Mathematics

Let ? denote von Mangoldt's function, and consider the averages A N f(x) = 1 N ??1?�n?�N f(x?�n)?(n). We prove sharp ??p -improving for these averages, and sparse bounds for the maximal function. The simplest inequality is that for sets F,G?�[0,N] there holds N ?? ??A N 1 F , 1 G ?�≪ |F|?�|G| N 2 (Log |F|?�|G| N 2 ) t , where t=2 , or assuming the Generalized Riemann Hypothesis, t=1 . The corresponding sparse bound is proved for the maximal function sup N A N 1 F . The inequalities for t=1 are sharp. The proof depends upon the Circle Method, and an interpolation argument of Bourgain.

We give an asymptotic formula for the number of D 4 quartic extensions of a function field with discriminant equal to some bound, essentially reproducing the analogous result over number fields due Cohen, Diaz y Diaz, and Olivier, but with a stronger error term. We also study the relative density of D 4 and S 4 quartic extensions of a function field and show that with mild conditions, the number of D 4 quartic extensions can far exceed the number of S 4 quartic extensions

Let $A\subset \N_{+}$ and by P A (n) denotes the number of partitions of an integer n into parts from the set A . The aim of this paper is to prove several result concerning the existence of integer solutions of Diophantine equations of the form P A (x)= P B (y) , where A,B are certain finite sets.

We associate a certain tensor product lattice to any primitive integer lattice and ask about its typical shape. These lattices are related to the tangent bundle of Grassmannians and their study is motivated by Peyre's programme on "freeness" for rational points of bounded height on Fano varieties.

Using equidistribution techniques from Arakelov theory as well as recent results obtained by Dimitrov, Gao, and Habegger, we deduce uniform results on the Manin-Mumford and the Bogomolov conjecture. For each given integer g?? , we prove that the number of torsion points lying on a smooth complex algebraic curve of genus g embedded into its Jacobian is uniformly bounded. Complementing other recent work of Dimitrov, Gao, and Habegger, we obtain a rather uniform version of the Mordell-Lang conjecture as well. In particular, the number of rational points on a smooth algebraic curve defined over a number field can be bounded solely in terms of its genus and the Mordell-Weil rank of its Jacobian.

Lafforgue and Genestier-Lafforgue have constructed the global and (semisimplified) local Langlands correspondences for arbitrary reductive groups over function fields. We apply equivariant localization arguments, inspired by work of Treumann-Venkatesh, to moduli spaces of shtukas, in order to prove properties of these correspondences regarding functoriality for cyclic base change. Globally, we establish the existence of functorial transfers of mod p automorphic forms through p -cyclic base change. Locally, we prove that Tate cohomology realizes cyclic base change functoriality in the Genestier-Lafforgue mod p local Langlands correspondence, verifying a function field version of a conjecture of Treumann-Venkatesh. The proofs draw upon new tools from representation theory, including parity sheaves and Smith-Treumann theory. In particular, we use these to establish a categorification of the base change homomorphism for Hecke algebras, in a joint appendix with Gus Lonergan.

Andrews' (k,i) -singular overpartition function C ¯ ¯ ¯ ¯ k,i (n) counts the number of overpartitions of n in which no part is divisible by k and only parts ?�±i(modk) may be overlined. In recent times, divisibility of C ¯ ¯ ¯ ¯ 3????(n) , C ¯ ¯ ¯ ¯ 4????(n) and C ¯ ¯ ¯ ¯ 6????(n) by 2 and 3 are studied for certain values of ??. In this article, we study divisibility of C ¯ ¯ ¯ ¯ 3????(n) , C ¯ ¯ ¯ ¯ 4????(n) and C ¯ ¯ ¯ ¯ 6????(n) by primes p?? . For all positive integer ??and prime divisors p?? of ??, we prove that C ¯ ¯ ¯ ¯ 3????(n) , C ¯ ¯ ¯ ¯ 4????(n) and C ¯ ¯ ¯ ¯ 6????(n) are almost always divisible by arbitrary powers of p . For s?�{3,4,6} , we next show that the set of those n for which C ¯ ¯ ¯ ¯ s?��?,??(n)??0(mod p k i ) is infinite, where k is a positive integer satisfying p k?? i ?��? . We further improve a result of Gordon and Ono on divisibility of ??-regular partitions by powers of certain primes. We also improve a result of Ray and Chakraborty on divisibility of ??-regular overpartitions by powers of certain primes.

In Cogdell et al., \it LMS Lecture Notes Series \bf 459, \rm 393--427 (2020), \rm the authors proved an analogue of Kronecker's limit formula associated to any divisor D which is smooth in codimension one on any smooth Kähler manifold X . In the present article, we apply the aforementioned Kronecker limit formula in the case when X is complex projective space $\CC\PP^n$ for n?? and D is a hyperplane, meaning the divisor of a linear form P D (z) for ${z} = (\mathcal{Z}_{j}) \in \CC\PP^n$. Our main result is an explicit evaluation of the Mahler measure of P D as a convergent series whose each term is given in terms of rational numbers, multinomial coefficients, and the L 2 -norm of the vector of coefficients of P D .

In the spirit of Lehmer's speculation that Ramanujan's tau-function never vanishes, it is natural to ask whether any given integer α is a value of ?(n) . For odd α , Murty, Murty, and Shorey proved that ?(n)?��?for sufficiently large n . Several recent papers have identified explicit examples of odd α which are not tau-values. Here we apply these results (most notably the recent work of Bennett, Gherga, Patel, and Siksek) to offer the first examples of even integers that are not tau-values. Namely, for primes ??we find that ?(n)?�{±2??: 3?��?<100}?�{±2 ??2 : 3?��?<100}?�{±2 ??3 : 3?��?<100 \rm with ?��?59}. Moreover, we obtain such results for infinitely many powers of each prime 3?��?<100 . As an example, for ??97 we prove that ?(n)?�{2??97 j : 1?�j??0(mod44)}?�{????97 j : j??}. The method of proof applies mutatis mutandis to newforms with residually reducible mod 2 Galois representation and is easily adapted to generic newforms with integer coefficients.

Let p be a prime number and let k be an algebraically closed field of characteristic p . A B T 1 group scheme over k is a finite commutative group scheme which arises as the kernel of p on a p -divisible (Barsotti--Tate) group. Our main result is that every B T 1 scheme group over k occurs as a direct factor of the p -torsion group scheme of the Jacobian of an explicit curve defined over F p . We also treat a variant with polarizations. Our main tools are the Kraft classification of B T 1 group schemes, a theorem of Oda, and a combinatorial description of the de Rham cohomology of Fermat curves.