Irreducible adjoint representations in prime power dimensions, with an application

We construct an infinite family of representations of finite groups with an irreducible adjoint action and we give an application to the question of lacunary of Frobenius traces in Galois representations.Comment: To appear in the Journal of the Ramanujan Mathematical Societ

The Average Number OF Divisors in Certain Arithmetic Sequences

In this paper we study the sum p≤xτ(np), where τ(n) denotes the number of divisors of n, and {np} is a sequence of integers indexed by primes. Under certain assumptions we show that the aforementioned sum is x as x → ∞. As an application, we consider the case where the sequence is given by the Fourier coefficients of a modular form

The trace of T2T_2 takes no repeated values

We prove that the trace of the Hecke operator $T_2$ acting on the vector space of cusp forms of level one takes no repeated values, except for 0, which only occurs when the space is trivial.Comment: To appear in Indagationes Mathematica

Summing Hecke Eigenvalues over Polynomials

In this paper we estimate sums of the form ∑n≤X|aSymmπ(|f(n)|)|, for symmetric power lifts of automorphic representations π attached to holomorphic forms and polynomials f(x)∈Z[x] of arbitrary degree. We give new upper bounds for these sums under certain natural assumptions on f. Our results are unconditional when deg(f)≤4. Moreover, we study the analogous sum over polynomials in several variables. We obtain an estimate for all cubic polynomials in two variables that define elliptic curves

The Trace of T2 Takes No Repeated Values

We prove that the trace of the Hecke operator T2 role= presentation style= box-sizing: border-box; margin: 0px; padding: 0px; display: inline-block; line-height: normal; font-size: 16.2px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative; \u3eT2 acting on the vector space of cusp forms of level one takes no repeated values, except for 0, which only occurs when the space is trivial

Follow-up analysis of federal process of care data reported from three acute care hospitals in rural Appalachia

Background: This investigation evaluated standardized process of care data collected on selected hospitals serving a remote rural section of westernmost North Carolina. Methods: Centers for Medicare and Medicaid Services data were analyzed retrospectively for multiple clinical parameters at Fannin Regional Hospital, Murphy Medical Center, and Union General Hospital. Data were analyzed by paired t-test for individual comparisons among the three study hospitals to compare the three facilities with each other, as well as with state and national average for each parameter. Results: Centers for Medicare and Medicaid Services “Hospital Compare” data from 2011 showed Fannin Regional Hospital to have significantly higher composite scores on standard­ized clinical process of care measures relative to the national average, compared with Murphy Medical Center (P = 0.01) and Union General Hospital (P = 0.01). This difference was noted to persist when Fannin Regional Hospital was compared with Union General Hospital using common state reference data (P = 0.02). When compared with national averages, mean process of care scores reported from Murphy Medical Center and Union General Hospital were both lower but not significantly different (−3.44 versus −6.07, respectively, P = 0.54). Conclusion: The range of process of care scores submitted by acute care hospitals in western North Carolina is considerable. Centers for Medicare and Medicaid Services “Hospital Compare” information suggests that process of care measurements at Fannin Regional Hospital are sig­nificantly higher than at either Murphy Medical Center or Union General Hospital, relative to state and national benchmarks. Further investigation is needed to determine what impact these differences in process of care may have on hospital volume and/or market share in this region. Additional research is planned to identify process of care trends in this demographic and geographically rural area

On the Equality Case of the Ramanujan Conjecture for Hilbert Modular Forms

The generalized Ramanujan Conjecture for cuspidal unitary automorphic representations π on GL(2) asserts that |av(π)| ≤ 2. We prove that this inequality is strict if π is generated by a CM Hilbert modular form of parallel weight two and v is a finite place of degree one. Equivalently, the Satake parameters of πv are necessarily distinct. We also give examples where the equality case does occur for primes of degree two

Special Frobenius Traces in Galois Representations

This thesis studies Frobenius traces in Galois representations from two different directions. In the first problem we explore how often they vanish in Artin-type representations. We give an upper bound for the density of the set of vanishing Frobenius traces in terms of the multiplicities of the irreducible components of the adjoint representation. Towards that, we construct an infinite family of representations of finite groups with an irreducible adjoint action. In the second problem we partially extend for Hilbert modular forms a result of Coleman and Edixhoven that the Hecke eigenvalues ap of classical elliptical modular newforms f of weight 2 are never extremal, i.e., ap is strictly less than 2[square root]p. The generalization currently applies only to prime ideals p of degree one, though we expect it to hold for p of any odd degree. However, an even degree prime can be extremal for f. We prove our result in each of the following instances: when one can move to a Shimura curve defined by a quaternion algebra, when f is a CM form, when the crystalline Frobenius is semi-simple, and when the strong Tate conjecture holds for a product of two Hilbert modular surfaces (or quaternionic Shimura surfaces) over a finite field.</p

Comparing Hecke Coefficients of Automorphic Representations

We prove a number of unconditional statistical results of the Hecke coefficients for unitary cuspidal representations of over number fields. Using partial bounds on the size of the Hecke coefficients, instances of Langlands functoriality, and properties of Rankin-Selberg -functions, we obtain bounds on the set of places where linear combinations of Hecke coefficients are negative. Under a mild functoriality assumption we extend these methods to . As an application, we obtain a result related to a question of Serre about the occurrence of large Hecke eigenvalues of Maass forms. Furthermore, in the cases where the Ramanujan conjecture is satisfied, we obtain distributional results of the Hecke coefficients at places varying in certain congruence or Galois classes