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.
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.
Papers in preparation
Local-global compatibility at $\ell=p$
for Iwahori level Siegel-Hilbert modular forms
$\mathscr{L}$-invariants for $\operatorname{GSp}(2n)$ and $\operatorname{GL}(n)$ under algebraic representations
The Witten zeta functions of some spherical varieties
with
Sam Evens
Expository
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.