arXiv subjects
Publications and source records attributed to Eknath Ghate.
We establish a rationality result for the twisted Asai L-values attached to a Bianchi cusp form and construct distributions interpolating these L-values. Using the method of abstract Kummer congruences, we then outline the main steps needed to show that these distributions come from a measure.
We make a zig-zag conjecture describing the reductions of irreducible crystalline two-dimensional representations of $G_{\mathbb{Q}_p}$ of half-integral slopes and exceptional weights. Such weights are two more than twice the slope mod $(p-1)$. We show that zig-zag holds for half-integral slopes at most $\frac{3}{2}$. We then explore the connection between zig-zag and local constancy results in the weight. First we show that known cases of zig-zag force local constancy to fail for small weights. Conversely, we explain how local constancy forces zig-zag to fail for some small weights and half-integral slopes at least $2$. However, we expect zig-zag to be qualitatively true in general. We end with some compatibility results between zig-zag and other results.
We describe Panchishkin's construction of the $p$-adic Rankin product $L$-function.
We compute the reductions of irreducible crystalline two-dimensional representations of $G_{\mathbf{Q}_p}$ of slope 1, for primes $p \geq 5$, and all weights. We describe the semisimplification of the reductions completely. In particular, we show that the reduction is often reducible. We also investigate whether the extension obtained is peu or très ramifiée, in the relevant reducible non-semisimple cases. The proof uses the compatibility between the $p$-adic and mod $p$ Local Langlands Correspondences, and involves a detailed study of the reductions of both the standard and non-standard lattices in certain $p$-adic Banach spaces.
We solve an elementary number theory problem on sums of fractional parts, using methods from group theory. We apply our result to deduce the finiteness of certain monodromy representations.
We describe the semi-simplification of the mod $p$ reduction of certain crystalline two dimensional local Galois representations of slopes in the interval $(1,2)$ and all weights. The proof uses the mod $p$ Local Langlands Correspondence for $GL_2(Q_p)$. We also give a complete description of the submodules generated by the second highest monomial in the mod $p$ symmetric power representations of $GL_2(F_p)$.
We formulate a question regarding uniform versions of "large Galois image properties" for modular abelian varieties of higher dimension, generalizing the well-known case of elliptic curves. We then answer our question affirmatively in the exceptional image case, and provide lower estimates for uniform bounds in the remaining cases.