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A classical construction of Katz gives a purely algebraic construction of Eisenstein–Kronecker series using the Gauß–Manin connection on the universal elliptic curve. This approach gives a systematic way to study algebraic and
$p$
-adic properties of real-analytic Eisenstein series. In the first part of this paper we provide an alternative algebraic construction of Eisenstein–Kronecker series via the Poincaré bundle. Building on this, we give in the second part a new conceptional construction of Katz’ two-variable
$p$
-adic Eisenstein measure through
$p$
-adic theta functions of the Poincaré bundle.
In this paper we prove the Rigidity Theorem for motives of rigid analytic varieties over a non-Archimedean valued field
$K$
. We prove this theorem both for motives with transfers and without transfers in a relative setting. Applications include the construction of étale realization functors, an upgrade of the known comparison between motives with and without transfers and an upgrade of the rigid analytic motivic tilting equivalence, extending them to
$\mathbb{Z}[1/p]$
-coefficients.
In this article we prove the explicit Mordell Conjecture for large families of curves. In addition, we introduce a method, of easy application, to compute all rational points on curves of quite general shape and increasing genus. The method bases on some explicit and sharp estimates for the height of such rational points, and the bounds are small enough to successfully implement a computer search. As an evidence of the simplicity of its application, we present a variety of explicit examples and explain how to produce many others. In the appendix our method is compared in detail to the classical method of Manin–Demjanenko and the analysis of our explicit examples is carried to conclusion.
Let
${\mathcal{X}}$
be a regular variety, flat and proper over a complete regular curve over a finite field such that the generic fiber
$X$
is smooth and geometrically connected. We prove that the Brauer group of
${\mathcal{X}}$
is finite if and only Tate’s conjecture for divisors on
$X$
holds and the Tate–Shafarevich group of the Albanese variety of
$X$
is finite, generalizing a theorem of Artin and Grothendieck for surfaces to arbitrary relative dimension. We also give a formula relating the orders of the group under the assumption that they are finite, generalizing the known formula for a surface.
Let
$F$
be a totally real field and let
$p$
be an odd prime which is totally split in
$F$
. We define and study one-dimensional ‘partial’ eigenvarieties interpolating Hilbert modular forms over
$F$
with weight varying only at a single place
$v$
above
$p$
. For these eigenvarieties, we show that methods developed by Liu, Wan and Xiao apply and deduce that, over a boundary annulus in weight space of sufficiently small radius, the partial eigenvarieties decompose as a disjoint union of components which are finite over weight space. We apply this result to prove the parity version of the Bloch–Kato conjecture for finite slope Hilbert modular forms with trivial central character (with a technical assumption if
$[F:\mathbb{Q}]$
is odd), by reducing to the case of parallel weight
$2$
. As another consequence of our results on partial eigenvarieties, we show, still under the assumption that
$p$
is totally split in
$F$
, that the ‘full’ (dimension
$1+[F:\mathbb{Q}]$
) cuspidal Hilbert modular eigenvariety has the property that many (all, if
$[F:\mathbb{Q}]$
is even) irreducible components contain a classical point with noncritical slopes and parallel weight
$2$
(with some character at
$p$
whose conductor can be explicitly bounded), or any other algebraic weight.
A result of Bleher, Chinburg, Greenberg, Kakde, Pappas, Sharifi and Taylor has initiated the topic of higher codimension Iwasawa theory. As a generalization of the classical Iwasawa main conjecture, they prove a relationship between analytic objects (a pair of Katz’s
$2$
-variable
$p$
-adic
$L$
-functions) and algebraic objects (two ‘everywhere unramified’ Iwasawa modules) involving codimension two cycles in a
$2$
-variable Iwasawa algebra. We prove a result by considering the restriction to an imaginary quadratic field
$K$
(where an odd prime
$p$
splits) of an elliptic curve
$E$
, defined over
$\mathbb{Q}$
, with good supersingular reduction at
$p$
. On the analytic side, we consider eight pairs of
$2$
-variable
$p$
-adic
$L$
-functions in this setup (four of the
$2$
-variable
$p$
-adic
$L$
-functions have been constructed by Loeffler and a fifth
$2$
-variable
$p$
-adic
$L$
-function is due to Hida). On the algebraic side, we consider modifications of fine Selmer groups over the
$\mathbb{Z}_{p}^{2}$
-extension of
$K$
. We also provide numerical evidence, using algorithms of Pollack, towards a pseudonullity conjecture of Coates–Sujatha.
Let
$p$
be an odd prime number and
$E$
an elliptic curve defined over a number field
$F$
with good reduction at every prime of
$F$
above
$p$
. We compute the Euler characteristics of the signed Selmer groups of
$E$
over the cyclotomic
$\mathbb{Z}_{p}$
-extension. The novelty of our result is that we allow the elliptic curve to have mixed reduction types for primes above
$p$
and mixed signs in the definition of the signed Selmer groups.
In this article we construct a p-adic three-dimensional eigenvariety for the group
$U$
(2,1)(
$E$
), where
$E$
is a quadratic imaginary field and
$p$
is inert in
$E$
. The eigenvariety parametrizes Hecke eigensystems on the space of overconvergent, locally analytic, cuspidal Picard modular forms of finite slope. The method generalized the one developed in Andreatta, Iovita and Stevens [
$p$
-adic families of Siegel modular cuspforms Ann. of Math. (2) 181, (2015), 623–697] by interpolating the coherent automorphic sheaves when the ordinary locus is empty. As an application of this construction, we reprove a particular case of the Bloch–Kato conjecture for some Galois characters of
$E$
, extending the results of Bellaiche and Chenevier to the case of a positive sign.
Let
$K$
be an algebraically closed field of prime characteristic
$p$
, let
$X$
be a semiabelian variety defined over a finite subfield of
$K$
, let
$\unicode[STIX]{x1D6F7}:X\longrightarrow X$
be a regular self-map defined over
$K$
, let
$V\subset X$
be a subvariety defined over
$K$
, and let
$\unicode[STIX]{x1D6FC}\in X(K)$
. The dynamical Mordell–Lang conjecture in characteristic
$p$
predicts that the set
$S=\{n\in \mathbb{N}:\unicode[STIX]{x1D6F7}^{n}(\unicode[STIX]{x1D6FC})\in V\}$
is a union of finitely many arithmetic progressions, along with finitely many
$p$
-sets, which are sets of the form
$\{\sum _{i=1}^{m}c_{i}p^{k_{i}n_{i}}:n_{i}\in \mathbb{N}\}$
for some
$m\in \mathbb{N}$
, some rational numbers
$c_{i}$
and some non-negative integers
$k_{i}$
. We prove that this conjecture is equivalent with some difficult diophantine problem in characteristic 0. In the case
$X$
is an algebraic torus, we can prove the conjecture in two cases: either when
$\dim (V)\leqslant 2$
, or when no iterate of
$\unicode[STIX]{x1D6F7}$
is a group endomorphism which induces the action of a power of the Frobenius on a positive dimensional algebraic subgroup of
$X$
. We end by proving that Vojta’s conjecture implies the dynamical Mordell–Lang conjecture for tori with no restriction.
We develop a theory of enlarged mixed Shimura varieties, putting the universal vectorial bi-extension defined by Coleman into this framework to study some functional transcendental results of Ax type. We study their bi-algebraic systems, formulate the Ax-Schanuel conjecture and explain its relation with the logarithmic Ax theorem and the Ax-Lindemann theorem which we shall prove. All these bi-algebraic and transcendental results extend their counterparts for mixed Shimura varieties. In the end we briefly discuss the André–Oort and Zilber–Pink type problems for enlarged mixed Shimura varieties.
Given an elliptic curve
$E$
over
$\mathbb{Q}$
, a celebrated conjecture of Goldfeld asserts that a positive proportion of its quadratic twists should have analytic rank 0 (respectively 1). We show that this conjecture holds whenever
$E$
has a rational 3-isogeny. We also prove the analogous result for the sextic twists of
$j$
-invariant 0 curves. For a more general elliptic curve
$E$
, we show that the number of quadratic twists of
$E$
up to twisting discriminant
$X$
of analytic rank 0 (respectively 1) is
$\gg X/\log ^{5/6}X$
, improving the current best general bound toward Goldfeld’s conjecture due to Ono–Skinner (respectively Perelli–Pomykala). To prove these results, we establish a congruence formula between
$p$
-adic logarithms of Heegner points and apply it in the special cases
$p=3$
and
$p=2$
to construct the desired twists explicitly. As a by-product, we also prove the corresponding
$p$
-part of the Birch and Swinnerton–Dyer conjecture for these explicit twists.
The congruent number elliptic curves are defined by
$E_{d}:y^{2}=x^{3}-d^{2}x$
, where
$d\in \mathbb{N}$
. We give a simple proof of a formula for
$L(\operatorname{Sym}^{2}(E_{d}),3)$
in terms of the determinant of the elliptic trilogarithm evaluated at some degree zero divisors supported on the torsion points on
$E_{d}(\overline{\mathbb{Q}})$
.
In this paper we prove a semistable version of the variational Tate conjecture for divisors in crystalline cohomology, showing that for
$k$
a perfect field of characteristic
$p$
, a rational (logarithmic) line bundle on the special fibre of a semistable scheme over
$k\unicode[STIX]{x27E6}t\unicode[STIX]{x27E7}$
lifts to the total space if and only if its first Chern class does. The proof is elementary, using standard properties of the logarithmic de Rham–Witt complex. As a corollary, we deduce similar algebraicity lifting results for cohomology classes on varieties over global function fields. Finally, we give a counter-example to show that the variational Tate conjecture for divisors cannot hold with
$\mathbb{Q}_{p}$
-coefficients.
In this paper we study the number of rational points on curves in an ensemble of abelian covers of the projective line: let
$\ell$
be a prime,
$q$
a prime power and consider the ensemble
${\mathcal{H}}_{g,\ell }$
of
$\ell$
-cyclic covers of
$\mathbb{P}_{\mathbb{F}_{q}}^{1}$
of genus
$g$
. We assume that
$q\not \equiv 0,1~\text{mod}~\ell$
. If
$2g+2\ell -2\not \equiv 0~\text{mod}~(\ell -1)\operatorname{ord}_{\ell }(q)$
, then
${\mathcal{H}}_{g,\ell }$
is empty. Otherwise, the number of rational points on a random curve in
${\mathcal{H}}_{g,\ell }$
distributes as
$\sum _{i=1}^{q+1}X_{i}$
as
$g\rightarrow \infty$
, where
$X_{1},\ldots ,X_{q+1}$
are independent and identically distributed random variables taking the values
$0$
and
$\ell$
with probabilities
$(\ell -1)/\ell$
and
$1/\ell$
, respectively. The novelty of our result is that it works in the absence of a primitive
$\ell$
th root of unity, the presence of which was crucial in previous studies.
We discuss the generalizations of the concept of Chebyshev’s bias from two perspectives. First, we give a general framework for the study of prime number races and Chebyshev’s bias attached to general L-functions satisfying natural analytic hypotheses. This extends the cases previously considered by several authors and involving, among others, Dirichlet L-functions and Hasse–Weil L-functions of elliptic curves over Q. This also applies to new Chebyshev’s bias phenomena that were beyond the reach of the previously known cases. In addition, we weaken the required hypotheses such as GRH or linear independence properties of zeros of L-functions. In particular, we establish the existence of the logarithmic density of the set
$ \{x \ge 2:\sum\nolimits_{p \le x} {\lambda _f}(p) \ge 0\}$
for coefficients (λf(p)) of general L-functions conditionally on a much weaker hypothesis than was previously known.
We generalize the Cohen–Lenstra heuristics over function fields to étale group schemes
$G$
(with the classical case of abelian groups corresponding to constant group schemes). By using the results of Ellenberg–Venkatesh–Westerland, we make progress towards the proof of these heuristics. Moreover, by keeping track of the image of the Weil-pairing as an element of
$\wedge ^{2}G(1)$
, we formulate more refined heuristics which nicely explain the deviation from the usual Cohen–Lenstra heuristics for abelian
$\ell$
-groups in cases where
$\ell \mid q-1$
; the nature of this failure was suggested already in the works of Malle, Garton, Ellenberg–Venkatesh–Westerland, and others. On the purely large random matrix side, we provide a natural model which has the correct moments, and we conjecture that these moments uniquely determine a limiting probability measure.
We develop the analog of crystalline Dieudonné theory for
$p$
-divisible groups in the arithmetic of function fields. In our theory
$p$
-divisible groups are replaced by divisible local Anderson modules, and Dieudonné modules are replaced by local shtukas. We show that the categories of divisible local Anderson modules and of effective local shtukas are anti-equivalent over arbitrary base schemes. We also clarify their relation with formal Lie groups and with global objects like Drinfeld modules, Anderson’s abelian
$t$
-modules and
$t$
-motives, and Drinfeld shtukas. Moreover, we discuss the existence of a Verschiebung map and apply it to deformations of local shtukas and divisible local Anderson modules. As a tool we use Faltings’s and Abrashkin’s theories of strict modules, which we review briefly.
Let
$X:=\mathbb{A}_{R}^{n}$
be the
$n$
-dimensional affine space over a discrete valuation ring
$R$
with fraction field
$K$
. We prove that any pointed torsor
$Y$
over
$\mathbb{A}_{K}^{n}$
under the action of an affine finite-type group scheme can be extended to a torsor over
$\mathbb{A}_{R}^{n}$
possibly after pulling
$Y$
back over an automorphism of
$\mathbb{A}_{K}^{n}$
. The proof is effective. Other cases, including
$X=\unicode[STIX]{x1D6FC}_{p,R}$
, are also discussed.
Let
$L/F$
be a quadratic extension of totally real number fields. For any prime
$p$
unramified in
$L$
, we construct a
$p$
-adic
$L$
-function interpolating the central values of the twisted triple product
$L$
-functions attached to a
$p$
-nearly ordinary family of unitary cuspidal automorphic representations of
$\text{Res}_{L\times F/F}(\text{GL}_{2})$
. Furthermore, when
$L/\mathbb{Q}$
is a real quadratic number field and
$p$
is a split prime, we prove a
$p$
-adic Gross–Zagier formula relating the values of the
$p$
-adic
$L$
-function outside the range of interpolation to the syntomic Abel–Jacobi image of generalized Hirzebruch–Zagier cycles.
We determine the parity of the Langlands parameter of a conjugate self-dual supercuspidal representation of
$\text{GL}(n)$
over a non-archimedean local field by means of the local Jacquet–Langlands correspondence. It gives a partial generalization of a previous result on the self-dual case by Prasad and Ramakrishnan.