Kimball Martin

On central critical values of the degree four L-functions for GSp (4): the fundamental lemma, II

We propose a new relative trace formula concerning the central critical values of the spinor

Test vectors and central L-values for GL(2)

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Periods and nonvanishing of central L-values for GL(2n)

-functions for GSp (4). The main result is a proof of the fundamental lemma for the unit element of the Hecke algebra. Our new relative trace formula has some significant advantages over the previous ones for the subsequent development.

A RELATIVE TRACE FORMULA FOR A COMPACT RIEMANN SURFACE

We determine local test vectors for Waldspurger functionals for GL2, in the case where both the representation of GL2 and the character of the degree two extension are ramied, with certain restrictions. We use this to obtain an explicit version of Waldspurger’s formula relating twisted central L-values of automorphic representations on GL2 with certain toric period integrals. As a consequence, we generalize an average value formula of Feigon and Whitehouse, and obtain some nonvanishing results. 1

Distinguishing graphs with zeta functions and generalized spectra

Let π be a cuspidal automorphic representation of PGL(2n) over a number field F, and η the quadratic idèle class character attached to a quadratic extension E/F. Guo and Jacquet conjectured a relation between the nonvanishing of L(1/2, π)L(1/2, π ⊗ η) for π of symplectic type and the nonvanishing of certain GL(n,E) periods. When n = 1, this specializes to a well-known result of Waldspurger. We prove this conjecture, and related global results, under some local hypotheses using a simple relative trace formula.We then apply these global results to obtain local results on distinguished supercuspidal representations, which partially establish a conjecture of Prasad and Takloo-Bighash.

Nonunique factorization and principalization in number fields

We study a relative trace formula for a compact Riemann surface with respect to a closed geodesic

Shalika periods on GL2(D) and GL4

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Shalika periods on GL 2 (D) and GL 4

. This can be expressed as a relation between the period spectrum and the ortholength spectrum of

A SYMPLECTIC CASE OF ARTIN'S CONJECTURE

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On central critical values of the degree four \(L\)-functions for \({\text {GSp}}\left( 4\right) \): a simple trace formula

. This provides a new proof of asymptotic results for both the periods of Laplacian eigenforms along