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Register a new userA framework for deterministic primality proving using elliptic curves with complex multiplication
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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.
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We give a deterministic algorithm that very quickly proves the primality or compositeness of the integers N in a certain sequence, using an elliptic curve E/Q with complex multiplication by the ring of integers of Q(sqrt(-7)). The algorithm uses O(log N) arithmetic operations in the ring Z/NZ, implying a bit complexity that is quasi-quadratic in log N. Notably, neither of the classical N-1 or N+1 primality tests apply to the integers in our sequence. We discuss how this algorithm may be applied, in combination with sieving techniques, to efficiently search for very large primes. This has allowed us to prove the primality of several integers with more than 100,000 decimal digits, the largest of which has more than a million bits in its binary representation. At the time it was found, it was the largest proven prime N for which no significant partial factorization of N-1 or N+1 is known.
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).
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.
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 $1$.
Let C/Q be the genus 3 Picard curve given by the affine model y^3=x^4-x. In this paper we compute its Sato-Tate group, show the generalized Sato-Tate conjecture for C, and compute the statistical moments for the limiting distribution of the normalized local factors of C.
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