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Let K be an imaginary quadratic field, and fix an odd prime p that does not divide the class number of K , and is excluded from a finite set of primes depending on K. In this paper we prove that Iwasawa's λ -invariant for the cyclotomic Z p -extension of K is greater than 1 if and only if the number of points on a certain reduced elliptic curve is divisible by p 2. [ABSTRACT FROM AUTHOR]

We define Jacobi forms with complex multiplication. Analogous to modular forms with complex multiplication, they are constructed from Hecke characters of the associated imaginary quadratic field. From this construction we obtain a Jacobi form which specializes to η (τ) 26 which we present to highlight an open question of Dyson and Serre. We give other examples and applications of Jacobi forms with complex multiplication including constructing theta blocks associated to elliptic curves with complex multiplication and new families of congruences and cranks for certain partition functions. [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]

Let K be an imaginary quadratic field of discriminant d K , and let n be a nontrivial integral ideal of K in which N is the smallest positive integer. Let Q N (d K) be the set of primitive positive definite binary quadratic forms of discriminant d K whose leading coefficients are relatively prime to N. We adopt an equivalence relation ∼ n on Q N (d K) so that the set of equivalence classes Q N (d K) / ∼ n can be regarded as a group isomorphic to the ray class group of K modulo n. We further establish an explicit isomorphism of Q N (d K) / ∼ n onto Gal (K n / K) in terms of Fricke invariants, where K n denotes the ray class field of K modulo n. This would be a certain extension of the classical composition theory of binary quadratic forms, originated and developed by Gauss and Dirichlet. [ABSTRACT FROM AUTHOR]

Abstract Let L be an extended ring class field of an imaginary quadratic field K other than Q (− 1) and Q (− 3). We show that there is a form class group induced from a congruence subgroup which describes the Galois group of L over K in a concrete way. We also construct a primitive generator of L over K as a real algebraic integer which can be applied to certain quadratic Diophantine equations. [ABSTRACT FROM AUTHOR]

There are several formulas for the number of orbits of the projective line under the action of subgroups of $$\mathrm{GL}_2$$ . We give an interpretation of two such formulas in terms of the geometry of elliptic curves, and prove a more general formula for a large class of congruence subgroups of Bianchi groups. Our formula involves the number of walks on a certain graph called an isogeny volcano. Underlying our results is a complete description of the group of extensions of a pair of CM elliptic curves, as well as the group of extensions of a pair of lattices in a quadratic field. [ABSTRACT FROM AUTHOR]

Let K be an imaginary quadratic field and O K be its ring of integers. Let h E be the Weber function on a certain elliptic curve E with complex multiplication by O K . We show that if N (>1) is an integer prime to 6, then the function h E alone generates the ray class field modulo N O K over K when evaluated at some N -torsion point of E , which would be a partial answer to the question mentioned in [10, p. 105] . [ABSTRACT FROM AUTHOR]

We give some explicit conditions under which the singular values of β-quotients generate ring class fields over imaginary quadratic fields. [ABSTRACT FROM AUTHOR]

We discuss class polynomials of multiple eta-quotients for an order of an imaginary quadratic field defined by using N -systems. We give an upper bound of the number of distinct class polynomials of multiple eta-quotient constructed from N -systems. Moreover we give a sufficient condition such that class polynomials are uniquely determined independent of a choice of N -system. [ABSTRACT FROM AUTHOR]

We give an algorithm to compute the generalized division polynomials for elliptic curves with complex multiplication. These polynomials can be used to generate the ray class fields of imaginary quadratic fields over the Hilbert class field with no restriction on the conductor. [ABSTRACT FROM AUTHOR]

For a family of G-fields K, we show an effective prime ideal theorem with probability 1. Namely, outside a density zero set, π(x, K)=Li(x)+O(x/(log x)2) for x≥(log dK)c for some c depending on G. Using the effective prime ideal theorem, we show that the exponents eK of the ideal class groups of a family of G-fields increase along with dK with probability 1. [ABSTRACT FROM PUBLISHER]

Abstract: Text: We relate non-critical special values of p-adic L-functions associated to algebraic Hecke characters of an imaginary quadratic number field with class number one to p-adic Eisenstein–Kronecker series constructed as the Coleman function, when the conductors of the algebraic Hecke characters are divisible by p. Video: For a video summary of this paper, please click here or visit http://youtu.be/AZemqgfp5pQ. [Copyright &y& Elsevier]

In this paper we construct elliptic curves over quadratic fields with prescribed torsion group and record rank. We do this using theoretical arguments and keeping lengthy computations to a minimum. [ABSTRACT FROM AUTHOR]

Abstract: Let g be a principal modulus with rational Fourier coefficients for a discrete subgroup of lying in between and for a positive integer N. Let K be an imaginary quadratic field. We introduce a relatively simple proof, without using Shimuraʼs canonical model, of the fact that the singular value of g generates the ray class field modulo N or the ring class field of the order of conductor N over K. Further, we construct a primitive generator of the ray class field of arbitrary modulus over K from Hasseʼs two generators. [Copyright &y& Elsevier]

Abstract: We establish several results towards the two-variable main conjecture of Iwasawa theory for elliptic curves without complex multiplication over imaginary quadratic fields, namely (i) the existence of an appropriate p-adic L-function, building on works of Hida and Perrin-Riou, (ii) the basic structure theory of the dual Selmer group, following works of Coates, Hachimori–Venjakob, et al., and (iii) the implications of dihedral or anticyclotomic main conjectures with basechange. The result of (i) is deduced from the construction of Hida and Perrin-Riou, which in particular is seen to give a bounded distribution. The result of (ii) allows us to deduce a corank formula for the p-primary part of the Tate–Shafarevich group of an elliptic curve in the -extension of an imaginary quadratic field. Finally, (iii) allows us to deduce a criterion for one divisibility of the two-variable main conjecture in terms of specializations to cyclotomic characters, following a suggestion of Greenberg, as well as a refinement via basechange. [Copyright &y& Elsevier]

Andrews, Dyson, and Hickerson showed that 2 $ q$ [ABSTRACT FROM AUTHOR]

Abstract: Let K be a quadratic imaginary number field with discriminant and class number one. Fix a prime which is unramified in K. Given an elliptic curve with complex multiplication by K, let be the representation which arises from the action of Galois on the Tate module. Herein it is shown that, for all but finitely many inert primes p, the image of a certain deformation of is “as large as possible”, that is, it is the full inverse image of a Cartan subgroup of . If p splits in K, then the same result holds as long as a certain Bernoulli–Hurwitz number is a p-adic unit which, in turn, is equivalent to a prime ideal not being a Wieferich place. The proof rests on the theory of elliptic units of Robert and Kubert–Lang, and on the two-variable main conjecture of Iwasawa theory for quadratic imaginary fields. [Copyright &y& Elsevier]

We prove that the relative class number of a nonabelian normal CM-field of degree 2pq (where p and q are two distinct odd primes) is always greater than four. Not only does this result solve the class number one problem for the nonabelian normal CM-fields of degree 42, but it has also been used elsewhere to solve the class number one problem for the nonabelian normal CM-fields of degree 84. [ABSTRACT FROM AUTHOR]

Let K be a number field and let E/K be an elliptic curve. If E has complex multiplication, we show that there is a positive lower bound for the canonical height of nontorsion points on E defined over the maximal abelian extension Kab of K. This is analogous to results of Amoroso, Dvornicich (2000) and Amoroso, Zannier (2000) for the multiplicative group. We also show that if E has nonintegral j-invariant (so that, in particular, E does not have complex multiplication), then there exists C > 0 such that there are only finitely many points P ∈ E(Kab) of canonical height less than C. This strengthens a result of Hindry and Silverman (1990). [ABSTRACT FROM AUTHOR]