arXiv subjects

Simon Marshall

Publications and source records attributed to Simon Marshall.

26 records · Page 2Linked to original sources

arXiv2011

Theta lifting and cohomology growth in p-adic towers

We use the theta lift to study the multiplicity with which certain automorphic representations of cohomological type occur in a family of congruence covers of an arithmetic manifold. When the family of covers is a so-called `p-adic congruence tower' we obtain sharp asymptotics for the number of representations which occur as lifts. When combined with theorems on the surjectivity of the theta lift due to Howe and Li, and Bergeron, Millson and Moeglin, this allows us to verify certain cases of a conjecture of Sarnak and Xue.

math.NT↗

arXiv2011

Zero repulsion in families of elliptic curve L-functions and an observation of S. J. Miller

We provide a theoretical explanation for an observation of S. J. Miller that if L(s,E) is an elliptic curve L-function for which L(1/2, E) is nonzero, then the lowest lying zero of L(s,E) exhibits a repulsion from the critical point which is not explained by the standard Katz-Sarnak heuristics. We establish a similar result in the case of first-order vanishing.

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arXiv2011

Unconditional bounds for the multiplicity of automorphic forms of cohomological type on GL_2

We prove an unconditional power saving for the dimension of the space of cohomological automorphic forms of fixed level and growing weight on GL_2 over any number field which is not totally real. Our proof involves the theory of p-adically completed cohomology developed by Calegari and Emerton, and a bound for the growth of coinvariants in certain finitely generated non-commutative Iwasawa modules.

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arXiv2011

On the torsion in the cohomology of arithmetic hyperbolic 3-manifolds

In this paper we consider the cohomology of a closed arithmetic hyperbolic 3-manifold with coefficients in the local system defined by the even symmetric powers of the standard representation of SL(2,C). The cohomology is defined over the integers and is a finite abelian group. We show that the order of the 2nd cohomology grows exponentially as the local system grows. We also consider the twisted Ruelle zeta function of a closed arithmetic hyperbolic 3-manifold and we express the leading coefficient of its Laurent expansion at the origin in terms of the orders of the torsion subgroups of the cohomology.

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arXiv2010

Bounds for multiplicities of automorphic forms of cohomological type on GL_2

We prove various results on the cohomology of arithmetic lattices arising from quaternion algebras over a number field with at least one complex place, including a strong restriction on the allowable weights of cuspidal cohomological automorphic forms and a conditional power saving for their multiplicities when the associated locally symmetric space is a hyperbolic 3-manifold.

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arXiv2010

Triple Product L Functions and Quantum Chaos on SL(2,C)

We extend the results of Watson, which link quantum unique ergodicity on arithmetic hyperbolic surfaces with subconvexity for the triple product L function, to the case of arithmetic hyperbolic three manifolds. We work with the full unitary dual of SL(2,C), and consider QUE for automorphic forms of arbitrary fixed weight and growing spectral parameter. We obtain our results by constructing microlocal lifts of nonspherical automorphic forms using representation theory, and quantifying the generalised triple product formula of Ichino in the case of complex places.

math.NT↗

arXiv2010

Mass Equidistribution for Automorphic Forms of Cohomological Type on GL_2

We extend Holowinsky and Soundararajan's proof of quantum unique ergodicity for holomorphic Hecke modular forms on SL(2,Z), by establishing it for automorphic forms of cohomological type on GL_2 over an arbitrary number field which satisfy the Ramanujan bounds. In particular, we have uncondtional theorems over totally real and imaginary quadratic fields. In the totally real case we show that our result implies the equidistribution of the zero divisors of holomorphic Hecke modular forms, generalising a result of Rudnick over Q.

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arXiv2005

Eigenvalues of Schroedinger operators with potential asymptotically homogeneous of degree -2

We strengthen and generalise a result of Kirsch and Simon on the behaviour of the function $N_L(E)$, the number of bound states of the operator $L = Δ+V$ in $\R^d$ below $-E$. Here $V$ is a bounded potential behaving asymptotically like $P(ω)r^{-2}$ where $P$ is a function on the sphere. It is well known that the eigenvalues of such an operator are all nonpositive, and accumulate only at 0. If the operator $Δ_{S^{d-1}}+P$ on the sphere has negative eigenvalues $-μ_1,...,-μ_n$ less than $-(d-2)^2/4$, we prove that $N_L(E)$ may be estimated as \[ N_L(E)) = \frac{\log(E^{-1})}{2π}\sum_{i=1}^n \sqrt{μ_i-(d-2)^2/4} +O(1); \] thus, in particular, if there are no such negative eigenvalues then $L$ has a finite discrete spectrum. Moreover, under some additional assumptions including that $d=3$ and that there is exactly one eigenvalue $-μ_1$ less than -1/4, with all others $> -1/4$, we show that the negative spectrum is asymptotic to a geometric progression with ratio $\exp(-2π/\sqrt{μ_1 - \qtr})$.

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