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Prime power fold cyclic branched covers along smoothly slice knots all bound rational homology balls. This phenomenon, however, does not characterize slice knots. In this paper, we give a new construction of non-slice knots that have the above property. The sliceness obstruction comes from computing twisted Alexander polynomials, and we introduce new techniques to simplify their calculation., Comment: 19 pages, 5 figures, Version 2: Minor changes to abstract and introduction. Added references on knots that are not concordant to their reverses

This paper defines two new extrinsic curvature quantities on the corner of a four-dimensional Riemannian manifold with corner. One of these is a pointwise conformal invariant, and the conformal transformation of the other is governed by a new linear second-order pointwise conformally invariant partial differential operator. The Gauss-Bonnet theorem is then stated in terms of these quantities., 14 pages

In this paper we study existence and lack thereof of closed embedded orientable co-dimension one totally geodesic submanifolds of minimal volume cusped orientable hyperbolic manifolds., 16 pages, 2 figure

Let $E/\mathbf{Q}$ be an elliptic curve of conductor $N$, let $p>3$ be a prime where $E$ has good ordinary reduction, and let $K$ be an imaginary quadratic field satisfying the Heegner hypothesis. In 1987, Perrin-Riou formulated an Iwasawa main conjecture for the Tate-Shafarevich group of $E$ over the anticyclotomic $\mathbf{Z}_p$-extension of $K$ in terms of Heegner points. In this paper, we give a proof of Perrin-Riou's conjecture under mild hypotheses. Our proof builds on Howard's theory of bipartite Euler systems and Wei Zhang's work on Kolyvagin's conjecture. In the case when $p$ splits in $K$, we also obtain a proof of the Iwasawa-Greenberg main conjecture for the $p$-adic $L$-functions of Bertolini-Darmon-Prasanna., 22 pages. Revised version, to appear in Algebra & Number Theory