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Lapid and Mao formulated a conjecture on an explicit formula of Whittaker–Fourier coefficients of automorphic forms on quasi-split reductive groups and metaplectic groups as an analogue of the Ichino–Ikeda conjecture. They also showed that this conjecture is reduced to a certain local identity in the case of unitary groups. In this article, we study the even unitary-group case. Indeed, we prove this local identity over p-adic fields. Further, we prove an equivalence between this local identity and a refined formal degree conjecture over any local field of characteristic zero. As a consequence, we prove a refined formal degree conjecture over p-adic fields and get an explicit formula of Whittaker–Fourier coefficients under certain assumptions.
The principal aim of this article is to attach and study $p$-adic $L$-functions to cohomological cuspidal automorphic representations $\Pi$ of $\operatorname {GL}_{2n}$ over a totally real field $F$ admitting a Shalika model. We use a modular symbol approach, along the global lines of the work of Ash and Ginzburg, but our results are more definitive because we draw heavily upon the methods used in the recent and separate works of all three authors. By construction, our $p$-adic $L$-functions are distributions on the Galois group of the maximal abelian extension of $F$ unramified outside $p\infty$. Moreover, we work under a weaker Panchishkine-type condition on $\Pi _p$ rather than the full ordinariness condition. Finally, we prove the so-called Manin relations between the $p$-adic $L$-functions at all critical points. This has the striking consequence that, given a unitary $\Pi$ whose standard $L$-function admits at least two critical points, and given a prime $p$ such that $\Pi _p$ is ordinary, the central critical value $L(\frac {1}{2}, \Pi \otimes \chi )$ is non-zero for all except finitely many Dirichlet characters $\chi$ of $p$-power conductor.
We improve upon the local bound in the depth aspect for sup-norms of newforms on $D^\times$, where $D$ is an indefinite quaternion division algebra over ${\mathbb {Q}}$. Our sup-norm bound implies a depth-aspect subconvexity bound for $L(1/2, f \times \theta _\chi )$, where $f$ is a (varying) newform on $D^\times$ of level $p^n$, and $\theta _\chi$ is an (essentially fixed) automorphic form on $\textrm {GL}_2$ obtained as the theta lift of a Hecke character $\chi$ on a quadratic field. For the proof, we augment the amplification method with a novel filtration argument and a recent counting result proved by the second-named author to reduce to showing strong quantitative decay of matrix coefficients of local newvectors along compact subsets, which we establish via $p$-adic stationary phase analysis. Furthermore, we prove a general upper bound in the level aspect for sup-norms of automorphic forms belonging to any family whose associated matrix coefficients have such a decay property.
It has been well established that congruences between automorphic forms have far-reaching applications in arithmetic. In this paper, we construct congruences for Siegel–Hilbert modular forms defined over a totally real field of class number 1. As an application of this general congruence, we produce congruences between paramodular Saito–Kurokawa lifts and non-lifted Siegel modular forms. These congruences are used to produce evidence for the Bloch–Kato conjecture for elliptic newforms of square-free level and odd functional equation.
We define variants of PEL type of the Shimura varieties that appear in the context of the arithmetic Gan–Gross–Prasad (AGGP) conjecture. We formulate for them a version of the AGGP conjecture. We also construct (global and semi-global) integral models of these Shimura varieties and formulate for them conjectures on arithmetic intersection numbers. We prove some of these conjectures in low dimension.
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
$L_p$
.
We introduce a new family of real-analytic modular forms on the upper-half plane. They are arguably the simplest class of ‘mixed’ versions of modular forms of level one and are constructed out of real and imaginary parts of iterated integrals of holomorphic Eisenstein series. They form an algebra of functions satisfying many properties analogous to classical holomorphic modular forms. In particular, they admit expansions in
$q,\overline{q}$
and
$\log |q|$
involving only rational numbers and single-valued multiple zeta values. The first nontrivial functions in this class are real-analytic Eisenstein series.
Waldspurger’s formula gives an identity between the norm of a torus period and an
$L$
-function of the twist of an automorphic representation on GL(2). For any two Hecke characters of a fixed quadratic extension, one can consider the two torus periods coming from integrating one character against the automorphic induction of the other. Because the corresponding
$L$
-functions agree, (the norms of) these periods—which occur on different quaternion algebras—are closely related. In this paper, we give a direct proof of an explicit identity between the torus periods themselves.
We construct the
$p$
-adic standard
$L$
-functions for ordinary families of Hecke eigensystems of the symplectic group
$\operatorname{Sp}(2n)_{/\mathbb{Q}}$
using the doubling method. We explain a clear and simple strategy of choosing the local sections for the Siegel Eisenstein series on the doubling group
$\operatorname{Sp}(4n)_{/\mathbb{Q}}$
, which guarantees the nonvanishing of local zeta integrals and allows us to
$p$
-adically interpolate the restrictions of the Siegel Eisenstein series to
$\operatorname{Sp}(2n)_{/\mathbb{Q}}\times \operatorname{Sp}(2n)_{/\mathbb{Q}}$
.
Since Rob Pollack and Glenn Stevens used overconvergent modular symbols to construct
$p$
-adic
$L$
-functions for non-critical slope rational modular forms, the theory has been extended to construct
$p$
-adic
$L$
-functions for non-critical slope automorphic forms over totally real and imaginary quadratic fields by the first and second authors, respectively. In this paper, we give an analogous construction over a general number field. In particular, we start by proving a control theorem stating that the specialisation map from overconvergent to classical modular symbols is an isomorphism on the small slope subspace. We then show that if one takes the modular symbol attached to a small slope cuspidal eigenform, then one can construct a ray class distribution from the corresponding overconvergent symbol, which moreover interpolates critical values of the
$L$
-function of the eigenform. We prove that this distribution is independent of the choices made in its construction. We define the
$p$
-adic
$L$
-function of the eigenform to be this distribution.
Let
$G$
be a semisimple Lie group with associated symmetric space
$D$
, and let
$\unicode[STIX]{x1D6E4}\subset G$
be a cocompact arithmetic group. Let
$\mathscr{L}$
be a lattice inside a
$\mathbb{Z}\unicode[STIX]{x1D6E4}$
-module arising from a rational finite-dimensional complex representation of
$G$
. Bergeron and Venkatesh recently gave a precise conjecture about the growth of the order of the torsion subgroup
$H_{i}(\unicode[STIX]{x1D6E4}_{k};\mathscr{L})_{\operatorname{tors}}$
as
$\unicode[STIX]{x1D6E4}_{k}$
ranges over a tower of congruence subgroups of
$\unicode[STIX]{x1D6E4}$
. In particular, they conjectured that the ratio
$\log |H_{i}(\unicode[STIX]{x1D6E4}_{k};\mathscr{L})_{\operatorname{tors}}|/[\unicode[STIX]{x1D6E4}:\unicode[STIX]{x1D6E4}_{k}]$
should tend to a nonzero limit if and only if
$i=(\dim (D)-1)/2$
and
$G$
is a group of deficiency
$1$
. Furthermore, they gave a precise expression for the limit. In this paper, we investigate computationally the cohomology of several (non-cocompact) arithmetic groups, including
$\operatorname{GL}_{n}(\mathbb{Z})$
for
$n=3,4,5$
and
$\operatorname{GL}_{2}(\mathscr{O})$
for various rings of integers, and observe its growth as a function of level. In all cases where our dataset is sufficiently large, we observe excellent agreement with the same limit as in the predictions of Bergeron–Venkatesh. Our data also prompts us to make two new conjectures on the growth of torsion not covered by the Bergeron–Venkatesh conjecture.
We construct the
$\unicode[STIX]{x1D6EC}$
-adic crystalline and Dieudonné analogues of Hida’s ordinary
$\unicode[STIX]{x1D6EC}$
-adic étale cohomology, and employ integral
$p$
-adic Hodge theory to prove
$\unicode[STIX]{x1D6EC}$
-adic comparison isomorphisms between these cohomologies and the
$\unicode[STIX]{x1D6EC}$
-adic de Rham cohomology studied in Cais [The geometry of Hida families I:
$\unicode[STIX]{x1D6EC}$
-adic de Rham cohomology, Math. Ann. (2017), doi:10.1007/s00208-017-1608-1] as well as Hida’s
$\unicode[STIX]{x1D6EC}$
-adic étale cohomology. As applications of our work, we provide a ‘cohomological’ construction of the family of
$(\unicode[STIX]{x1D711},\unicode[STIX]{x1D6E4})$
-modules attached to Hida’s ordinary
$\unicode[STIX]{x1D6EC}$
-adic étale cohomology by Dee [
$\unicode[STIX]{x1D6F7}$
–
$\unicode[STIX]{x1D6E4}$
modules for families of Galois representations, J. Algebra 235 (2001), 636–664], and we give a new and purely geometric proof of Hida’s finiteness and control theorems. We also prove suitable
$\unicode[STIX]{x1D6EC}$
-adic duality theorems for each of the cohomologies we construct.
We construct an Euler system—a compatible family of global cohomology classes—for the Galois representations appearing in the geometry of Hilbert modular surfaces. If a conjecture of Bloch and Kato on injectivity of regulator maps holds, this Euler system is nontrivial, and we deduce bounds towards the Iwasawa main conjecture for these Galois representations.
We prove that for
$d\in \left\{ 2,3,5,7,13 \right\}$
and
$K$
a quadratic (or rational) field of discriminant
$D$
and Dirichlet character
$\chi $
, if a prime
$p$
is large enough compared to
$D$
, there is a newform
$f\in {{S}_{2}}({{\Gamma }_{0}}(d{{p}^{2}}))$
with sign
$(+1)$
with respect to the Atkin–Lehner involution
${{w}_{{{p}^{2}}}}$
such that
$L(f\otimes \chi ,1)\ne 0$
. This result is obtained through an estimate of a weighted sum of twists of
$L$
-functions that generalises a result of Ellenberg. It relies on the approximate functional equation for the
$L$
-functions
$L(f\otimes \chi ,\cdot )$
and a Petersson trace formula restricted to Atkin–Lehner eigenspaces. An application of this nonvanishing theorem will be given in terms of existence of rank zero quotients of some twisted jacobians, which generalises a result of Darmon and Merel.
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 use the method of Ash and Stevens to prove the existence of small slope
$p$
-adic families of cohomological modular forms for an indefinite quaternion algebra
$B$
. We prove that the Jacquet–Langlands correspondence relating modular forms on
$\text{G}{{\text{L}}_{\text{2}}}/\mathbb{Q}$
and cohomomological modular forms for
$B$
is compatible with the formation of
$p$
-adic families. This result is an analogue of a theorem of Chenevier concerning definite quaternion algebras.
We develop a derivative version of the relative trace formula on
$\text{PGL}\left( 2 \right)$
studied in our previous work, and derive an asymptotic formula of an average of central values (derivatives) of automorphic
$L$
-functions for Hilbert cusp forms. As an application, we prove the existence of Hilbert cusp forms with non-vanishing central values (derivatives) such that the absolute degrees of their Hecke fields are arbitrarily large.
Following Jacquet, Lapid and Rogawski, we regularize trilinear periods. We use the regularized trilinear periods to compute Fourier–Jacobi periods of residues of Eisenstein series on metaplectic groups, which has an application to the Gan–Gross–Prasad conjecture.
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.