In this paper, we prove that a GL(2n)-eigenvariety is ´etale over the (pure) weight space
at non-critical Shalika points, and construct multi-variable p-adic L-functions varying over
the resulting Shalika components. Our constructions hold in tame level 1 and Iwahori
level at p, and give p-adic variation of L-values (of regular algebraic cuspidal automorphic
representations of GL(2n) admitting Shalika models) over the whole pure weight space. In
the case of GL(4), these results have been used by Loeffler and Zerbes to prove cases of
the Bloch–Kato conjecture for GSp(4).
Our main innovations are: (a) the introduction and systematic study of ‘Shalika refinements’ of local representations of GL(2n), and evaluation of their attached local twisted
zeta integrals; and (b) the p-adic interpolation of representation-theoretic branching laws
for GL(n)×GL(n) inside GL(2n). Using (b), we give a construction of multi-variable p-adic
functionals on the overconvergent cohomology groups for GL(2n), interpolating the zeta
integrals of (a). We exploit the resulting non-vanishing of these functionals to prove our
main arithmetic applications.
We prove that the trace of the Hecke operator $T_2$ acting on the space of weight $2k$ modular forms of level 1 takes no repeated values as $k$ varies. This answers a question of Vilardi and Xue, which arose in relation to Maeda's conjecture. Our proof combines 2-adic results from our previous paper (Newton Polygons of Hecke Operators) and a new application of classical bounds on linear forms on logarithms in the context of exponential sums with more than 2 terms.
In this computational paper we verify a truncated version of the Buzzard-Calegari conjecture on the Newton polygon of the Hecke operator $T_2$ for all large enough weights. We first develop a formula for computing $p$-adic valuations of exponential sums, which we then implement to compute 2-adic valuations of traces of Hecke operators acting on spaces of cusp forms. Finally, we verify that if Newton polygon of the Buzzard-Calegari polynomial has a vertex at $n\leq 15$, then it agrees with the Newton polygon of $T_2$ up to $n$.
In the first part of the paper, we show that any cohomological Hilbert
modular cusp form which is non-critical and of nearly finite slope at
$p$, in the sense that the local representation is not supercuspidal at
any place above $p$, belongs to a unique $p$-adic family of maximal
dimension. Further, using automorphic symbols for Hilbert modular
varieties, we construct a functorial map sending a finite slope
overconvergent cohomology class to a distribution over the Galois
group of the maximal abelian $p$-ramified extension. These evaluation
maps yield, in a unified fashion, both $p$-adic $L$-functions for nearly
finite slope families and their improved counterparts.
In the second part of the paper we prove the exceptional zero
conjecture at the central point for the Greenberg-Benois $p$-adic
$\mathscr{L}$-invariant in the case of Hilbert modular forms which are
in addition Iwahori spherical at places above p and have trivial
central character. In the case of a multiple exceptional zero we go
beyond the Greenberg-Stevens method and use the interplay between
partial finite slope families and partially improved p-adic
L-functions to establish the vanishing of many Taylor coefficients of
the $p$-adic $L$-function of the family.
We prove a number of statistical properties of Hecke coefficients for unitary cuspidal representations on $\operatorname{GL}(2)$ over number fields (unconditionally) and on $\operatorname{GL}(n)$ over number fields (conditionally, either assuming the Ramanujan conjecture, or the functoriality of $\pi\otimes\pi^\vee$). Using partial bounds on Hecke coefficients, properties of Rankin-Selberg $L$-functions, and instances of Langlands functoriality, we obtain bounds on the set of places where (linear combinations of) Hecke coefficients are bounded above (or below). We furthermore prove a number of consequences: we obtain an improved answer to a question of Serre about the occurrence of large Hecke eigenvalues of Maass forms ($|a_p|>1$ for density at least $0.00135$ set of primes), we prove the existence of negative Hecke coefficients over arbitrary number fields, and we obtain distributional results on the Hecke coefficients $a_v$ when $v$ varies in certain congruence or Galois classes. E.g., if $E$ is an elliptic curve without CM we show that $a_p(E)<0$ for a density $\geq \frac{1}{8}$ of primes $p\equiv a\pmod{n}$, or density $\geq \frac{1}{16}$ of primes of the form $p=m^2+27n^2$.
We compute the arithmetic $\mathscr{L}$-invariants (of Greenberg-Benois) of twists of symmetric powers
of $p$-adic Galois representations attached to Iwahori level Hilbert modular forms (under some
technical conditions). Our method uses the automorphy of symmetric powers and the study of analytic
Galois representations on $p$-adic families of automorphic forms over symplectic and unitary
groups. Combining these families with some explicit plethysm in the representation theory of $\operatorname{GL}(2)$,
we construct global Galois cohomology classes with coefficients in the symmetric powers and provide
formulae for the $\mathscr{L}$-invariants in terms of logarithmic derivatives of Hecke eigenvalues.
We classify the cohomology classes of Lagrangian 4-planes $\mathbb{P}^4$ in a smooth manifold $X$ deformation equivalent to a Hilbert scheme of 4 points on a $K3$ surface, up to the monodromy action. Classically, the cone of effective curves on a $K3$ surface $S$ is generated by nonegative classes $C$, for which $(C,C)\geq0$, and nodal classes $C$, for which $(C,C)=-2$; Hassett and Tschinkel conjecture that the cone of effective curves on a holomorphic symplectic variety $X$ is similarly controlled by ``nodal'' classes $C$ such that $(C,C)=-\gamma$, for $(\cdot,\cdot)$ now the Beauville-Bogomolov form, where $\gamma$ classifies the geometry of the extremal contraction associated to $C$. In particular, they conjecture that for $X$ deformation equivalent to a Hilbert scheme of $n$ points on a $K3$ surface, the class $C=\ell$ of a line in a smooth Lagrangian $n$-plane $\mathbb{P}^n$ must satisfy $(\ell,\ell)=-\frac{n+3}{2}$. We prove the conjecture for $n=4$ by computing the ring of monodromy invariants on $X$, and showing there is a unique monodromy orbit of Lagrangian $4$-planes.
We prove local-global compatibility (up to a quadratic twist) of
Galois representations associated to holomorphic Hilbert-Siegel
modular forms in many cases (induced from Borel or Klingen
parabolic), and as a corollary we obtain a conjecture of Skinner and Urban. For Siegel modular forms, when the local
representation is an irreducible principal series we get local-global compatibility without a twist. We achieve this by proving
a version of rigidity (strong multiplicity one) for $\operatorname{GSp}(4)$
using, on the one hand the doubling method to compute the
standard $L$-function, and on the other hand the explicit
classification of the irreducible local representations of
$\operatorname{GSp}(4)$ over $p$-adic fields.
Mathematical Research Letters19
(2012), no. 5, 1–10
We prove local-global
compatibility for holomorphic Siegel modular forms with Iwahori
level. In previous work we proved a weaker version of this
result (up to a quadratic twist) and one of the goals of this
article is to remove this quadratic twist by different methods, using $p$-adic families. We further study the
local Galois representation at $p$ for nonregular holomorphic
Siegel modular forms. Then we apply the results to the setting
of modular forms on $\operatorname{GL}(2)$ over a quadratic imaginary field
and prove results on the local Galois representation $\ell$, as well as crystallinity results at $p$.
We define and compute higher rank analogs of Pandharipande–Thomas
stable pair invariants in primitive classes for $K3$ surfaces. Higher
rank stable pair invariants for Calabi–Yau threefolds have been
defined by Sheshmani using moduli of pairs of the form
$\mathcal{O}^n\to \mathcal{F}$ for $\mathcal{F}$
purely one-dimensional and computed via wall-crossing
techniques. These invariants may be thought of as virtually counting
embedded curves decorated with a $(n−1)$-dimensional linear system. We
treat invariants counting pairs $\mathcal{O}^n\to\mathcal{E}$ on a K3 surface for $\mathcal{E}$ an
arbitrary stable sheaf of a fixed numerical type (“coherent systems”), whose first Chern class is primitive, and
fully compute them geometrically. The ordinary stable pair theory of
$K3$ surfaces is treated by Maulik-Pandharipande-Thomas; there they prove the KKV conjecture in
primitive classes by showing the resulting partition functions are
governed by quasimodular forms. We prove a “higher” KKV conjecture by
showing that our higher rank partition functions are modular forms.
We show that Galois representations attached by Harris-Taylor-Soudry
to regular algebraic cuspidal automorphic representations on
$\operatorname{GL}(2)$ over a quadratic imaginary field (when the
central character is a base-change from $\mathbb{Q}$) is crystalline
at places above good primes $p$ as long as the Satake parameters at places above
$p$ are distinct.
We describe theorems and computational methods for verifying the Birch and Swinnerton-Dyer conjectural formula for specific elliptic curves over $ \mathbb{Q}$ of analytic ranks 0 and $ 1$. We apply our techniques to show that if $ E$ is a non-CM elliptic curve over $ \mathbb{Q}$ of conductor $ \leq 1000$ and rank 0 or $ 1$, then the Birch and Swinnerton-Dyer conjectural formula for the leading coefficient of the $ L$-series is true for $ E$, up to odd primes that divide either Tamagawa numbers of $ E$ or the degree of some rational cyclic isogeny with domain $ E$. Since the rank part of the Birch and Swinnerton-Dyer conjecture is a theorem for curves of analytic rank 0 or $ 1$, this completely verifies the full conjecture for these curves up to the primes excluded above.
I explain Tate's proof of the fact that if the Birch and
Swinnerton-Dyer conjecture is true for an abelian variety over a
number field, it is also true for any isogenous abelian variety.