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In this paper, we study transcendence theory for Thakur multizeta values in positive characteristic. We prove an analogue of the strong form of Goncharov’s conjecture. The same result is also established for Carlitz multiple polylogarithms at algebraic points.
We consider two families of arithmetic divisors defined on integral models of Shimura curves. The first was studied by Kudla, Rapoport and Yang, who proved that if one assembles these divisors in a formal generating series, one obtains the
$q$
-expansion of a modular form of weight 3/2. The present work concerns the Shimura lift of this modular form: we identify the Shimura lift with a generating series comprising divisors arising in the recent work of Kudla and Rapoport regarding cycles on Shimura varieties of unitary type. In the prequel to this paper, the author considered the geometry of the two families of cycles. These results are combined with the Archimedean calculations found in this work in order to establish the theorem. In particular, we obtain new examples of modular generating series whose coefficients lie in arithmetic Chow groups of Shimura varieties.
We study the distribution of 2-Selmer ranks in the family of quadratic twists of an elliptic curve
$\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}E$
over an arbitrary number field
$K$
. Under the assumption that
${\rm Gal}(K(E[2])/K) \ {\cong }\ S_3$
, we show that the density (counted in a nonstandard way) of twists with Selmer rank
$r$
exists for all positive integers
$r$
, and is given via an equilibrium distribution, depending only on a single parameter (the ‘disparity’), of a certain Markov process that is itself independent of
$E$
and
$K$
. More generally, our results also apply to
$p$
-Selmer ranks of twists of two-dimensional self-dual
${\bf F}_p$
-representations of the absolute Galois group of
$K$
by characters of order
$p$
.
We apply the endoscopic classification of automorphic forms on
$U(3)$
to study the growth of the first Betti number of congruence covers of a Picard modular surface. As a consequence, we establish a case of a conjecture of Sarnak and Xue on cohomology growth.
A conjecture of Manin predicts the distribution of rational points on Fano varieties. We provide a framework for proofs of Manin’s conjecture for del Pezzo surfaces over imaginary quadratic fields, using universal torsors. Some of our tools are formulated over arbitrary number fields. As an application, we prove Manin’s conjecture over imaginary quadratic fields
$K$
for the quartic del Pezzo surface
$S$
of singularity type
${\boldsymbol{A}}_{3}$
with five lines given in
${\mathbb{P}}_{K}^{4}$
by the equations
${x}_{0}{x}_{1}-{x}_{2}{x}_{3}={x}_{0}{x}_{3}+{x}_{1}{x}_{3}+{x}_{2}{x}_{4}=0$
.
We explain how the André–Oort conjecture for a general Shimura variety can be deduced from the hyperbolic Ax–Lindemann conjecture, a good lower bound for Galois orbits of special points and the definability, in the
$o$
-minimal structure
${ \mathbb{R} }_{\mathrm{an} , \mathrm{exp} } $
, of the restriction to a fundamental set of the uniformizing map of a Shimura variety. These ingredients are known in some important cases. As a consequence a proof of the André–Oort conjecture for projective special subvarieties of
${ \mathcal{A} }_{6}^{N} $
for an arbitrary integer
$N$
is given.
We prove modularity of some two-dimensional,
$\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}2$
-adic Galois representations over a totally real field that are nearly ordinary at all places above
$2$
and that are residually dihedral. We do this by employing the strategy of Skinner and Wiles, using Hida families, together with the
$2$
-adic patching method of Khare and Wintenberger. As an application we deduce modularity of some elliptic curves over totally real fields that have good ordinary or multiplicative reduction at places above
$2$
.
Let
$A/K$
be an abelian variety over a function field of characteristic
$p>0$
and let
$\ell $
be a prime number (
$\ell =p$
allowed). We prove the following: the parity of the corank
$r_\ell $
of the
$\ell $
-discrete Selmer group of
$A/K$
coincides with the parity of the order at
$s=1$
of the Hasse–Weil
$L$
-function of
$A/K$
. We also prove the analogous parity result for pure
$\ell $
-adic sheaves endowed with a nice pairing and in particular for the congruence Zeta function of a projective smooth variety over a finite field. Finally, we prove that the full Birch and Swinnerton-Dyer conjecture is equivalent to the Artin–Tate conjecture.
We analyze the relationship between two compactifications of the moduli space of maps from curves to a Grassmannian: the Kontsevich moduli space of stable maps and the Marian–Oprea–Pandharipande moduli space of stable quotients. We construct a moduli space which dominates both the moduli space of stable maps to a Grassmannian and the moduli space of stable quotients, and equip our moduli space with a virtual fundamental class. We relate the virtual fundamental classes of all three moduli spaces using the virtual push-forward formula. This gives a new proof of a theorem of Marian–Oprea–Pandharipande: that enumerative invariants defined as intersection numbers in the stable quotient moduli space coincide with Gromov–Witten invariants.
We develop some basic homological theory of hopfological algebra as defined by Khovanov [Hopfological algebra and categorification at a root of unity: the first steps, Preprint (2006), arXiv:math/0509083v2]. Several properties in hopfological algebra analogous to those of usual homological theory of DG algebras are obtained.
As the simplest case of Langlands functoriality, one expects the existence of the symmetric power
$S^n(\pi )$
, where
$\pi $
is an automorphic representation of
${\rm GL}(2,{\mathbb{A}})$
and
${\mathbb{A}}$
denotes the adeles of a number field
$F$
. This should be an automorphic representation of
${\rm GL}(N,{\mathbb{A}})$
(
$N=n+1)$
. This is known for
$n=2,3$
and
$4$
. In this paper we show how to deduce the general case from a recent result of J.T. on deformation theory for ‘Schur representations’, combined with expected results on level-raising, as well as another case (a particular tensor product) of Langlands functoriality. Our methods assume
$F$
totally real, and the initial representation
$\pi $
of classical type.
We show that some hypergeometric monodromy groups in
${\rm Sp}(4,\mathbf{Z})$
split as free or amalgamated products and hence by cohomological considerations give examples of Zariski dense, non-arithmetic monodromy groups of real rank
$2$
. In particular, we show that the monodromy group of the natural quotient of the Dwork family of quintic threefolds in
$\mathbf{P}^{4}$
splits as
$\mathbf{Z}\ast \mathbf{Z}/5\mathbf{Z}$
. As a consequence, for a smooth quintic threefold
$X$
we show that the group of autoequivalences
$D^{b}(X)$
generated by the spherical twist along
${\mathcal{O}}_{X}$
and by tensoring with
${\mathcal{O}}_{X}(1)$
is an Artin group of dihedral type.
In this paper we generalize the work of Harris–Soudry–Taylor and construct the compatible systems of two-dimensional Galois representations attached to cuspidal automorphic representations of cohomological type on
${\rm GL}_2$
over a CM field with a suitable condition on their central characters. We also prove a local-global compatibility statement, up to semi-simplification.
While investigating the Doi–Naganuma lift, Zagier defined integral weight cusp forms
$f_D$
which are naturally defined in terms of binary quadratic forms of discriminant
$D$
. It was later determined by Kohnen and Zagier that the generating function for the function
$f_D$
is a half-integral weight cusp form. A natural preimage of
$f_D$
under a differential operator at the heart of the theory of harmonic weak Maass forms was determined by the first two authors and Kohnen. In this paper, we consider the modularity properties of the generating function of these preimages. We prove that although the generating function is not itself modular, it can be naturally completed to obtain a half-integral weight modular object.
Generalizing previous results of Deligne–Serre and Taylor, Galois representations are attached to cuspidal automorphic representations of unitary groups whose Archimedean component is a holomorphic limit of discrete series. The main ingredient is a construction of congruences, using the Hasse invariant, that is independent of
$q$
-expansions.
We formulate a conjecture which generalizes Darmon’s ‘refined class number formula’. We discuss relations between our conjecture and the equivariant leading term conjecture of Burns. As an application, we give another proof of the ‘except
$2$
-part’ of Darmon’s conjecture, which was first proved by Mazur and Rubin.
We prove the existence of certain rationally rigid triples in
${E}_{8}(p)$
for good primes
$p$
(i.e.
$p>5$
) thereby showing that these groups occur as Galois groups over the field of rational numbers. We show that these triples arise from rigid triples in the algebraic group and prove that they generate an interesting subgroup in characteristic zero. As a byproduct of the proof, we derive a remarkable symmetry between the character table of a finite reductive group and that of its dual group. We also give a short list of possible overgroups of regular unipotent elements in simple exceptional groups.
We endow certain GKZ-hypergeometric systems with a natural structure of a mixed Hodge module, which is compatible with the mixed Hodge module structure on the Gauß–Manin system of an associated family of Laurent polynomials. As an application we show that the underlying perverse sheaf of a GKZ-system with rational parameter has quasi-unipotent local monodromy.
We classify all (abstract) homomorphisms from the group
$\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}{\sf PGL}_{r+1}(\mathbf{C})$
to the group
${\sf Bir}(M)$
of birational transformations of a complex projective variety
$M$
, provided that
$r\geq \dim _\mathbf{C}(M)$
. As a byproduct, we show that: (i)
${\sf Bir}(\mathbb{P}^n_\mathbf{C})$
is isomorphic, as an abstract group, to
${\sf Bir}(\mathbb{P}^m_\mathbf{C})$
if and only if
$n=m$
; and (ii)
$M$
is rational if and only if
${\sf PGL}_{\dim (M)+1}(\mathbf{C})$
embeds as a subgroup of
${\sf Bir}(M)$
.