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Register a new userIntegrality properties in the Moduli Space of Elliptic Curves: CM Case
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Authors
Stefan Schmid
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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.
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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.
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.
We investigate the arithmetic of algebraic curves on coarse moduli spaces for special linear rank two local systems on surfaces with fixed boundary traces. We prove a structure theorem for morphisms from the affine line into the moduli space. We show that the set of integral points on any nondegenerate algebraic curve on the moduli space can be effectively determined.
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