We describe a vanishing result on the cohomology of a cochain complex associated to the moduli of chains of finite subgroup schemes on elliptic curves. These results have applications to algebraic topology, in particular to the study of power operations for Morava E-theory at height 2.

We prove that if one can predict the least significant bit of the Diffie–Hellman secret keys for elliptic curves with non-negligible advantage on a polynomial fraction of all curves over a given finite field Fp, then one can compute the entire Diffie–Hellman secret on a polynomial fraction of all curves over the same finite field. Our method combines rapid mixing properties of certain isogeny g...

In this paper families of elliptic curves admitting a rational isogeny of degree 3 are studied. It is known that the 3-torsion in the class group of the field defined by the points in the kernel of such an isogeny is related to the rank of the elliptic curve. Families in which almost all the curves have rank at least 3 are constructed. In some cases this provides lower bounds for the number of ...

We investigate special structures due to automorphisms in isogeny graphs of principally polarized abelian varieties, and surfaces particular. give theoretical experimental results on the spectral statistical properties <mml:mr...

We provide examples of abelian surfaces over number fields K whose reductions at almost all good primes possess an isogeny prime degree $$\ell $$ ℓ rational the residue field, but which themselves do not admit a K-rational -isogeny. This builds on work Cullinan and Sutherland. When $$K=\mathbb {Q}$$ = Q , we identify certain weight-2 newforms f with quadratic Fourier coefficients associated mod...