Motivated by recent work of Florian Pop, we study the connections between three notions of equivalence of function fields: isomorphism, elementary equivalence, and the condition that each of a pair of fields can be embedded in the other, which we call isogeny. Some of our results are purely geometric: we give an isogeny classification of Severi-Brauer varieties and quadric surfaces. These resul...

Let p be a prime and K a number field of degree p. We count the number of elliptic curves, up to K-isomorphism, having a prescribed property, where this property is either that the curve contains a fixed torsion group as a subgroup, or that it has an isogeny of prescribed degree. We also study the following question: for a given n such that |Y0(n)(Q)| > 0, does every elliptic curve over K with ...

We present signature schemes whose security relies on computational assumptions relating to isogeny graphs of supersingular elliptic curves. We give two schemes, both of them based on interactive identification protocols. The first identification protocol is due to De Feo, Jao and Plût. The second one, and the main contribution of the paper, makes novel use of an algorithm of Kohel, Lauter, Pet...

Strikingly, this is not true when E has a single rational point of order two. In this case E has a degree two isogeny φ : E → E ′ and an associated Selmer group Selφ(E/Q). Work of Xiong shows that if E does not have a cyclic 4-isogeny de ned over Q(E[2]), then the distribution of the ranks of Selφ(E /Q) as d varies among the squarefree integers less than X tends to the distribution Max{0,N (0, ...

This paper gives an explicit formula for the size of the isogeny class of a Hilbert-Blumenthal abelian variety over a finite field. More precisely, let OL be the ring of integers in a totally real field dimension g over Q, let N0 and N be relatively prime square-free integers, and let k be a finite field of characteristic relatively prime to both N0N and disc(L,Q). Finally, let (X/k, ι, α) be a...

We construct six infinite series of families of pairs of curves (X, Y ) of arbitrarily high genus, defined over number fields, together with an explicit isogeny JX → JY splitting multiplication by 2, 3, or 4. The families are derived from Cassou–Noguès and Couveignes’ explicit classification of pairs (f, g) of polynomials such that f(x1)− g(x2) is reducible.

The standard references for this section are [Gro91], [Kol90] and [McC91]. Let E be an elliptic curve over Q of conductor N . Let K = Q( √ −D), where −D is a fundamental discriminant, D #= 3, 4, and all prime factors of N are split in K, i.e. (N) = NN̄ for an ideal N of the ring of integers OK of K with OK/N $ Z/NZ. We call such a discriminant a Heegner discriminant for E/Q. By the modularity th...

This paper focuses on isogeny representations, defined as ways to evaluate isogenies and verify membership the language of isogenous supersingular curves (the set triples $$D,E_1,E_2$$ with a cyclic degree D between $$E_1$$ $$E_2$$ ). The tasks evaluating verifying are fundamental for isogeny-based cryptography. Our main contribution is design suborder representation, new representation targett...