Related papers: Subconvexity for $L$-Functions on $\mathrm{GL}_3$ …

We prove an asymptotic formula for a special case of the Gauss hypergeometric function which arises in explicit formulas for moments of Maass form symmetric square L-functions. The resulting formula is uniform in several variables, which is…

Assume that the generalized Ramanujan conjecture holds on the automorphic $L$-function $L(s, \pi)$ on $\GL_d$ over $\mathbb{Q}$ with $d\geq 3$, we can obtain a small log-saving non-trivial bound on the second integral moment of $L(1/2+it,…

The main result of this article proves the nonvanishing of cuspidal cohomology for $GL(n)$ over a number field which is Galois over its maximal totally real subfield. The proof uses the internal structure of a strongly-pure weight that can…

Given a Dirichlet character $\chi$ modulo $q$ and its associated $L$-function, $L(s,\chi)$, we provide an explicit version of Burgess' estimate for $|L(s, \chi)|$. We use partial summation to provide bounds along the vertical lines $\Re{s}…

We study Artin $L$-functions on a finite $2$-dimensional complex $X_\Gamma$ arising from PGL$_3$ attached to finite-dimensional representations $\rho$ of its fundamental group. Some key properties, such as rationality, functional equation,…

We derive integral representations for the Rankin-Selberg L-functions on GL(3) x GL(1) and GL(3) x GL(2) by a process of unipotent averaging at archimedean places. A key feature of our result is that it allows one to fix the choice of test…

Let $L$ be a solvable Lie algebra of dimension less than or equal to 4 over finite fields. We compute and record, in explicit symbolic form, the zeta functions enumerating subalgebras or ideals of $L$, and study their properties. We also…

We consider Galilean limit of conformal algebra for spin-2 and spin-3 fields, and study the gauge theory of these algebras. We analyze the equations of motion and obtain the spectra of the theories.

We present an overview of bounds on zeros of $L$-functions and obtain some improvements under weak conjectures related to the Goldbach problem.

In this paper, we obtain a subconvexity result for the Rankin-Selberg L-function in both levels. The new feature in this result is applying an amplification method of Duke-Friedlander-Iwaniec to a double Petersson-Kuznetsov trace formula.…

The problem of finding upper bounds for L-functions at the edge of the critical strip has a long and interesting history. Here, the situation for classical L-functions such as Dirichlet L-functions is relatively well understood. The reason…

We generalize our previous method on subconvexity problem for $\mathrm{GL}_2 \times \mathrm{GL}_1$ with cuspidal representations to Eisenstein series, and deduce a Burgess-like subconvex bound for Hecke characters, i.e., the bound…

Fix $n \geq 2$ an integer, and let $F$ be a totally real number field. We derive estimates for the finite parts of the $L$-functions of irreducible cuspidal $\operatorname{GL}_n({\bf{A}}_F)$-automorphic representations twisted by class…

In three dimensions, an abelian gauge field is related by duality to a free, periodic scalar field. Though usually considered on Euclidean space, this duality can be extended to a general three-manifold M, in which case topological features…

In this paper we prove a hybrid subconvexity bound for class group $L$-functions associated to a quadratic extension $K/\mathbb{Q}$ (real or imaginary). Our proof relies on relating the class group $L$-functions to Eisenstein series…

We formulate for function fields an analog of Serre's conjecture on the modularity of 2-dimensional irreducible mod l representations of the absolute Galois group of Q: our analog is not restricted to 2-dimensional represntations. While the…

We show that M\"uller's $L\log L$ bound $$F(Du)\geq 0,\,Du\in L^p_{\mathrm{loc}}(\mathbb{R}^n)\implies F(Du)\in L\log L_{\mathrm{loc}}(\mathbb{R}^n)$$ for $F =\det$ and $p=n$ holds for quasiconcave $F$ which are homogeneous of degree $p>1$.…

We give a soft proof of a uniform upper bound for the local factors in the triple product formula, sufficient for deducing effective and general forms of quantum unique ergodicity (QUE) from subconvexity.

In this paper, we extend the results of Michel-Venkatesh and Hu-Michel-Nelson to establish an upper bound for triple product and Rankin-Selberg L-functions of the form $$L(\pi_1 \otimes \pi_2 \otimes…

We improve upon the local bound in the depth aspect for sup-norms of newforms on $D^\times$ where $D$ is an indefinite quaternion division algebra over $\mathbb{Q}$. Our sup-norm bound implies a depth-aspect subconvexity bound for $L(1/2, f…