AF-algebra given by periodic Bratteli diagram with the incidence matrix A in GL(n, Z). For a given polynomial p(x) in Z[x] we assign to G(A) a finite abelian group Z^n/p(A) Z^n. It is shown that if p(0)=1 or p(0)=-1 and Z[x]/(p(x)) is a principal ideal domain, then Z^n/p(A) Z^n is an invariant of the strong stable isomorphism class of G(A). For n=2 and p(x)=x-1 we conjecture a formula linking values of the invariant and torsion subgroup of elliptic curves with complex multiplication.
In this paper we construct parameterizations of elliptic curves over the rationals which have many consecutive integral multiples. Using these parameterizations, we perform searches in GMP and Magma to find curves with points of small height, curves with many integral multiples of a point, curves with high multiples of a point integral, and over two hundred curves with more than one hundred integral points. In addition, a novel and complete classification of self-descriptive numbers is constructed by bounding the number of zeros such a number must contain.
An elliptic curve $E$ over $mathbb{Q}$ is said to be good if $N_{E}^{6}<max!left{ leftvert c_{4}^{3}rightvert ,c_{6}^{2}right} $ where $N_{E}$ is the conductor of $E$ and $c_{4}$ and $c_{6}$ are the invariants associated to a global minimal model of $E$. In this article, we generalize Massers Theorem on the existence of infinitely many good elliptic curves with full $2$-torsion. Specifically, we prove via constructive methods that for each of the fifteen torsion subgroups $T$ allowed by Mazurs Torsion Theorem, there are infinitely many good elliptic curves $E$ with $E!left(mathbb{Q}right) _{text{tors}}cong T$.
Let $C$ be a smooth, absolutely irreducible genus-$3$ curve over a number field $M$. Suppose that the Jacobian of $C$ has complex multiplication by a sextic CM-field $K$. Suppose further that $K$ contains no imaginary quadratic subfield. We give a bound on the primes $mathfrak{p}$ of $M$ such that the stable reduction of $C$ at $mathfrak{p}$ contains three irreducible components of genus $1$.
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