Higher rank stable pairs on K3 surfaces

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 \cite{shesh1,shesh2} using moduli of pairs of the form \O^n\into \F for \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 \O^n\into \E on a \K3 surface for \E an arbitrary stable sheaf of a fixed numerical type ("coherent systems" in the language of \cite{KY}) whose first Chern class is primitive, and fully compute them geometrically. The ordinary stable pair theory of \K3 surfaces is treated by \cite{MPT}; 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

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

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

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

p-adic families and Galois representations for GS_p(4) and GL(2)

In this brief paper, 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 paper 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 GL(2) over a quadratic imaginary field and prove results on the local Galois representation ℓ, as well as crystallinity results at p

p-ADIC FAMILIES AND GALOIS REPRESENTATIONS FOR GSp(4) AND GL(2)

Abstract. In this brief article 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 GL(2) over a quadratic imaginary field and prove results on the local Galois representation ℓ, as well as crystallinity results at p

Galois representations for holomorphic Siegel modular forms

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 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 GSp(4) over p-adic fields; then we use the existence of a globally generic Hilbert–Siegel modular form weakly equivalent to the original and we refer to Sorensen (Mathematica 15:623–670, 2010) for local–global compatibility in that case