Fix coprime s, t ≥ 1. We re-prove, without Ehrhart reciprocity, a conjecture of Armstrong (recently verified by Johnson) that the finitely many simultaneous (s, t)-cores have average size 1 24 (s−1)(t−1)(s+t+1), and that the subset of self-conjugate cores has the same average (first shown by Chen-Huang-Wang). We similarly prove a recent conjecture of Fayers that the average weighted by an inverse stabilizer-giving the "expected size of the t-core of a random s-core"-is 1 24 (s − 1)(t 2 − 1). We also prove Fayers' conjecture that the analogous self-conjugate average is the same if t is odd, but instead 1 24 (s − 1)(t 2 + 2) if t is even. In principle, our explicit methods-or implicit variants thereof-extend to averages of arbitrary powers.The main new observation is that the stabilizers appearing in Fayers' conjectures have simple formulas in Johnson's z-coordinates parameterization of (s, t)-cores.We also observe that the z-coordinates extend to parameterize general t-cores. As an example application with t := s + d, we count the number of (s, s + d, s + 2d)-cores for coprime s, d ≥ 1, verifying a recent conjecture of Amdeberhan and Leven.Date: January 12, 2016. v4: added reference [10] and updated [11]; shorter version available at Electronic J. Combin. 23(1) (2016), #P1.4. v3: minor improvements and clarifications throughout, including implicit variant details; added some references and remarks on finite beta-sets, maximal cores, and poset method. v2: extended z-coordinates of Johnson to parameterize all t-cores, not just (s, t)-cores; applied this to a conjecture of Amdeberhan-Leven (v3 mentions later independent proof by Johnson); added some references. Version for the arXiv, with 34 pages, 3 figures. Shorter version with 28 pages (currently available at https://www.overleaf.com/read/ xmzxdgbdcpnq), with fewer details, to be submitted for publication. Comments welcome on either. 1 arXiv:1507.04290v4 [math.CO]

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Fix a smooth cubic form F/Q in 6 variables. For N F (X) := #{x ∈ [−X, X] 6 : F (x) = 0}, the "randomness" prediction "N F (X) = (c HL + o(1)) • X 3 as X → ∞" of Hardy-Littlewood may fail. Nonetheless, Hooley suggested a modified prediction accounting for "special structured loci" on the projective variety V := V (F ) ⊆ P 5 Q . A weighted version of N F (X) essentially decomposes as a sum of adelic data over hyperplane sections V c ⊆ V , generically with nonzero discriminant D(c). Assuming standard conjectures for the Hasse-Weil L-functions L(s, V c ) over {c ∈ Z 6 : D(c) = 0}, Hooley proved the bound N F (X) = O (X 3+ ), essentially for any given diagonal F . Now assume (1) standard conjectures for each L(s, V c ), for certain tensor L-functions thereof, and for L(s, V ); (2) standard predictions (of Random Matrix Theory type) for the mean values of 1/L(s) and 1/L(s 1 )L(s 2 ) over certain geometric families; (3) a quantitative form of the Square-free Sieve Conjecture for D; and (4) an effective bound on the local variation (in c) of the local factors L p (s, V c ), in the spirit of Krasner's lemma.Under (1)-( 4), we establish (away from the Hessian of F ) a weighted, localized version of Hooley's prediction for diagonal F -and hence the Hasse principle for V /Q.Still under (1)-( 4), we conclude that asymptotically 100% of integers a / ∈ {4, 5} mod 9 lie in {x 3 + y 3 + z 3 : x, y, z ∈ Z}-and a positive fraction lie in {x 3 + y 3 + z 3 : x, y, z ∈ Z ≥0 }. ContentsConventions and notes 1. Rough goals and background 2. An asymptotic variance analysis for sums of three cubes 3. Main introduction 4. An overview of key technical aspects 5. A framework for restriction and separation 6. Bounding the local near-zero loci of any given polynomial 7. "Derivations" of "RMT-type predictions" and "intermediary conjectures" 8. Clarifying the dominant ranges in our "generic HLH error" 9. An absolute bound via RMT-type second moments 10. Identifying predictable cancellation via RMT-type first moments Appendix A. Complementary commentary on conjectural aspects of the paper Appendix B. New oscillatory integral bounds sensitive to real geometry Appendix C. Bounding the locus of unbounded bad sums to a prime modulus Appendix D. New boundedness criteria for bad sums to prime-power moduli Appendix E. Possibilities for enlarging or deforming the delta method Appendix F. Speculation on other problems

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Inspired by the Odlyzko root discriminant and Golod-Shafarevich p-group bounds, Martinet (1978) asked whether an imaginary quadratic number field K/Q must always have an infinite Hilbert 2-class field tower when the class group of K has 2-rank 4, or equivalently when the discriminant of K has 5 prime factors. No negative results are known. Benjamin (