arXiv subjects

Valentin Blomer

Publications and source records attributed to Valentin Blomer.

At least 19 recordsLinked to original sources

arXiv2026

Bilinear forms with Kloosterman sums via quadratic characters

We prove new bounds for bilinear forms with Kloosterman sums, valid for all moduli c. In the critical range where the summation length is the square root of the modulus, the saving over the trivial bound is c^{-1/32}, improving on all previous approaches even for prime moduli. This is based on a new connection to quadratic character sums. Applications to moments of twisted L-functions and to the large sieve for exceptional Maass forms are given.

math.NT↗

arXiv2026

Moments and joint nonvanishing of symplectic $L$-functions

We compute an asymptotic formula for a moment involving the spinor and the standard $L$-functions for holomorphic Siegel cusp forms of degree two and large weight $k$. Applications include simultaneous non-vanishing statements and lower bounds for second moments.

math.NT↗

arXiv2026

Joint Linnik problems

We prove a conjecture of Michel--Venkatesh on joinings of distinct Linnik problems, in the setting of simultaneous quaternionic embeddings of imaginary quadratic fields having sufficiently many small split primes. This splitting condition is known to hold for all but $O((\log\log X)^{1 + o(1)})$ discriminants up to $X$. We also treat a non-equivariant form of this conjecture proposed by Aka--Einsiedler--Shapira, which in particular applies to the classical Gauss construction joining Linnik points on the sphere with CM points on the modular surface.

math.NT↗

arXiv2026

The cost of cyclic permutations and remainder sums in the Euclidean algorithm

We discuss a modification to the Gries-Mills block swapping scheme for in-place rotation with average costs of 1.85 moves per element and worst case performance still at 3 moves per element. Analysis of the average case relies on the asymptotic behavior of the sum of remainders in the Euclidean algorithm.

cs.DS↗

arXiv2025

A higher rank shifted convolution problem with applications to L-functions

While several instances of shifted convolution problems for GL(3) x GL(2) have been solved, the case where one factor is the classical divisor function and one factor is a GL(3) Fourier coefficient has remained open. We solve this case in the present paper. The proof involves two intertwined applications of different types of delta symbol methods. As an application we establish an asymptotic formula for central values of L-functions for a GL(3) automorphic form twisted by Dirichlet characters to moduli q < Q.

math.NT↗

arXiv2025

Analytic properties of representation zeta functions of groups of type $\mathsf{A_2}$

We study analytic properties of the representation zeta functions of arithmetic groups of type $\mathsf{A}_2$, such as $\textrm{SL}_3(\mathbb{Z})$. In particular, we uncover further poles of these functions and determine a natural boundary for their meromorphic continuation beyond their abscissa of convergence. We analyse both the number field and function field case.

math.NT↗

arXiv2025

Generation of Hecke fields by squares of cyclotomic twists of modular $L$-values

Let $f$ be a non-CM elliptic newform without a quadratic inner twist, $p$ an odd prime and $\chi$ a Dirichlet character of $p$-power order and sufficiently large $p$-power conductor. We show that the compositum $\mathbb{Q}_{f}(\chi)$ of the Hecke fields associated to $f$ and $\chi$ is generated by the square of the absolute value of the corresponding central $L$-value $L^{\rm alg}(1/2, f \otimes \chi)$ over $\mathbb{Q}(\mu_p)$. The proof is based among other things on techniques used for the recent resolution of unipotent mixing conjecture by the first and third named authors.

math.NT↗

arXiv2024

Triple sums of Kloosterman sums and the discrepancy of modular inverses

We investigate the distribution of modular inverses modulo positive integers $c$ in a large interval. We provide upper and lower bounds for their box, ball and isotropic discrepancy, thereby exhibiting some deviations from random point sets. The analysis is based, among other things, on a new bound for a triple sum of Kloosterman sums.

math.NT↗

arXiv2024

Zeros of $L$-functions in families near the critical line

We combine the relative trace formula with analytic methods to obtain zero density estimate for $L$-functions in various families of automorphic representations for $\mathrm{GL}(m)$. Applications include strong bounds for the average analytic rank of these $L$-functions at the central point and average equidistribution results for the imaginary parts of the zeros.

math.NT↗

arXiv2024

Moments of symmetric square L-functions on GL(3)

We give an asymptotic formula with power saving error term for the twisted first moment of symmetric square L-functions on GL(3) in the level aspect. As applications, we obtain non-vanishing results as well as lower bounds of the expected order of magnitude for all even moments, supporting the random matrix model for a unitary ensemble. Besides the GL(3) Kuznetsov formula, the ingredients include detailed local computations at ramified places, including root numbers and orthonormalization of oldforms and Eisenstein series.

math.NT↗

arXiv2024

Local integral transforms and global spectral decomposition

We establish an explicit global spectral decomposition of shifted convolution sums and the second moment of automorphic $L$-functions for Maass forms with explicit integral transforms as well as explicit inversion formulae over every local field.

math.NT↗

arXiv2024

Local analysis of the Kuznetsov formula and the density conjecture

We prove Sarnak's spherical density conjecture for the principal congruence subgroup of SL(n, Z) of arbitrary level. Applications include a complete version of Sarnak's optimal lifting conjecture for principal congruence subgroups of SL(n, Z), as well as a transfer of the density theorem to certain co-compact situations. The main ingredients are new lower bounds for Whittaker functions and strong estimates for the cardinality of ramified Kloosterman sets.

math.NT↗

arXiv2024

A GL(3) converse theorem via a "beyond endoscopy" approach

We give a new proof of the converse theorem for Maass forms on ${\rm GL}(3)$ using a technique that is inspired by Langlands' philosophy of "beyond endoscopy", thereby implementing these ideas for the first time in a higher rank setting.

math.NT↗

arXiv2023

Hyperbolic lattice point counting in unbounded rank

We use spectral analysis to give an asymptotic formula for the number of matrices in SL(n, Z) of height at most T with strong error terms, far beyond the previous known, both for small and large rank.

math.NT↗

arXiv2023

The unipotent mixing conjecture

We show that shifted pairs of discrete or continuous low-lying horocycles equidistribute in the product space of two modular curves.

math.NT↗

arXiv2022

The mixing conjecture under GRH

We prove the Mixing Conjecture of Michel--Venkatesh for the class group action on Heegner points of large discriminant on compact arithmetic surfaces attached to maximal orders in rational quaternion algebras. The proof is conditional on the Generalized Riemann Hypothesis, and when the division algebra is indefinite we furthermore assume the Ramanujan conjecture. Our methods, which provide an effective rate, are based on the spectral theory of automorphic forms and their $L$-functions, as well as techniques in classical analytic number theory.

math.NT↗

arXiv2022

Correlations of values of random diagonal forms

We study the value distribution of diagonal forms in k variables and degree d with random real coefficients and positive integer variables, normalized so that mean spacing is one. We show that the l-correlation of almost all such forms is Poissonian when k is large enough depending on l and d.

math.NT↗

arXiv2022

The density conjecture for principal congruence subgroups

We prove Sarnak's density conjecture for the principal congruence subgroup of SL_n(Z) of squarefree level and discuss various arithmetic applications. The ingredients include new bounds for local Whittaker functions and Kloosterman sums.

math.NT↗