Let [Formula: see text] be an integer such that has genus text], and let a field of characteristic or relatively prime to text]. In this paper, we explicitly classify the isogeny graphs all rational elliptic curves admit nontrivial over We achieve by introducing parameterized families defined which have following two properties for fixed text]: are isogenous there integers text]-invariants give...

Let $E$ be an elliptic curve over $Bbb{Q}$ with the given Weierstrass equation $ y^2=x^3+ax+b$. If $D$ is a squarefree integer, then let $E^{(D)}$ denote the $D$-quadratic twist of $E$ that is given by $E^{(D)}: y^2=x^3+aD^2x+bD^3$. Let $E^{(D)}(Bbb{Q})$ be the group of $Bbb{Q}$-rational points of $E^{(D)}$. It is conjectured by J. Silverman that there are infinitely many primes $p$ for which $...

Let q be a prime and −D < −4 be an odd fundamental discriminant such that q splits in Q( √ −D). For f a weight zero Hecke-Maass newform of level q and Θχ the weight one theta series of level D corresponding to an ideal class group character χ of Q( √ −D), we establish a hybrid subconvexity bound for L(f×Θχ, s) at s = 1/2 when q D for 0 < η < 1. With this circle of ideas, we show that the Heegne...

We explore the possibilities for Galois representation [Formula: see text] attached to a weight-one newform be residually reducible, i.e. Hecke eigenvalues congruent those of Eisenstein series. A special role is played by series level text], where quadratic character associated with an imaginary field discriminant respect which dihedral type. prove congruences, modulus divides either class numb...

We study the elliptic algebras Qn,k(E,τ) introduced by Feigin and Odesskii as a generalization of Sklyanin algebras. They form family quadratic parametrized coprime integers n>k≥1, an curve E, point τ∈E. consider compare several different definitions provide proofs various statements about them made Odesskii. For example, we show that Qn,k(E,0), Qn,n−1(E,τ) are polynomial rings on n variables. ...