Dinakar Ramakrishnan

Modularity of the Rankin-Selberg L-series, and multiplicity one for SL(2)

Let f, g be primitive cusp forms, holomorphic or otherwise, on the upper half-plane H of levels N,M respectively, with (unitarily normalized) L-functions L(s, f) = [equation] and L(s, g) = [equation]. When p does not divide N (resp. M), the inverse roots αp, βp (resp. α′p, β′p ) are nonzero with sum ap (resp. bp). For every p prime to NM, set Lp(s, f × g) = [(1 − αpα′pp−s)(1 − αpβ′pp−s)(1 − βpα′pp−s)(1 − βpβ′pp−s)]^−1. Let L∗(s, f × g) denote the (incomplete Euler) product of Lp(s, f × g) over all p not dividing NM. This is closely related to the convolution L-series [sum over n≥1] a[sub]n b[sub] n n^−s, whose miraculous properties were first studied by Rankin and Selberg.

Modularity of solvable Artin representations of GO(4)-type

This is an updated version of ANT-0253. Let F be a number field with absolute Galois group G. We associate, to each continuous, solvable C-representation of G of GO(4)-type, an automorphic form P of GL(4)/F with the same L-function. As a consequence we exhibit an infinite class of primitive, 16-dimensional representations for which the Artin conjecture holds.

On the Exceptional Zeros of Rankin–Selberg L-Functions

The main objects of study in this article are two classes of Rankin–Selberg L-functions, namely L(s,f×g) and L(s, sym2(g)× sym2(g)), where f,g are newforms, holomorphic or of Maass type, on the upper half plane, and sym2(g) denotes the symmetric square lift of g to GL(3). We prove that in general, i.e., when these L-functions are not divisible by L-functions of quadratic characters (such divisibility happening rarely), they do not admit any LandauSiegel zeros. Such zeros, which are real and close to s=1, are highly mysterious and are not expected to occur. There are corollaries of our result, one of them being a strong lower bound for special value at s=1, which is of interest both geometrically and analytically. One also gets this way a good bound onthe norm of sym2(g).

A cuspidality criterion for the functorial product on GL(2) × GL(3) with a cohomological application

For all cusp forms π on GL(3) and π′ on GL(2) over a number field F, H. Kim and F. Shahidi have functorially associated an automorphic form Formula on GL(6) such that L(s, Π) agrees with the Rankin-Selberg L-function of the pair (π, π′). First we establish a criterion as to when Π is cuspidal. Then we apply it to construct non-self-dual, nonmonomial cuspidal cohomology classes for suitable congruence subgroups of SL(3, ℤ). We also analyze the Galois image of certain related l-adic representations.

Self-dual representations of division algebras and Weil groups: A contrast

Irreducible selfdual representations of any group fall into two classes: those which carry a symmetric bilinear form, and the others which carry an alternating bilinear form. The Langlands correspondence, which matches the irreducible representations

Consequences of the Gross–Zagier formulae: Stability of average L-values, subconvexity, and non-vanishing mod p

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Irreducibility and Cuspidality

of the Weil group of a local field

Lifting orthogonal representations to spin groups and local root numbers

k

Increasing the number of fibered faces of arithmetic hyperbolic 3-manifolds

of dimension

Eisenstein Series of Weight One, q-Averages of the 0-Logarithm and Periods of Elliptic Curves

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