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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.
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.
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 study a kernel function of the twisted symmetric square
$L$
-function of elliptic modular forms. As an application, several exact special values of the
$L$
-function are computed.
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 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.
This paper draws connections between the double shuffle equations and structure of associators; Hain and Matsumoto’s universal mixed elliptic motives; and the Rankin–Selberg method for modular forms for
$\text{SL}_{2}(\mathbb{Z})$
. We write down explicit formulae for zeta elements
$\unicode[STIX]{x1D70E}_{2n-1}$
(generators of the Tannaka Lie algebra of the category of mixed Tate motives over
$\mathbb{Z}$
) in depths up to four, give applications to the Broadhurst–Kreimer conjecture, and solve the double shuffle equations for multiple zeta values in depths two and three.
Let
$F/\mathbf{Q}$
be a totally real field and
$K/F$
a complex multiplication (CM) quadratic extension. Let
$f$
be a cuspidal Hilbert modular new form over
$F$
. Let
${\it\lambda}$
be a Hecke character over
$K$
such that the Rankin–Selberg convolution
$f$
with the
${\it\theta}$
-series associated with
${\it\lambda}$
is self-dual with root number 1. We consider the nonvanishing of the family of central-critical Rankin–Selberg
$L$
-values
$L(\frac{1}{2},f\otimes {\it\lambda}{\it\chi})$
, as
${\it\chi}$
varies over the class group characters of
$K$
. Our approach is geometric, relying on the Zariski density of CM points in self-products of a Hilbert modular Shimura variety. We show that the number of class group characters
${\it\chi}$
such that
$L(\frac{1}{2},f\otimes {\it\lambda}{\it\chi})\neq 0$
increases with the absolute value of the discriminant of
$K$
. We crucially rely on the André–Oort conjecture for arbitrary self-product of the Hilbert modular Shimura variety. In view of the recent results of Tsimerman, Yuan–Zhang and Andreatta–Goren–Howard–Pera, the results are now unconditional. We also consider a quaternionic version. Our approach is geometric, relying on the general theory of Shimura varieties and the geometric definition of nearly holomorphic modular forms. In particular, the approach avoids any use of a subconvex bound for the Rankin–Selberg
$L$
-values. The Waldspurger formula plays an underlying role.
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.
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.
We construct an Euler system attached to a weight 2 modular form twisted by a Grössencharacter of an imaginary quadratic field
$K$
, and apply this to bounding Selmer groups.
Let
${\mathcal{K}}$
be an imaginary quadratic field. Let
${\rm\Pi}$
and
${\rm\Pi}^{\prime }$
be irreducible generic cohomological automorphic representation of
$\text{GL}(n)/{\mathcal{K}}$
and
$\text{GL}(n-1)/{\mathcal{K}}$
, respectively. Each of them can be given two natural rational structures over number fields. One is defined by the rational structure on topological cohomology, and the other is given in terms of the Whittaker model. The ratio between these rational structures is called a Whittaker period. An argument presented by Mahnkopf and Raghuram shows that, at least if
${\rm\Pi}$
is cuspidal and the weights of
${\rm\Pi}$
and
${\rm\Pi}^{\prime }$
are in a standard relative position, the critical values of the Rankin–Selberg product
$L(s,{\rm\Pi}\times {\rm\Pi}^{\prime })$
are essentially algebraic multiples of the product of the Whittaker periods of
${\rm\Pi}$
and
${\rm\Pi}^{\prime }$
. We show that, under certain regularity and polarization hypotheses, the Whittaker period of a cuspidal
${\rm\Pi}$
can be given a motivic interpretation, and can also be related to a critical value of the adjoint
$L$
-function of related automorphic representations of unitary groups. The resulting expressions for critical values of the Rankin–Selberg and adjoint
$L$
-functions are compatible with Deligne’s conjecture.
We compute the
$p$
-adic
$L$
-functions of evil Eisenstein series, showing that they factor as products of two Kubota–Leopoldt
$p$
-adic
$L$
-functions times a logarithmic term. This proves in particular a conjecture of Glenn Stevens.
Following Jacquet, Lapid and Rogawski, we define a regularized period of an automorphic form on
$\text{GL}_{n+1}\times \text{GL}_{n}$
along the diagonal subgroup
$\text{GL}_{n}$
and express it in terms of the Rankin–Selberg integral of Jacquet, Piatetski-Shapiro and Shalika. This extends the theory of Rankin–Selberg integrals to all automorphic forms on
$\text{GL}_{n+1}\times \text{GL}_{n}$
.
We describe algorithms for computing central values of twists of
$L$
-functions associated to Hilbert modular forms, carry out such computations for a number of examples, and compare the results of these computations to some heuristics and predictions from random matrix theory.
Let k be a positive integer such that k≡3 mod 4, and let N be a positive square-free integer. In this paper, we compute a basis for the two-dimensional subspace Sk/2(Γ0(4N),F) of half-integral weight modular forms associated, via the Shimura correspondence, to a newform F∈Sk−1(Γ0(N)), which satisfies . This is accomplished by using a result of Waldspurger, which allows one to produce a basis for the forms that correspond to a given F via local considerations, once a form in the Kohnen space has been determined.