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Let f and g be two cuspidal modular forms and let
${\mathcal {F}}$
be a Coleman family passing through f, defined over an open affinoid subdomain V of weight space
$\mathcal {W}$
. Using ideas of Pottharst, under certain hypotheses on f and g, we construct a coherent sheaf over
$V \times \mathcal {W}$
that interpolates the Bloch–Kato Selmer group of the Rankin–Selberg convolution of two modular forms in the critical range (i.e, the range where the p-adic L-function
$L_p$
interpolates critical values of the global L-function). We show that the support of this sheaf is contained in the vanishing locus of
$L_p$
.
Fix an odd prime p. Let
$\mathcal{D}_n$
denote a non-abelian extension of a number field K such that
$K\cap\mathbb{Q}(\mu_{p^{\infty}})=\mathbb{Q}, $
and whose Galois group has the form
$ \text{Gal}\big(\mathcal{D}_n/K\big)\cong \big(\mathbb{Z}/p^{n'}\mathbb{Z}\big)^{\oplus g}\rtimes \big(\mathbb{Z}/p^n\mathbb{Z}\big)^{\times}\ $
where g > 0 and
$0 \lt n'\leq n$
. Given a modular Galois representation
$\overline{\rho}:G_{\mathbb{Q}}\rightarrow \text{GL}_2(\mathbb{F})$
which is p-ordinary and also p-distinguished, we shall write
$\mathcal{H}(\overline{\rho})$
for the associated Hida family. Using Greenberg’s notion of Selmer atoms, we prove an exact formula for the algebraic λ-invariant
\begin{equation}
\lambda^{\text{alg}}_{\mathcal{D}_n}(f) \;=\; \text{the number of zeroes of }
\text{char}_{\Lambda}\big(\text{Sel}_{\mathcal{D}_n^{\text{cy}}}\big(f\big)^{\wedge}\big)
\end{equation}
at all
$f\in\mathcal{H}(\overline{\rho})$
, under the assumption
$\mu^{\text{alg}}_{K(\mu_p)}(f_0)=0$
for at least one form f0. We can then easily deduce that
$\lambda^{\text{alg}}_{\mathcal{D}_n}(f)$
is constant along branches of
$\mathcal{H}(\overline{\rho})$
, generalising a theorem of Emerton, Pollack and Weston for
$\lambda^{\text{alg}}_{\mathbb{Q}(\mu_{p})}(f)$
.
For example, if
$\mathcal{D}_{\infty}=\bigcup_{n\geq 1}\mathcal{D}_n$
has the structure of a p-adic Lie extension then our formulae include the cases where: either (i)
$\mathcal{D}_{\infty}/K$
is a g-fold false Tate tower, or (ii)
$\text{Gal}\big(\mathcal{D}_{\infty}/K(\mu_p)\big)$
has dimension ≤ 3 and is a pro-p-group.
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.
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.
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
$E_{d}(\overline{\mathbb{Q}})$
.
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.
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.
In this paper, we prove an ‘explicit reciprocity law’ relating Howard’s system of big Heegner points to a two-variable
$p$
-adic
$L$
-function (constructed here) interpolating the
$p$
-adic Rankin
$L$
-series of Bertolini–Darmon–Prasanna in Hida families. As applications, we obtain a direct relation between classical Heegner cycles and the higher weight specializations of big Heegner points, refining earlier work of the author, and prove the vanishing of Selmer groups of CM elliptic curves twisted by 2-dimensional Artin representations in cases predicted by the equivariant Birch and Swinnerton-Dyer conjecture.
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.
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.
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.
We give a formula relating the order of the Brauer group of a surface fibered over a curve over a finite field to the order of the Tate–Shafarevich group of the Jacobian of the generic fiber. The formula implies that the Brauer group of a smooth and proper surface over a finite field is a square if it is finite.
We prove a general formula for the
$p$
-adic heights of Heegner points on modular abelian varieties with potentially ordinary (good or semistable) reduction at the primes above
$p$
. The formula is in terms of the cyclotomic derivative of a Rankin–Selberg
$p$
-adic
$L$
-function, which we construct. It generalises previous work of Perrin-Riou, Howard, and the author to the context of the work of Yuan–Zhang–Zhang on the archimedean Gross–Zagier formula and of Waldspurger on toric periods. We further construct analytic functions interpolating Heegner points in the anticyclotomic variables, and obtain a version of our formula for them. It is complemented, when the relevant root number is
$+1$
rather than
$-1$
, by an anticyclotomic version of the Waldspurger formula. When combined with work of Fouquet, the anticyclotomic Gross–Zagier formula implies one divisibility in a
$p$
-adic Birch and Swinnerton-Dyer conjecture in anticyclotomic families. Other applications described in the text will appear separately.
We establish a connection between motivic cohomology classes over the Siegel threefold and non-critical special values of the degree-four
$L$
-function of some cuspidal automorphic representations of
$\text{GSp}(4)$
. Our computation relies on our previous work [On higher regulators of Siegel threefolds I: the vanishing on the boundary, Asian J. Math. 19 (2015), 83–120] and on an integral representation of the
$L$
-function due to Piatetski-Shapiro.
We give an explicit description of the stable reduction of superelliptic curves of the form yn=f(x) at primes
$\mathfrak{p}$
whose residue characteristic is prime to the exponent n. We then use this description to compute the local L-factor and the exponent of conductor at
$\mathfrak{p}$
of the curve.
We extend the
$p$
-adic Gross–Zagier formula of Bertolini et al. [Generalized Heegner cycles and
$p$
-adic Rankin
$L$
-series, Duke Math. J.162(6) (2013), 1033–1148] to the semistable non-crystalline setting, and combine it with our previous work [Castella, On the
$p$
-adic variation of Heegner points, Preprint, 2014, arXiv:1410.6591] to obtain a derivative formula for the specializations of Howard’s big Heegner points [Howard, Variation of Heegner points in Hida families, Invent. Math.167(1) (2007), 91–128] at exceptional primes in the Hida family.
Suppose E is an elliptic curve over
$\Bbb Q$
, and p>3 is a split multiplicative prime for E. Let q ≠ p be an auxiliary prime, and fix an integer m coprime to pq. We prove the generalised Mazur–Tate–Teitelbaum conjecture for E at the prime p, over number fields
$K\subset \Bbb Q\big(\mu_{{q^{\infty}}},\;\!^{q^{\infty}\!\!\!\!}\sqrt{m}\big)$
such that p remains inert in
$K\cap\Bbb Q(\mu_{{q^{\infty}}})^+$
. The proof makes use of an improved p-adic L-function, which can be associated to the Rankin convolution of two Hilbert modular forms of unequal parallel weight.
We study the growth of
$\unicode[STIX]{x0428}$
and
$p^{\infty }$
-Selmer groups for isogenous abelian varieties in towers of number fields, with an emphasis on elliptic curves. The growth types are usually exponential, as in the ‘positive
${\it\mu}$
-invariant’ setting in the Iwasawa theory of elliptic curves. The towers we consider are
$p$
-adic and
$l$
-adic Lie extensions for
$l\neq p$
, in particular cyclotomic and other
$\mathbb{Z}_{l}$
-extensions.
In a recent important paper, Hoffstein and Hulse [Multiple Dirichlet series and shifted convolutions, arXiv:1110.4868v2] generalized the notion of Rankin–Selberg convolution
$L$
-functions by defining shifted convolution
$L$
-functions. We investigate symmetrized versions of their functions, and we prove that the generating functions of certain special values are linear combinations of weakly holomorphic quasimodular forms and “mixed mock modular” forms.