We show that for primes
Let
 Award ID(s):
 2152780
 NSFPAR ID:
 10513780
 Publisher / Repository:
 American Mathematical Society
 Date Published:
 Journal Name:
 Mathematics of Computation
 Volume:
 92
 Issue:
 340
 ISSN:
 00255718
 Page Range / eLocation ID:
 805 to 837
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
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$N, p \geq 5$ with$N \equiv 1 \bmod p$ , the class number of$\mathbb {Q}(N^{1/p})$ is divisible by$p$ . Our methods are via congruences between Eisenstein series and cusp forms. In particular, we show that when$N \equiv 1 \bmod p$ , there is always a cusp form of weight$2$ and level$\Gamma _0(N^2)$ whose$\ell$ th Fourier coefficient is congruent to$\ell + 1$ modulo a prime above$p$ , for all primes$\ell$ . We use the Galois representation of such a cusp form to explicitly construct an unramified degree$p$ extension of$\mathbb {Q}(N^{1/p})$ . 
We show that for any even logconcave probability measure
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For each odd integer
$n \geq 3$ , we construct a rank3 graph$\Lambda _n$ with involution$\gamma _n$ whose real$C^*$ algebra$C^*_{\scriptscriptstyle \mathbb {R}}(\Lambda _n, \gamma _n)$ is stably isomorphic to the exotic Cuntz algebra$\mathcal E_n$ . This construction is optimal, as we prove that a rank2 graph with involution$(\Lambda ,\gamma )$ can never satisfy$C^*_{\scriptscriptstyle \mathbb {R}}(\Lambda , \gamma )\sim _{ME} \mathcal E_n$ , and Boersema reached the same conclusion for rank1 graphs (directed graphs) in [Münster J. Math.10 (2017), pp. 485–521, Corollary 4.3]. Our construction relies on a rank1 graph with involution$(\Lambda , \gamma )$ whose real$C^*$ algebra$C^*_{\scriptscriptstyle \mathbb {R}}(\Lambda , \gamma )$ is stably isomorphic to the suspension$S \mathbb {R}$ . In the Appendix, we show that the$i$ fold suspension$S^i \mathbb {R}$ is stably isomorphic to a graph algebra iff$2 \leq i \leq 1$ . 
Proving the “expectationthreshold” conjecture of Kahn and Kalai [Combin. Probab. Comput. 16 (2007), pp. 495–502], we show that for any increasing property
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Let
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