186 papers
This paper introduces canonical classes in Selmer groups, derived from special cycles in unitary Shimura varieties and generalizing Heegner points, which are conjectured to play a central role in verifying the Beilinson–Bloch–Kato conjectures in rank 1.
This paper establishes a purity theorem for -adic local systems on rigid-analytic varieties with semistable formal models by proving that such systems are semistable if and only if their restrictions to the irreducible components of the special fiber are semistable, utilizing analytic prismatic -crystals and a newly derived prismatic purity theorem.
This paper generalizes a recent result by Alladi and Johnson by proving that for any set of prime numbers with a natural density, the weighted sum of the Möbius and distinct prime factor functions over integers whose smallest prime factor lies in that set converges to zero, while also providing an effective estimate for the rate of this convergence.
This paper establishes a p-adic analogue of the arithmetic Gan-Gross-Prasad conjectures for unitary groups by constructing a cyclotomic p-adic L-function and proving a formula linking its first derivative to p-adic heights of Selmer classes derived from arithmetic diagonal cycles, thereby advancing the p-adic Beilinson-Bloch-Kato conjecture.
This paper establishes effective versions of Bilu's equidistribution theorem for Galois orbits of small-height points in the -dimensional algebraic torus by developing a general Fourier analysis framework that quantifies convergence rates based on the regularity of mildly regular test functions, thereby extending prior results by Petsche and D'Andrea et al.
This paper constructs canonical sets of right coset representatives for the congruence subgroups , , and to prove that their corresponding fundamental domains are connected, utilizing a study of the projective line and a multiplicity function that is shown to be one less than a more computable function .
This paper further investigates the function used to construct a canonical connected fundamental domain for , establishing identities to match its cusps with known classes and detailing the domain's boundary arcs and gluing patterns to aid in understanding the modular curve .
This paper derives asymptotic formulas for the sums of the smallest integer such that is divisible by , considering both all integers up to and the subset of -free integers.
This paper clarifies the ring-theoretic and homological properties of the tilt of extensions between perfectoid rings, building upon Faltings' almost étale extensions and Scholze's tilting operations to advance the construction of big Cohen-Macaulay algebras.
This paper investigates the natural action of the Galois group on the class group of a Galois number field through a direct approach, utilizing the class groups of overrings and the arithmetic of normsets to elucidate the relationship between these algebraic structures.
This paper demonstrates that the optimal homothetic quadratic discrepancy for planar convex bodies does not necessarily follow a single order of growth, but can instead exhibit prescribed polynomial-order oscillations between logarithmic and square-root rates depending on the geometry of the boundary.
This paper computes minimal zero-free regions for the Riemann zeta-function to prove the existence of a prime between consecutive perfect th powers for , specifically establishing this result for and a specific subset of integers for , while quantifying the progress toward Legendre's conjecture.
This paper verifies the local analogue of Jiang's conjecture regarding the upper bound of geometric wavefront sets for Arthur type representations of split classical -adic groups with large , thereby establishing related upper bound conjectures by combining Waldspurger's endoscopic transfer with recent wavefront set computations.
This paper generalizes the Duffin–Schaeffer theorem to real residue systems by establishing a zero-one law for inhomogeneous approximation based on the parameter's membership in specific sets, utilizing novel results on the distribution of primes in Bohr sets and the equidistribution of circle rotations.
This paper provides concise proofs for recent permutation polynomial results and establishes a general framework for constructing complete permutation polynomials over finite fields by combining Zieve's fiber criterion with the AGW criterion, specifically utilizing fiber decompositions over the cube roots of unity.
This paper demonstrates that for asymptotic bases of order 2 with weaker representation function growth rates, the properties of having a divergent representation function, being decomposable into two bases, and containing a minimal basis are independent, thereby refining earlier results by Erdős and Nathanson through an inductive construction on exponentially growing intervals.
This paper determines the nonemptiness and classifies the genera of isomorphism classes of polarized superspecial abelian varieties over with specific Frobenius and polarization kernel conditions by reducing the problem to the arithmetic study of hermitian lattices.
This paper proves Lubin's conjecture in new cases by demonstrating that over -tame extensions, the consistent sequences associated with a commuting pair of noninvertible and invertible height-one formal power series form a crystalline character of weight 1.
This paper investigates the multifractal properties of the exceptional set in the Lüroth expansion where the maximal run-length function exhibits linear growth, specifically establishing the Hausdorff dimension for sets defined by prescribed lower and upper limits of the ratio between the run-length and the number of digits.
This paper classifies the -adic Galois images of elliptic curves over with mod non-split Cartan images using -adic Hodge theory, providing a novel algorithm for the potentially supersingular case and deriving global consequences for elliptic curves over that sharpen bounds on their adelic images.