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A recent paper of Shekhar compares the ranks of elliptic curves E1 and E2 for which there is an isomorphism E1[p] ∼ E2[p]asGal(.../Q)- modules, where p is a prime of good ordinary reduction for both curves. In this paper we prove an analogous result in the case where p is a prime of good supersingular reduction. [ABSTRACT FROM AUTHOR]

Let K be a number field with ring of integers 𝓞K, and let {fk}k∈N be a sequence of monic polynomials in 𝓞K[x] such that for every n ∈ ℕ, the composition f(n) = f1 ◦ f2 ◦ . . . ◦ fn is irreducible. In this paper we show that if the size of the Galois group of f(n) is large enough (in a precise sense) as a function of n, then the set of primes p ⊆ 𝓞K such that every f(n) is irreducible modulo p has density zero. Moreover, we prove that the subset of polynomial sequences such that the Galois group of f(n) is large enough has density 1, in an appropriate sense, within the set of all polynomial sequences. [ABSTRACT FROM AUTHOR]

Watkins' conjecture asserts that for a rational elliptic curve E the degree of the modular parametrization is divisible by 2r, where r is the rank of E. In this paper, we prove that if the modular degree is odd, then E has rank zero. Moreover, we prove that the conjecture holds for all rank two rational elliptic curves of prime conductor and positive discriminant. [ABSTRACT FROM AUTHOR]

We use the Wilf-Zeilberger method to prove identities between Mahler measures of polynomials. In particular, we offer a new proof of a formula due to Lal'ın, and we show how to translate the identity into a formula involving elliptic dilogarithms. This work settles a challenge problem proposed by Kontsevich and Zagier in their paper, "Periods". [ABSTRACT FROM AUTHOR]

In a 1998 paper, Kaneko and Zagier explain unpublished work of Atkin which exhibits an infinite sequence of polynomials with the property that when suitable polynomials are reduced mod p for a prime p, one gets the locus of supersingular elliptic curves. Here we generalize this phenomenon by considering the continued fraction expansions of modular and quasimodular forms. [ABSTRACT FROM AUTHOR]

In this paper we give a sufficient condition on the class numbers of distinct quadratic imaginary fields so that on a given CM elliptic curve over Q with fixed modular parameterization, the Heegner points associated to (the maximal orders of) these quadratic imaginary fields are linearly independent. This extends results of Rosen and Silverman from the non-CM to the CM case. [ABSTRACT FROM AUTHOR]

In this paper we prove a relation between the period of an elliptic curve and the period of its real and imaginary quadratic twists. This relation is often misstated in the literature. [ABSTRACT FROM AUTHOR]

In this paper we will prove some cases of the Fontaine-Mazur conjecture. Let $p$ be an odd prime and let $G_{mathbb {Q},{p}}$ be the Galois group over $mathbb {Q}$ of the maximal unramified-outside-$p$ extension of $mathbb {Q}$. We show that under certain hypotheses, the universal deformation of the action of $G_{mathbb {Q},{p}}$ on the $2$-torsion of an elliptic curve defined over $mathbb {Q}$ has finite image. We compute the associated universal deformation ring, and we show in the process that $hat {S}_4$ caps $mathbb {Q}$ for the prime $2$, where $hat {S}_4$ is the double cover of $S_4$ whose Sylow $2$-subgroups are generalized quaternion groups. [ABSTRACT FROM AUTHOR]