In the previous chapter, we showed that Brandt matrices for an order in a definite quaternion
algebra B contain a wealth of arithmetic. In the special case where\, disc\, B= p disc B= p is …

In this thesis we present the theory of supersingular elliptic curves and provide extensive
examples of their properties. We develop necessary theory of orders in number fields and …

Let p be a prime, and let E/Fp2 be a supersingular elliptic curve. Then it is straightforward to
show that EndFp (E) is isomorphic to a maximal order within Qp,∞, the unique quaternion …

arXiv:math/0404538v1 [math.NT] 29 Apr 2004 Page 1 arXiv:math/0404538v1 [math.NT] 29 Apr
2004 ON THE CORRESPONDECE BETWEEN SUPERSINGULAR ELLIPTIC CURVES AND …

The graph method simultaneously uses the theory of quaternion algebras, elliptic curves
and modular forms in order to determine all supersingular points in a given characteristic …

Let O be a maximal order in the quaternion algebra Bp over Q ramified at p and∞. The
paper is about the computational problem: construct a supersingular elliptic curve E over Fp …

Let K/Q be a quadratic extension and let 0 be the ring of integers in K. Any elliptic curve
having complex multiplication by 0 and supersingular reduction mod I has as its full …

We prove the existence of infinitely many supersingular primes for certain abelian surfaces
with quaternionic multiplication extending the result of N. Elkies for elliptic curves over …

A recent paper of Shekhar compares the ranks of elliptic curves $ E_1 $ and $ E_2 $ for
which there is an isomorphism $ E_1 [p]\simeq E_2 [p] $ as $\mathrm {Gal}(\bar {\mathbf …

Let $ p $ be a prime, and let $\mathcal {M} $ denote the space of weight two modular forms
on $\Gamma _ {0}(p) $ all of whose Fourier coefficients are integral, except possibly for the …