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Register a new userIntegrality properties in the Moduli Space of Elliptic Curves: Isogeny Case
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Authors
Stefan Schmid
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For a fixed $j$-invariant $j_0$ of an elliptic curve without complex multiplication we bound the number of $j$-invariants $j$ that are algebraic units and such that elliptic curves corresponding to $j$ and $j_0$ are isogenous. Our bounds are effective. We also modify the problem slightly by fixing a singular modulus $alpha$ and looking at all $j$ such that $j-alpha$ is an algebraic unit and such that elliptic curves corresponding to $j$ and $j_0$ are isogenous. The number of such $j$ can again be bounded effectively.
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A result of Habegger shows that there are only finitely many singular moduli such that $j$ or $j-alpha$ is an algebraic unit. The result uses Dukes Equidistribution Theorem and is thus not effective. For a fixed $j$-invariant $alpha in bar{mathbb{Q}}$ of an elliptic curve without complex multiplication, we prove that there are only finitely many singular moduli $j$ such that $j-alpha$ is an algebraic unit. The difference to the work of Habegger is that we give explicit bounds.
Computing endomorphism rings of supersingular elliptic curves is an important problem in computational number theory, and it is also closely connected to the security of some of the recently proposed isogeny-based cryptosystems. In this paper we give a new algorithm for computing the endomorphism ring of a supersingular elliptic curve $E$ that runs, under certain heuristics, in time $O((log p)^2p^{1/2})$. The algorithm works by first finding two cycles of a certain form in the supersingular $ell$-isogeny graph $G(p,ell)$, generating an order $Lambda subseteq operatorname{End}(E)$. Then all maximal orders containing $Lambda$ are computed, extending work of Voight. The final step is to determine which of these maximal orders is the endomorphism ring. As part of the cycle finding algorithm, we give a lower bound on the set of all $j$-invariants $j$ that are adjacent to $j^p$ in $G(p,ell)$, answering a question in arXiv:1909.07779.
Let E/Q be an elliptic curve and p a rational prime of good ordinary reduction. For every imaginary quadratic field K/Q satisfying the Heegner hypothesis for E we have a corresponding line in E(K)otimes Q_p, known as a shadow line. When E/Q has analytic rank 2 and E/K has analytic rank 3, shadow lines are expected to lie in E(Q)otimes Q_p. If, in addition, p splits in K/Q, then shadow lines can be determined using the anticyclotomic p-adic height pairing. We develop an algorithm to compute anticyclotomic p-adic heights which we then use to provide an algorithm to compute shadow lines. We conclude by illustrating these algorithms in a collection of examples.
This is an introduction to a probabilistic model for the arithmetic of elliptic curves, a model developed in a series of articles of the author with Bhargava, Kane, Lenstra, Park, Rains, Voight, and Wood. We discuss the theoretical evidence for the model, and we make predictions about elliptic curves based on corresponding theorems proved about the model. In particular, the model suggests that all but finitely many elliptic curves over $mathbb{Q}$ have rank $le 21$, which would imply that the rank is uniformly bounded.
In this paper, $p$ and $q$ are two different odd primes. First, We construct the congruent elliptic curves corresponding to $p$, $2p$, $pq$, and $2pq,$ then, in the cases of congruent numbers, we determine the rank of the corresponding congruent elliptic curves.
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