Inventiones mathematicae最新文献

We combine transcendental methods and the modular approaches to the (ABC) conjecture to show that the largest prime factor of (n^{2}+1) is at least of size ((log _{2} n)^{2}/log _{3}n) where (log _{k}) is the (k)-th iterate of the logarithm. This gives a substantial improvement on the best available estimates, which are essentially of size (log _{2} n) going back to work of Chowla in 1934. Using the same ideas, we also obtain significant progress on subexpoential bounds for the (ABC) conjecture, which in a case gives the first improvement on a result by Stewart and Yu dating back over two decades. Central to our approach is the connection between Shimura curves and the (ABC) conjecture developed by the author.

In enumerative geometry, Virasoro constraints were first conjectured in Gromov-Witten theory with many new recent developments in the sheaf theoretic context. In this paper, we rephrase the sheaf theoretic Virasoro constraints in terms of primary states coming from a natural conformal vector in Joyce’s vertex algebra. This shows that Virasoro constraints are preserved under wall-crossing. As an application, we prove the conjectural Virasoro constraints for moduli spaces of torsion-free sheaves on any curve and on surfaces with only ((p,p)) cohomology classes by reducing the statements to the rank 1 case.

Abstract

Fix a prime number (p) and let (E/F) be a CM extension of number fields in which (p) splits relatively. Let (pi ) be an automorphic representation of a quasi-split unitary group of even rank with respect to (E/F) such that (pi ) is ordinary above (p) with respect to the Siegel parabolic subgroup. We construct the cyclotomic (p) -adic (L) -function of (pi ) , and a certain generating series of Selmer classes of special cycles on Shimura varieties. We show, under some conditions, that if the vanishing order of the (p) -adic (L) -function is 1, then our generating series is modular and yields explicit nonzero classes (called Selmer theta lifts) in the Selmer group of the Galois representation of (E) associated with (pi ) ; in particular, the rank of this Selmer group is at least 1. In fact, we prove a precise formula relating the (p) -adic heights of Selmer theta lifts to the derivative of the (p) -adic (L) -function. In parallel to Perrin-Riou’s (p) -adic analogue of the Gross–Zagier formula, our formula is the (p) -adic analogue of the arithmetic inner product formula recently established by Chao Li and the second author.

We prove the existence of nonconstant harmonic maps of optimal regularity from an arbitrary closed manifold ((M^{n},g)) of dimension (n>2) to any closed, non-aspherical manifold (N) containing no stable minimal two-spheres. In particular, this gives the first general existence result for harmonic maps from higher-dimensional manifolds to a large class of positively curved targets. In the special case of the round spheres (N=mathbb{S}^{k}), (kgeqslant 3), we obtain a distinguished family of nonconstant harmonic maps (Mto mathbb{S}^{k}) of index at most (k+1), with singular set of codimension at least 7 for (k) sufficiently large. Furthermore, if (3leqslant nleqslant 5), we show that these smooth harmonic maps stabilize as (k) becomes large, and correspond to the solutions of an eigenvalue optimization problem on (M), generalizing the conformal maximization of the first Laplace eigenvalue on surfaces.

Let (X) be a smooth (p)-adic formal scheme. We show that integral crystalline local systems on the generic fiber of (X) are equivalent to prismatic (F)-crystals over the analytic locus of the prismatic site of (X). As an application, we give a prismatic proof of Fontaine’s (mathrm {C}_{{mathrm {crys}}})-conjecture, for general coefficients, in the relative setting, and allowing ramified base fields. Along the way, we also establish various foundational results for the cohomology of prismatic (F)-crystals, including various comparison theorems, Poincaré duality, and Frobenius isogeny.

We construct a canonical family of even periodic (mathbb{E}_{infty})-ring spectra, with exactly one member of the family for every prime (p) and chromatic height (n). At height 1 our construction is due to Snaith, who built complex (K)-theory from (mathbb{CP}^{infty}). At height 2 we replace (mathbb{CP}^{infty}) with a (p)-local retract of (mathrm{BU} langle 6 rangle ), producing a new theory that orients elliptic, but not generic, height 2 Morava (E)-theories.

In general our construction exhibits a kind of redshift, whereby (mathrm{BP}langle n-1 rangle ) is used to produce a height (n) theory. A familiar sequence of Bocksteins, studied by Tamanoi, Ravenel, Wilson, and Yagita, relates the (K(n))-localization of our height (n) ring to work of Peterson and Westerland building (E_{n}^{hSmathbb{G}^{pm}}) from (mathrm{K}(mathbb{Z},n+1)).

Let (X) be a smooth, complex Fano 4-fold, and (rho _{X}) its Picard number. We show that if (rho _{X}>12), then (X) is a product of del Pezzo surfaces. The proof relies on a careful study of divisorial elementary contractions (fcolon Xto Y) such that (dim f(operatorname{Exc}(f))=2), together with the author’s previous work on Fano 4-folds. In particular, given (fcolon Xto Y) as above, under suitable assumptions we show that (S:=f(operatorname{Exc}(f))) is a smooth del Pezzo surface with (-K_{S}=(-K_{Y})_{|S}).

Abstract

A (C^{infty }) smooth surface diffeomorphism admits an SRB measure if and only if the set ({ x, limsup _{n}frac{1}{n}log |d_{x}f^{n}|>0}) has positive Lebesgue measure. Moreover the basins of the ergodic SRB measures are covering this set Lebesgue almost everywhere. We also obtain similar results for (C^{r}) surface diffeomorphisms with (+infty >r>1) .

We construct a non-full exceptional collection of maximal length consisting of line bundles on the blow-up of the projective plane in 10 general points. As a consequence, the orthogonal complement of this collection is a universal phantom category. This provides a counterexample to a conjecture of Kuznetsov and to a conjecture of Orlov.

Let (k) be a number field and (G) be a finite group. Let (mathfrak{F}_{k}^{G}(Q)) be the family of number fields (K) with absolute discriminant (D_{K}) at most (Q) such that (K/k) is normal with Galois group isomorphic to (G). If (G) is the symmetric group (S_{n}) or any transitive group of prime degree, then we unconditionally prove that for all (Kin mathfrak{F}_{k}^{G}(Q)) with at most (O_{varepsilon }(Q^{varepsilon })) exceptions, the (L)-functions associated to the faithful Artin representations of (mathrm{Gal}(K/k)) have a region of holomorphy and non-vanishing commensurate with predictions by the Artin conjecture and the generalized Riemann hypothesis. This result is a special case of a more general theorem. As applications, we prove that:

1. (1)

   there exist infinitely many degree (n) (S_{n})-fields over ℚ whose class group is as large as the Artin conjecture and GRH imply, settling a question of Duke;

2. (2)

   for a prime (p), the periodic torus orbits attached to the ideal classes of almost all totally real degree (p) fields (F) over ℚ equidistribute on (mathrm{PGL}_{p}(mathbb{Z})backslash mathrm{PGL}_{p}(mathbb{R})) with respect to Haar measure;

3. (3)

   for each (ell geq 2), the (ell )-torsion subgroups of the ideal class groups of almost all degree (p) fields over (k) (resp. almost all degree (n) (S_{n})-fields over (k)) are as small as GRH implies; and

4. (4)

   an effective variant of the Chebotarev density theorem holds for almost all fields in such families.