Given a smooth geometrically connected variety X defined over a number field K and an & eacute;tale torsor V-* U over a Zariski-open U of X, we investigate the problem of which adelic points of X can be approximated by adelic points that lift to a (twist of a) V. The question has long been investigated in the literature when U = X, but less so in the general case. We introduce a Brauer-Manin obstruction to the problem, and provide an example where this obstruction is nontrivial and purely transcendental. This answers in the negative a question posed by Harari at a 2019 workshop. Our example is also an explicit example of a nontrivial transcendental Brauer-Manin obstruction on a smooth compactification of a quotient SLn/G, with G constant metabelian.
In local relative p-adic Hodge theory, we show that the Galois cohomology of a finite-height crystalline representation (up to a twist) is essentially computed via the (Fontaine-Messing) syntomic complex with coefficients in the associated F-isocrystal. In global applications, for smooth (p-adic formal) schemes, we establish a comparison between the syntomic complex with coefficients in a locally free Fontaine-Laffaille module and the p-adic nearby cycles of the associated & eacute;tale local system on the (rigid) generic fibre.
We prove an injectivity theorem for the cohomology of the Du Bois complexes of varieties with isolated singularities. We use this to deduce vanishing statements for the cohomologies of higher Du Bois complexes of such varieties. Besides some extensions and conjectures in the non-isolated case, we also provide analogues for intersection complexes.
Let K be a finitely generated field over . Let X -> C -> B be a family of nontrivial elliptic surfaces over K such that the configuration of singular fibers of Xb -> Cb is the same for each closed point b is an element of |B Let r be the minimum of the Mordell-Weil rank in this family. Then we show that the locus inside |B where the Mordell-Weil rank is at least r + 1 is a sparse subset. In this way we prove Cowan's conjecture on the average Mordell-Weil rank of elliptic surfaces over and prove a similar result for elliptic surfaces over arbitrary number fields.
We study the Galois groups G f of degree 2n reciprocal (a.k.a. palindromic) polynomials f of height at most H, finding that G f falls short of the maximal possible group S2 Sn for a proportion of all bounded above and below by constant multiples of H-1 log H, whether or not f is required to be monic. This answers a 1998 question of Davis, Duke and Sun and extends Bhargava's 2023 resolution of van der Waerden's 1936 conjecture on the corresponding question for general polynomials. Unlike in that setting, the dominant contribution comes not from reducible polynomials but from those f for which (-1)n f (1) f (-1) is a square, causing G f to lie in an index-2 subgroup.
Recently, G. Navarro introduced a new conjecture that unifies the Alperin Weight Conjecture and the Glauberman correspondence into a single statement. In this paper, we reduce this problem to simple groups and prove it for several classes of groups and blocks. Our reduction can be divided into two steps. First, we show that assuming the so-called Inductive (Blockwise) Alperin Weight Condition for finite simple groups, we obtain an analogous statement for arbitrary finite groups, that is, an automorphism-equivariant version of the Alperin Weight Conjecture inducing isomorphisms of modular character triples. Then, we show that the latter implies Navarro's conjecture for each finite group.
We answer various questions concerning the distribution of extensions of a given central simple algebra K over a number field. Specifically, we give asymptotics for the count of inner Galois extensions L/K of fixed degree and center with bounded discriminant. We also relate the distribution of outer extensions of K to the distribution of field extensions of its center Z(K). This paper generalizes the study of asymptotics of field extensions to the noncommutative case in an analogous manner to the program initiated by Deschamps and Legrand to extend inverse Galois theory to division algebras.
We define a reduction covariant for the representations a la Vinberg associated to stably graded Lie algebras. We then give an analogue of the LLL algorithm for the odd split special orthogonal group and show how this can be combined with our theory to effectively reduce the coefficients of vectors in a representation connected to 2-descent for odd hyperelliptic curves.
For certain families of L-functions, we prove that if each L-function in the family has only real zeros in a fixed yet arbitrarily small neighborhood of s=1, then one may considerably improve upon the known results on Landau-Siegel zeros. Sarnak and the third author proved a similar result under much more restrictive hypotheses.
Let $\Gamma\subseteq PSL_2({\bf R})$ be a finite volume Fuchsian group. The hyperbolic circle problem is the estimation of the number of elements of the $\Gamma$-orbit of $z$ in a hyperbolic circle around $w$ of radius $R$, where $z$ and $w$ are given points of the upper half plane and $R$ is a large number. An estimate with error term $e^{{2\over 3}R}$ is known, and this has not been improved for any group. Petridis and Risager proved that in the special case $\Gamma =PSL_2({\bf Z})$ taking $z=w$ and averaging over $z$ locally the error term can be improved to $e^{\left({7\over {12}}+\epsilon\right)R}$. Here we show such an improvement for the local $L^2$-norm of the error term. Our estimate is $e^{\left({9\over {14}}+\epsilon\right)R}$, which is better than the pointwise bound $e^{{2\over 3}R}$ but weaker than the bound of Petridis and Risager for the local average.
Let A be a non-CM simple abelian variety over a number field K. For a place v of K such that A has good reduction at v, let F(A,v) denote the Frobenius field generated by the corresponding Frobenius eigenvalues. Assuming A has connected monodromy groups, we show that the set of places v such that F(A,v) is isomorphic to a fixed number field has upper Dirichlet density zero. Assuming the GRH, we give a power saving upper bound for the number of such places.
We prove a Chevalley formula to multiply the motivic Chern classes of Schubert cells in a generalized flag manifold GAP by the class of any line bundle L lambda. Our formula is given in terms of the lambda-chains of Lenart and Postnikov. Its proof relies on a change of basis formula in the affine Hecke algebra due to Ram, and on the Hecke algebra action on torus-equivariant K-theory of the complete flag manifold GAB via left Demazure-Lusztig operators. We revisit some wall-crossing formulae for the stable envelopes in T*(GAB). We use our Chevalley formula, and the equivalence between motivic Chern classes of Schubert cells and K-theoretic stable envelopes in T *(GAB), to give formulae for the change of polarization, and for the change of slope for stable envelopes. We prove several additional applications, including Serre, star, and Dynkin, dualities of the Chevalley coefficients, new formulae for the Whittaker functions, and for the Hall-Littlewood polynomials. We also discuss positivity properties of Chevalley coefficients, and properties of the coefficients arising from multiplication by minuscule weights.
We prove the existence of "murmurations" in the family of holomorphic modular forms of level $1$ and weight $k\to\infty$, that is, correlations between their root numbers and Hecke eigenvalues at primes growing in proportion to the analytic conductor. This is the first demonstration of murmurations in an archimedean family.
We prove that under certain explicit conditions, the Mahler measure of a three-variable polynomial can be expressed in terms of elliptic curve $L$-values and Bloch-Wigner dilogarithmmic values, conditionally on Beilinson's conjecture. In some cases, these dilogarithmic values simplify to Dirichlet $L$-values. The proof involves a construction of an element in $K_4^{(3)}$ of a smooth projective curve over a number field. This generalizes a result of Lal\'in for the polynomial $z + (x+1)(y+1)$. We apply our method to several other Mahler measure identities conjectured by Boyd and Brunault.
We prove that the set of rational points on a nonisotrivial curves of genus at least 2 over a global function field is equal to the set of adelic points cut out by the Brauer-Manin obstruction.
For an algebraically closed non-archimedean extension $C/\mathbb{Q}_p$, we define a Tannakian category of $p$-adic Hodge structures over $C$ that is a local, $p$-adic analog of the global, archimedean category of $\mathbb{Q}$-Hodge structures in complex geometry. In this setting the filtrations of classical Hodge theory must be enriched to lattices over a complete discrete valuation ring, Fontaine's integral de Rham period ring $B^+_\mathrm{dR}$, and a pure $p$-adic Hodge structure is then a $\mathbb{Q}_p$-vector space equipped with a $B^+_\mathrm{dR}$-lattice satisfying a natural condition analogous to the transversality of the complex Hodge filtration with its conjugate. We show $p$-adic Hodge structures are equivalent to a full subcategory of basic objects in the category of admissible pairs, a toy category of cohomological motives over $C$ that is equivalent to the isogeny category of rigidified Breuil-Kisin-Fargues modules and closely related to Fontaine's $p$-adic Hodge theory over $p$-adic subfields. As an application, we characterize basic admissible pairs with complex multiplication in terms of the transcendence of $p$-adic periods. This generalizes an earlier result for one-dimensional formal groups and is an unconditional, local, $p$-adic analog of a global, archimedean characterization of CM motives over $\mathbb{C}$ conditional on the standard conjectures, the Hodge conjecture, and the Grothendieck period conjecture (known unconditionally for abelian varieties by work Cohen and Shiga and Wolfart).
We show the failure of the integral Tate conjecture for the product of a smooth odd-dimensional projective hypersurface with certain smooth projective varieties. To do this, we use a similar specialization argument developed by Gabber (2002) and Colliot-Th & eacute;l & egrave;ne (2019), now applied to Schreieder's refined unramified cohomology (2023). The results thus obtained give an interpretation of Shen's result (2021) in terms of refined unramified cohomology. We avoid the need to work over the complex numbers and in turn, all results hold over general algebraically closed fields of characteristic not 2.
In this paper, we study the moduli space of unipotent Weil-Deligne representations and characterise which irreducible components are smooth. We also study a certain class of unions of irreducible components, and prove that they are Cohen-Macaulay at points $(\Phi, N)$ with $\Phi$ regular semisimple. We apply the smoothness results proved earlier to show that a certain space of ordinary automorphic forms is a locally generically free module over the corresponding global deformation ring.
We investigate a notion of "higher modularity" for elliptic curves over function fields. Given such an elliptic curve $E$ and an integer $r\geq 1$, we say that $E$ is $r$-modular when there is an algebraic correspondence between a stack of $r$-legged shtukas, and the $r$-fold product of $E$ considered as an elliptic surface. The (known) case $r=1$ is analogous to the notion of modularity for elliptic curves over $\mathbf{Q}$. Our main theorem is that if $E/\mathbf{F}_q(t)$ is a nonisotrivial elliptic curve whose conductor has degree 4, then $E$ is 2-modular. Ultimately, the proof uses properties of K3 surfaces. Along the way we prove a result of independent interest: A K3 surface admits a finite morphism to a Kummer surface attached to a product of elliptic curves if and only if its Picard lattice is rationally isometric to the Picard lattice of such a Kummer surface.
We consider tautological bundles and their exterior and symmetric powers on the Quot scheme over the projective line. We prove and conjecture several statements regarding the vanishing of their higher cohomology, and we describe their spaces of global sections via tautological constructions. To this end, we make use of the embedding of the Quot scheme as an explicit local complete intersection in the product of two Grassmannians, studied by Strømme. This allows us to construct resolutions with vanishing cohomology for the tautological bundles and their exterior and symmetric powers.
