We prove Mazur and Rubin's $\Lambda$-adic Gross–Zagier conjecture (under some restrictive hypotheses), which relates Heegner points in towers of number fields to the 2-variable p-adic L-function. The result generalizes Perrin-Riou's p-adic Gross–Zagier theorem.

We show that for the primes l = 2, 3, 5, 7 or 13, there do not exist any non-zero abelian varieties over $\mathbb{Q}$ that have good reduction at every prime different from l and are semi-stable at l. We show that any semi-stable abelian variety over $\mathbb{Q}$ with good reduction outside l = 11 is isogenous to a power of the Jacobian variety of the modular curve X0(11). In addition, we show that for l = 2, 3 and 5, there do not exist any non-zero abelian varieties over $\mathbb{Q}$ with good reduction outside l that acquire semi-stable reduction at l over a tamely ramified extension.

Recall that a Borel measure $\mu$ on $\mathbb{R}$ is said to be extremal if $\mu$-almost every number in $\mathbb{R}$ is not very well approximable. In this paper, we investigate extremality (implied by the exponentially fast decay (efd) property) of conformal measures induced by regular infinite conformal iterated function systems. We then give particular attention to the class of such systems generated by the continued fractions algorithm with restricted entries. It is proved that if the index set of entries has bounded gaps, then the corresponding conformal measure satisfies the efd property and is extremal. Also a class of examples of index sets with unbounded gaps is provided for which the corresponding conformal measure also satisfies the efd property and is extremal.

We define here an analogue, for a semi-stable group scheme whose generic fiber is an abelian variety, of M. J. Taylor's class-invariant homomorphism (defined for abelian schemes), and we give a geometric description of it. Then we extend a result of Taylor, Srivastav, Agboola and Pappas concerning the kernel of this homomorphism in the case of an elliptic curve.

Let Mc be the moduli space of semistable torsion-free sheaves of rank 2 with Chern classes c1 = 0 and c2 = c over a K3 surface with generic polarization. When $c=2n\ge 4$ is even, Mc is a singular projective variety which admits a symplectic form, called the Mukai form, on the smooth part. A natural question raised by O'Grady asks if there exists a desingularization on which the Mukai form extends everywhere nondegenerately. In this paper we show that such a desingularization does not exist for many even integers c by computing the stringy Euler numbers.

We prove an interpolation lemma with multiplicities on a commutative algebraic group. This statement is ‘dual’ to zero estimates used in transcendental number theory. The special case where no multiplicities are involved has been proved by Masser.

For any finite group G we define the moduli space of pointed admissible G-covers and the concept of a G-equivariant cohomological field theory (G-CohFT), which, when G is the trivial group, reduces to the moduli space of stable curves and a cohomological field theory (CohFT), respectively. We prove that taking the ‘quotient’ by G reduces a G-CohFT to a CohFT. We also prove that a G-CohFT contains a G-Frobenius algebra, a G-equivariant generalization of a Frobenius algebra, and that the ‘quotient’ by G agrees with the obvious Frobenius algebra structure on the space of G-invariants, after rescaling the metric. We then introduce the moduli space of G-stable maps into a smooth, projective variety X with G action. Gromov–Witten-like invariants of these spaces provide the primary source of examples of G-CohFTs. Finally, we explain how these constructions generalize (and unify) the Chen–Ruan orbifold Gromov–Witten invariants of $[X/G]$ as well as the ring $H^{\bullet}(X,G)$ of Fantechi and Göttsche.

Let F be a non-Archimedean local field. Let $\mathcal{G}_n^{\rm et}(F)$ be the set of equivalence classes of irreducible, n-dimensional representations of the Weil group $\mathcal{W}_F$ of F which are essentially tame. Let $\mathcal{A}_n^{\rm et}(F)$ be the set of equivalence classes of irreducible, essentially tame, supercuspidal representations of GLn(F). The Langlands correspondence induces a canonical bijection $\mathcal{L}:\mathcal{G}_n^{\rm et}(F) \to \mathcal{A}_n^{\rm et}(F)$. We continue the programme of describing this map in terms of explicit descriptions of the sets $\mathcal{G}_n^{\rm et}(F)$ and $\mathcal{A}_n^{\rm et}(F)$. These descriptions are in terms of admissible pairs $(E/F, \xi)$, consisting of a tamely ramified field extension $E/F$ of degree n and a quasicharacter $\xi$ of $E^\times$ subject to certain technical conditions. If Pn(F) is the set of isomorphism classes of admissible pairs of degree n, we have explicit bijections $P_n(F) \cong \mathcal{G}_n^{\rm et}(F)$ and $P_n(F) \cong \mathcal{A}_n^{\rm et}(F)$. In an earlier paper we showed that, if $\sigma \in \mathcal{G}_n^{\rm et}(F)$ corresponds to an admissible pair $(E/F,\xi)$, then $\mathcal{L}(\sigma)$ corresponds to the admissible pair $(E/F,\mu\xi)$, for a certain tamely ramified character $\mu$ of $E^\times$. In this paper, we determine the character $\mu$ when $E/F$ is totally ramified.

We show that the norm of a Bethe vector in the $sl_{r+1}$ Gaudin model is equal to the Hessian of the corresponding master function at the corresponding critical point. In particular the Bethe vectors corresponding to non-degenerate critical points are non-zero vectors. This result is a byproduct of functorial properties of Bethe vectors studied in this paper. As another byproduct of functoriality we show that the Bethe vectors form a basis in the tensor product of several copies of first and last fundamental $sl_{r+1}$-modules.

We classify quadruples $(M, g, m, \tau)$ in which (M, g) is a compact Kähler manifold of complex dimension m > 2 and $\tau$ is a nonconstant function on M such that the conformally related metric $g/\tau^{2}$, defined wherever $\tau \ne 0$, is an Einstein metric. It turns out that M then is the total space of a holomorphic $\mathbb{C}{\rm P}^1$ bundle over a compact Kähler–Einstein manifold (N, h). The quadruples in question constitute four disjoint families: one, well known, with Kähler metrics g that are locally reducible; a second, discovered by Bérard Bergery (1982), and having $\tau \ne 0$ everywhere; a third one, related to the second by a form of analytic continuation, and analogous to some known Kähler surface metrics; and a fourth family, present only in odd complex dimensions $m \ge 9$. Our classification uses a moduli curve, which is a subset $\mathcal{C}$, depending on m, of an algebraic curve in $\mathbb{R}^2$. A point (u, v) in $\mathcal{C}$ is naturally associated with any $(M, g, m, \tau)$ having all of the above properties except for compactness of M, replaced by a weaker requirement of ‘vertical’ compactness. One may in turn reconstruct M, g and $\tau$ from (u, v) coupled with some other data, among them a Kähler–Einstein base (N, h) for the $\mathbb{C}{\rm P}^1$ bundle M. The points (u, v) arising in this way from $(M, g, m, \tau)$ with compactM form a countably infinite subset of \mathcal{C}$.