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For a modular elliptic curve A = X 0 (4 9) and its quadratic twists A (M) , we give equivalent conditions such that the 2 -Selmer group S 2 (A (M) / ℚ) is minimal, namely, it is of order 2. One of these conditions is described by the L -value L (A (M) / ℚ , 1). The other conditions are described by quadratic and biquadratic residue symbol, so explicit and computable (and one can compute the density of M). 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. [ABSTRACT FROM AUTHOR]

We analyze complex multiplication for Jacobians of curves of genus 3, as well as the resulting Shimura class groups and their subgroups corresponding to Galois conjugation over the reflex field. We combine our results with numerical methods to find CM fields K for which there exist both hyperelliptic and non-hyperelliptic curves whose Jacobian has complex multiplication by \mathbb {Z}_K. More precisely, we find all sextic CM fields K in the LMFDB for which (heuristically) Jacobians of both types with CM by \mathbb {Z}_K exist. There turn out to be 14 such fields among the 547,156 sextic CM fields that the LMFDB contains. We determine invariants of the corresponding curves, and in the simplest case we also give an explicit defining equation. [ABSTRACT FROM AUTHOR]

A complex algebraic curve is said to have many automorphisms if it cannot be deformed nontrivially together with its automorphism group. Oort asked whether the jacobian of any such curve has complex multiplication. We answer this question completely when the curve is hyperelliptic. For this, we first classify all hyperelliptic curves with many automorphisms, finding 3 infinite sequences and 15 separate isomorphism classes. Of these, all members of the infinite sequences and 5 of the isolated curves have a jacobian with complex multiplication, but the remaining 10 curves do not. The methods used include a criterion of Wolfart, a representation theoretic criterion of streit and the non-integrality of the j-invariants of certain quotient curves of genus 1. For the hardest cases, we use an ad hoc developed computational criterion depending on the splitting fields of the characteristic polynomials of Frobenius acting on the Tate module of the Jacobian. [ABSTRACT FROM AUTHOR]

Let C be a smooth, projective, genus g ≥ 2 curve, defined over C. Then C has many automorphisms if its corresponding moduli point p ∈ Mg has a neighborhood U in the complex topology, such that all curves corresponding to points in U \ p have strictly fewer automorphisms than C. We compute completely the list of superelliptic curves C for which the superelliptic automorphism is normal in the automorphism group Aut (C) and C has many automorphisms. For each of these curves, we determine whether its Jacobian has complex multiplication. As a consequence, we prove the converse of Streit's complex multiplication criterion for these curves. [ABSTRACT FROM AUTHOR]

Let P and Q be two points on an elliptic curve defined over a number field K. For α ∈ End (E) , define B α to be the O K -integral ideal generated by the denominator of x (α (P) + Q) . Let O be a subring of End (E) , that is a Dedekind domain. We will study the sequence { B α } α ∈ O . We will show that, for all but finitely many α ∈ O , the ideal B α has a primitive divisor when P is a non-torsion point and there exist two endomorphisms g ≠ 0 and f so that f (P) = g (Q) . This is a generalization of previous results on elliptic divisibility sequences. [ABSTRACT FROM AUTHOR]

Let E / F be an elliptic curve defined over a number field F. Suppose that E has complex multiplication over F ¯ , i.e. End F ¯ (E) ⊗ ℚ is an imaginary quadratic field. With the aid of CM theory, we find elliptic curves whose quadratic twists have a constant root number. [ABSTRACT FROM AUTHOR]

An isogeny graph is a graph whose vertices are principally polarizable abelian varieties and whose edges are isogenies between these varieties. In his thesis, Kohel describes the structure of isogeny graphs for elliptic curves and shows that one may compute the endomorphism ring of an elliptic curve defined over a finite field by using a depth-first search (DFS) algorithm in the graph. In dimension 2, the structure of isogeny graphs is less understood and existing algorithms for computing endomorphism rings are very expensive. In this article, we show that, under certain conditions, the problem of determining the endomorphism ring can also be solved in genus 2 with a DFS-based algorithm. We consider the case of genus-2 Jacobians with complex multiplication, with the assumptions that the real multiplication subring has class number one and is locally maximal at ℓ , for ℓ a fixed prime. We describe the isogeny graphs in that case, by considering cyclic isogenies of degree ℓ , under the assumption that there is an ideal l of norm ℓ in K 0 which is generated by a totally positive algebraic integer. The resulting algorithm is implemented over finite fields, and examples are provided. To the best of our knowledge, this is the first DFS-based algorithm in genus 2. [ABSTRACT FROM AUTHOR]

In 1987, Gordon gave an integer primality condition similar to the familiar test based on Fermat’s little theorem, but based instead on the arithmetic of elliptic curves with complex multiplication. We prove the existence of infinitely many composite numbers simultaneously passing all elliptic curve primality tests assuming a weak form of a standard conjecture on the bound on the least prime in (special) arithmetic progressions. Our results are somewhat more general than both the 1999 dissertation of the first author (written under the direction of the third author) and a 2010 paper on Carmichael numbers in a residue class written by Banks and the second author. [ABSTRACT FROM PUBLISHER]

Fix m greater than one and let E be an elliptic curve over Q with complex multiplication. We formulate conjectures on the density of primes p (congruent to one modulo m) for which the pth Fourier coefficient of E is an mth power modulo p; often these densities differ from the naive expectation of 1/m. We also prove our conjectures for m dividing the number of roots of unity lying in the CM field of E; the most involved case is m = 4 and complex multiplication by Q(i). [ABSTRACT FROM AUTHOR]