what are the examples of elliptic curves defined over $\mathbb{Q}$ with supersingular reduction at a prime $p$ and having a $p$-isogeny over $\mathbb{Q}$ ?
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$\begingroup$ Have you tried taking $p$ to be small so that $X_0(p)$ has genus 0, and looking for rational points on $X_0(p)$ that lie in the supersingular locus? $\endgroup$– David LoefflerCommented Aug 28, 2013 at 14:47
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$\begingroup$ Self p-isogeny or p-isogeny to possibly some other curve? I guess self won't work because it will be CM and p will have to split in the CM field to get the p-isogeny, but then the reduction at p will be ordinary. If it's isogenous to another curve, then David's suggestion is a good one. $\endgroup$– Felipe VolochCommented Aug 28, 2013 at 15:10
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2 Answers
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In fact, this cannot happen: an elliptic curve over $\mathbb{Q}_p$ is supersingular if and only if its associated mod $p$ Galois representation is irreducible, but if it is irreducible as a representation of $\mathbb{F}_p[G_{\mathbb{Q}_p}]$ then it is certainly irreducible as a representation of $\mathbb{F}_p[G_{\mathbb{Q}}]$ and thus it has no $p$-isogenies.
This argument doesn't work over larger fields: if $K$ is an extension of $\mathbb{Q}_p$ it's no longer true that $E/K$ is supersingular if and only if $E[p]$ is irreducible as a $\mathbb{F}_p[G_{K}]$-module, cf. this MSE question. In particular, one can have an elliptic curve $E / \mathbb{Q}$ with a $p$-isogeny and bad reduction at $p$, and a number field $F / \mathbb{Q}$ such that $E$ has good supersingular reduction at all primes of $F$ above $p$; my previous suggestion to look for points on $X_0(p)$ gives lots of examples of this.
answered Aug 28, 2013 at 15:31
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$\begingroup$ for the same problem can we replace the p isogeny over Q with p isogeny over Q($ \mu_{p} $) and have examples ? ( where $ \mu_{p} $ is a primitive p-1 th root of unity.) $\endgroup$– SumanCommented Aug 28, 2013 at 18:25
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1$\begingroup$ @Suman, a rough computation tells me that the ramification necessary for this is not $p-1$ but $p+1$. This is getting a bit technical, so if you’re strongly interested in this, just e-mail me directly (find my e-mail on my home page), and we can discuss. $\endgroup$– LubinCommented Aug 29, 2013 at 1:00
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I felt that there couldn’t be such an isogeny, but when I saw @DavidLoeffler’s answer, I realized that I had an argument, too. The formal group of the elliptic curve would be of height two, defined over $\mathbb Z_p$, but such things don’t have $p$-isogenies: the quotient formal group is definable only over a suitably ramified extension.
