{"id":1826,"date":"2011-09-11T00:25:56","date_gmt":"2011-09-11T00:25:56","guid":{"rendered":"http:\/\/math.columbia.edu\/~dejong\/wordpress\/?p=1826"},"modified":"2011-09-11T00:25:56","modified_gmt":"2011-09-11T00:25:56","slug":"shioda-cycles-ii","status":"publish","type":"post","link":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/?p=1826","title":{"rendered":"Shioda cycles, II"},"content":{"rendered":"

This post won’t make sense if you haven’t read Shioda cycles, I<\/a>.<\/p>\n

Let X be a hypersurface of even degree d in P<\/strong>^3_{F_p} such that the determinant of the geometric frobenius acting on H^2 has a positive sign. Assuming the Tate conjecture (which we will do throughout this post), we can find our Shioda cycle by listing all the low degree curves in P<\/strong>^3_{F_p} and for each of them checking whether the curve lies on X and if so whether it gives a Shioda cycle. Now although this is a common recipe for finding a Shioda cycle if it should exist, it isn’t the kind of pattern I am looking for. (Moreover, you’d be hard pressed to argue that this recipe is uniform over all primes p because after all the lists will change with p.)<\/p>\n

Now, I have a suggestion for a recipe that could work (which only means I can’t prove it doesn’t work). I am not saying or conjecturing that it does work, although I do have some very special cases where I can show that it works (basically families of surfaces related to families of abelian surfaces). A while ago I wrote a preprint about this (you can find it on my web page), but I think I can explain it here in a few sentences.<\/p>\n

Namely, suppose that F = F(X_0, …, X_3) ∈ Z<\/strong>[X_0, X_1, X_2, X_3; A_I] is the universal polynomial of degree d, i.e., the coefficients A_I of F are variables where I = (i_0, i_1, i_2, i_3) with i_0 + i_1 + i_2 + i_3 = d. For every collection of values a = (a_I), a_I ∈ F_p we obtain a hypersurface X(a) in P<\/strong>^3_{F_p} by setting A_I equal to a_I in F. Now, suppose that we have a polynomial<\/p>\n

G(X_0, …, X_3, Y_0, …, Y_3) ∈ Z<\/strong>[X_i; Y_j; A_I]<\/p><\/blockquote>\n

For each a = (a_I) as above we can consider the intersection of X(a) with<\/p>\n

G(X_0, …, X_3, X_0^p, …, X_3^p)|_{A_I = a_I} = 0<\/p><\/blockquote>\n

i.e., we replace Y_j by X_j^p and A_I by a_I. Let’s call this intersection Z(a). Then my suggestion is to look for a Shioda cycle among the irreducible components<\/em> of Z(a). In other words, given the even integer d, is there a polynomial G as above, such that, if X(a) is a surface which should have a Shioda cycle, then one of the irreducible components of Z_a is a Shioda cycle?<\/p>\n

Actually in my write-up I (a) only require this to work in most of the cases where we expect a Shioda cycle, and (b) allow G also to depend on more variables which get replaced by X_j^{p^n}.<\/p>\n

You might think it would be more natural to consider a system of polynomials such as G and ask them, after being mangled as above, to actually cut out a Shioda cycle. It seemed to me at the time of writing the preprint that this might be too strong a requirement, but I actually do not know how to disprove even this statement.<\/p>\n

There are variant constructions we could use, e.g., we could allow variables Z_{ij} that get replaced by<\/p>\n

(1\/p)[(X_i + X_j)^p – X_i^p – X_j^p] mod p<\/p><\/blockquote>\n

if you know what I mean. The _meta_ question I have is whether anything like this can be true? Can you think of a (heuristic) argument showing this cannot work?<\/p>\n

For example, if you could show that the (minimal) degrees of Shioda cycles tends to infinity rapidly with p then we would get a contradiction. However one can prove, assuming the Tate conjecture is true, an upper bound of the degree of Shioda cycles occuring in the family (unfortunately I don’t remember the shape of the formula I got when I worked it out) which shows this kind of argument won’t contradict my suggestion.<\/p>\n","protected":false},"excerpt":{"rendered":"

This post won’t make sense if you haven’t read Shioda cycles, I. Let X be a hypersurface of even degree d in P^3_{F_p} such that the determinant of the geometric frobenius acting on H^2 has a positive sign. Assuming the … Continue reading →<\/span><\/a><\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[],"class_list":["post-1826","post","type-post","status-publish","format-standard","hentry","category-uncategorized"],"_links":{"self":[{"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/1826","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=1826"}],"version-history":[{"count":23,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/1826\/revisions"}],"predecessor-version":[{"id":1849,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/1826\/revisions\/1849"}],"wp:attachment":[{"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=1826"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=1826"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=1826"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}