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The purpose of this paper is to show how the modular method together with different techniques can be used to prove non-existence of primitive non-trivial solutions of the equation x 2 + d y 6 = z p for square-free values 1 ≤ d ≤ 2 0. The key ingredients are: the approach presented in [A. Pacetti and L. V. Torcomian, ℚ -curves, Hecke characters and some Diophantine equations, Math. Comp. 91(338) (2022) 2817–2865] (in particular its recipe for the space of modular forms to be computed) together with the use of the symplectic method (as developed in [E. Halberstadt and A. Kraus, Courbes de Fermat: Résultats et problèmes, J. Reine Angew. Math. 548 (2002) 167–234], although we give a variant over ramified extensions needed in our applications) to discard solutions and the use of a second Frey curve, aiming to prove large image of residual Galois representations. [ABSTRACT FROM AUTHOR]

In this paper, we extend a uniformity result of Dimitrov et al. [Uniformity in Mordell-Lang for curves, Ann. of Math. (2) 194(1) (2021) 237–298] to dimension two and use it to get a uniform bound on the cardinality of the set of all quadratic points for non-hyperelliptic non-bielliptic curves which only depend on the Mordell–Weil rank, the genus of the curve and the degree of the number field. [ABSTRACT FROM AUTHOR]

For K / F a finite Galois extension of number fields, the relative Pólya group Po (K / F) is the subgroup of the ideal class group of K generated by all the strongly ambiguous ideal classes in K / F. The notion of Ostrowski quotient Ost (K / F) , as the cokernel of the capitulation map into Po (K / F) , has been recently introduced in [E. Shahoseini, A. Rajaei and A. Maarefparvar, Ostrowski quotients for finite extensions of number fields, Pacific J. Math. 321 (2022) 415–429, doi: 10.2140/pjm.2022.321.415]. In this paper, using some results of [C. D. González-Avilés, Capitulation, ambiguous classes and the cohomology of the units, J. Reine Angew. Math. 613 (2007) 75–97], we find a new approach to define Po (K / F) and Ost (K / F) which is the main motivation for us to investigate analogous notions in the elliptic curve setting. For E an elliptic curve defined over F , we define the Ostrowski quotient Ost (E , K / F) and the coarse Ostrowski quotient Ost c (E , K / F) of E relative to K / F , for which in the latter group we do not take into account primes of bad reduction. Our main result is a nontrivial structure theorem for the group Ost c (E , K / F) and we analyze this theorem, in some details, for the class of curves E over quadratic extensions K / F. [ABSTRACT FROM AUTHOR]

A Carmichael number is a composite number such that divides for all integers coprime to . Korselt discovered that is a Carmichael number if and only if is square-free and for each prime divisor of . Let , a -number is defined to be a composite number , such that and for each prime . The set of all such that is a -number is called the Korselt set of and we denote this set by . In this paper, we investigate some properties of when is a power of a prime. [ABSTRACT FROM AUTHOR]

In a recent paper, Sun posed six conjectures on the relations between T (a 1 , a 2 , ... , a k ; n) and N (a 1 , a 2 , ... , a k ; n) , where T (a 1 , a 2 , ... , a k ; n) denotes the number of representations of n as a 1 x 1 (x 1 + 1) 2 + a 2 x 2 (x 2 + 1) 2 + ⋯ + a k x k (x k + 1) 2 , where a 1 , a 2 , ... , a k are positive integers, n , x 1 , x 2 , ... , x k are arbitrary nonnegative integers, and N (a 1 , a 2 , ... , a k ; n) denotes the number of representations of n as a 1 x 1 2 + a 2 x 2 2 + ⋯ + a k x k 2 , where this time x 1 , x 2 , ... , x k are integers. In this paper, we prove Sun's six conjectures by using Ramanujan's theta function identities. [ABSTRACT FROM AUTHOR]

The classical Menon's identity [P. K. Menon, On the sum ∑ (a − 1 , n) [ (a , n) = 1 ] , J. Indian Math. Soc. (N.S.) 29 (1965) 155–163] states that ∑ a ∈ ℤ n ∗ gcd (a − 1 , n) = φ (n) σ 0 (n) , where for a positive integer n , ℤ n ∗ is the group of units of the ring ℤ n = ℤ / n ℤ , gcd (,) represents the greatest common divisor, φ (n) is the Euler's totient function and σ k (n) = ∑ d | n d k is the divisor function. In this paper, we generalize Menon's identity with Dirichlet characters in the following way: ∑ a ∈ ℤ n ∗ b 1 , ... , b k ∈ ℤ n gcd (a − 1 , b 1 , ... , b k , n) χ (a) = φ (n) σ k n d , where k is a non-negative integer and χ is a Dirichlet character modulo n whose conductor is d. Our result can be viewed as an extension of Zhao and Cao's result [Another generalization of Menon's identity, Int. J. Number Theory13(9) (2017) 2373–2379] to k > 0. It can also be viewed as an extension of Sury's result [Some number-theoretic identities from group actions, Rend. Circ. Mat. Palermo58 (2009) 99–108] to Dirichlet characters. [ABSTRACT FROM AUTHOR]

In this paper, the convolution sum Σi+25j=n σ(i)σ(j) is evaluated for all n ∈ ℕ based on some identities due to Berndt, Lemire and Williams. [ABSTRACT FROM AUTHOR]

Let D be a positive integer which is not a square. Further, let (u1,v1) be the least positive integer solution of the Pell equation u2−Dv2=1, and let h(4D) denote the class number of binary quadratic primitive forms of discriminant 4D. If D satisfies 2∤D and v1h(4D)≡0 (modD), then D is called an exceptional number. In this paper, under the assumption that there have no exceptional numbers, we prove that the equation xy+yx=z2 has no positive integer solutions (x,y,z) satisfy gcd(x,y)=1 and 2∤xy. [ABSTRACT FROM AUTHOR]

The goal of this paper is to prove several identities for sums of the type where is the number of divisors of and is the number of representations of as a sum of two squares. Some of our identities involve weighted divisor functions, and most of our identities are new. [ABSTRACT FROM AUTHOR]

In this paper, we prove an analog of Ramanujan's "Most Beautiful Identity". This analog is closely related to Ramanujan's beautiful results involving the cubic continued fraction. [ABSTRACT FROM AUTHOR]

We verify the Hardy–Littlewood conjecture on primes in quadratic progressions on average. The results in the present paper significantly improve those of a previous paper by the authors [3]. [ABSTRACT FROM AUTHOR]

Reeder and Yu have recently given a new construction of a class of supercuspidal representations called epipelagic representations [M. Reeder and J.-K. Yu, Epipelagic representations and invariant theory, J. Amer. Math. Soc.27(2) (2014) 437–477, MR 3164986]. We explicitly calculate the Local Langlands Correspondence for certain families of epipelagic representations of unitary groups, following the general construction of Kaletha [Epipelagic L -packets and rectifying characters, Invent. Math.202(1) (2015) 1–89, MR 3402796]. The interesting feature of our computation is that we find simplifications within L -packets of the two novel invariants introduced in the above-mentioned paper of Kaletha, the toral invariant and the admissible L-embedding. [ABSTRACT FROM AUTHOR]

There are two types of Belyi's Theorems for curves defined over finite fields of characteristic p , namely the Wild and the Tame p -Belyi Theorems. In this paper, we discuss them in the language of function fields. In particular, we provide a constructive proof for the existence of a pseudo-tame element introduced in [Y. Sugiyama and S. Yasuda, Belyi's theorem in characteristic two, Compos. Math. 156(2) (2020) 325–339], which leads to a self-contained proof for the Tame 2 -Belyi Theorem. Moreover, we provide unified and simple proofs for Belyi's Theorems unlike the known ones that use technical results from Algebraic Geometry. [ABSTRACT FROM AUTHOR]

In this paper, I generalize Holzer's theorem for semi-diagonal equation. [ABSTRACT FROM AUTHOR]

Let gcd (j , k) be the greatest common divisor of the integers j and k. In this paper, we give several interesting asymptotic formulas for weighted averages of the gcd -sum function f (gcd (j , k)) and the function ∑ d | k , d s | j (f ∗ μ) (d) for any positive integers j and k , namely ∑ k ≤ x 1 k r + 1 ∑ j = 1 k j r f (gcd (j , k)) and ∑ k ≤ x 1 k s (r + 1) ∑ j = 1 k s j r ∑ d | k d s | j (f ∗ μ) (d) with any fixed integer s > 1 and any arithmetical function f. We also establish mean value formulas for the error terms of asymptotic formulas for partial sums of gcd -sum functions f (gcd (j , k)). [ABSTRACT FROM AUTHOR]

For positive integers m , n , k with m ≥ 2 and ⌈ n / 2 ⌉ ≤ k ≤ n , let E { m } (m , n , k) be the sum of multiple zeta values of depth k and weight m n with arguments m or 2 m , i.e. E { m } (m , n , k) = ∑ | α | = n α j = 1 or 2 ζ (m α 1 , m α 2 , ... , m α k). In this paper, we are going to evaluate E { m } (m , n , k). As an application, we produce the stuffle relations from k identical Riemann zeta values ζ (2 m) as well as k identical Riemann zeta values ζ (2 m) and ζ (m). [ABSTRACT FROM AUTHOR]

Let N be a sufficiently large integer. In this paper, we prove that almost all sufficiently large even integer n ∈ [ N − 6 U , N + 6 U ] can be represented as n = p 1 2 + p 2 2 + p 3 3 + p 4 3 + p 5 4 + p 6 4 , p i 2 − N 6 ≤ U , i = 1 , 2 , p i 3 − N 6 ≤ U , i = 3 , 4 , p i 4 − N 6 ≤ U , i = 5 , 6 , where U = N 1 − δ + 𝜀 with δ = 1 / 4 4. [ABSTRACT FROM AUTHOR]

The Minkowski question mark function ? : [ 0 , 1 ] → [ 0 , 1 ] is a continuous, strictly increasing, one-to-one and onto function that has derivative zero almost everywhere. Key to these facts are the basic properties of continued fractions. Thus ? (x) is a naturally occurring number theoretic singular function. This paper generalizes the question mark function to the 216 triangle partition (TRIP) maps. These are multidimensional continued fractions which generate a family of almost all known multidimensional continued fractions. We show for each TRIP map that there is a natural candidate for its analog of the Minkowski question mark function. We then show that the analog is singular for 96 of the TRIP maps and show that 60 more are singular under an assumption of ergodicity. [ABSTRACT FROM AUTHOR]

In this paper, we confirm several conjectures of Sun on Hankel-type determinants for some combinatorial sequences including Franel numbers, Domb numbers and Apéry numbers. For any nonnegative integer n, define fn:=∑k=0nnk3,Dn:=∑k=0nnk22kk2(n−k)n−k,bn:=∑k=0nnk2n+kk,An:=∑k=0nnk2n+kk2. For n=0,1,2,…, we show that 6−n|fi+j|0≤i,j≤n and 12−n|Di+j|0≤i,j≤n are positive odd integers, and 10−n|bi+j|0≤i,j≤n and 24−n|Ai+j|0≤i,j≤n are always integers. [ABSTRACT FROM AUTHOR]

In this paper we investigate the factorization of trinomials of the form xn + cxn-1 + d ∈ ℤ[x]. We then use these results about trinomials to prove results about the factorization of polynomials of the form xn + c(xn-1 +⋯+ x + 1) ∈ ℤ[x]. [ABSTRACT FROM AUTHOR]

A subset A of a given finite abelian group G is called (k,l)-sum-free if the sum of k (not necessarily distinct) elements of A does not equal the sum of l (not necessarily distinct) elements of A. We are interested in finding the maximum size λk,l(G) of a (k,l)-sum-free subset in G. A (2,1)-sum-free set is simply called a sum-free set. The maximum size of a sum-free set in the cyclic group ℤn was found almost 40 years ago by Diamanda and Yap; the general case for arbitrary finite abelian groups was recently settled by Green and Ruzsa. Here we find the value of λ3,1(ℤn). More generally, a recent paper by Hamidoune and Plagne examines (k,l)-sum-free sets in G when k - l and the order of G are relatively prime; we extend their results to see what happens without this assumption. [ABSTRACT FROM AUTHOR]

Let Nq be the number of solutions to the equation \[ (x_1+ \cdots + x_n)^2 = ax_1 \cdots x_n \] over the finite field 픽q = 픽ps. L. Carlitz found formulas for Nq when n = 3 or 4. In an earlier paper, we found formulas for Nq when d = gcd(n - 2, q - 1) = 1 or 2 or 3 or 4; and when there exists an ℓ such that pℓ ≡ -1 (mod d). In our other paper, the cases d = 7 or 14, p ≡ 2 or 4 (mod 7) were considered. Recently, we obtained formulas for Nq when d = 8. In this paper, we find formulas for Nq when d = 2t, t ≥ 4, p ≡ 3 or 5 (mod 8) or p ≡ 9 (mod 16). [ABSTRACT FROM AUTHOR]

We study a recursively defined sequence which is constructed using the least common multiple. Several authors have conjectured that every term of that sequence is or a prime. In this paper we show that this claim is connected to a strong version of Linnik's theorem, which is still unproved. We also study a generalization that replaces the first term by any positive integer. Under this variation some composite numbers may appear now. We give a full characterization of these numbers. [ABSTRACT FROM AUTHOR]

Let f be a Hecke–Maass cusp form of weight zero for S L 2 (ℤ) and λ f (n) be the nth Fourier coefficient. For almost all , we have ∑ n ≤ x λ f 2 (n) e (n) ≪ x 0. 8 1 2 5 + , which improves the result of Acharya [Exponential sums of squares of Fourier coefficients of cusp forms, Proc. Indian Acad. Sci. Math. Sci. 130 (2020) 24], who showed an upper bound larger than x 0. 8 2 9 7. For all α > 1 of type τ < ∞ , we also show that ∑ 1 ≤ n ≤ x λ f 2 ([ α n + β ]) − α − 1 ∑ 1 ≤ n ≤ [ α x + β ] λ f 2 (n) ≪ x 1 3 / 1 6 + + x 1 − 3 / 4 (τ + 1) + , where ℬ α , β : = { [ α n + β ] } n = 1 ∞. This result relies heavily on a generalized double sum (see Theorem 1.3). [ABSTRACT FROM AUTHOR]

Ramanujan's formula for the number r24(n) of representations of a positive integer n as a sum of 24 squares asserts that where and Ramanujan's function τ(n) is given by In this paper, we determine a class of formulae analogous to Ramanujan's formula for r24(n). [ABSTRACT FROM AUTHOR]

In this short paper, it is proved that every pair of large positive odd integers satisfying some necessary conditions can be represented in the form of a pair of one prime, two prime squares and k powers of 2. In particular, we have k -- 332. [ABSTRACT FROM AUTHOR]

We give a q -analogue of the supercongruence (I.2) of Van Hamme. As a conclusion, we confirm a recent conjecture of Swisher. We also give a q -analogue of the corresponding π series (I.1) along with some similar results. [ABSTRACT FROM AUTHOR]

The goal of this paper is to construct explicitly a p-adic family of representations (which are dihedral representations), to construct their attached (φ, Γ)-modules by writing down explicit matrices for φ and for the action of Γ, and finally to determine which of these are trianguline. [ABSTRACT FROM AUTHOR]

In this paper, we resolve a problem raised by Nathanson concerning the maximal solutions to functional equations naturally arising from multiplication of quantum integers ([4]). Together with our results obtained in [11], which treats the case where the field of coefficients is ℚ, this provides a complete description of the maximal solutions to these functional equations and their support bases P in characteristic zero setting. [ABSTRACT FROM AUTHOR]

In this paper, we will prove a Weyl law for the modular surface weighted by the first Fourier coefficient of the Maass cusp forms. Our error term corresponds to the best known error term in the Weyl law and improves a previous result of Kuznetsov. [ABSTRACT FROM AUTHOR]

In this paper, we study certain asymptotic properties of global fields. We consider the set of Tsfasman-Vlăduţ invariants of infinite global fields and answer some natural questions arising from their work. In particular, we prove the existence of infinite global fields having finitely many strictly positive invariants at given places, and the existence of infinite number fields with certain prescribed invariants being zero. We also give precisions on the deficiency of infinite global fields and on the primes decomposition in those fields. [ABSTRACT FROM AUTHOR]

Let E be an elliptic curve over ℚ. We know that the ring of endomorphisms of its reduction modulo an ordinary prime p is an order of the quadratic imaginary field generated by the Frobenius element πp. However, except in the trivial case of complex multiplication, very little is known about the fields that appear as algebras of endomorphisms when p varies. In this paper, we study the endomorphism ring by looking at the arithmetic of $a_p^2-4p$, the discriminant of the characteristic polynomial of πp. In particular, we give a precise asymptotic for the function counting the number of primes p up to x such that $a_p^2-4p$ is square-free and in certain congruence class fixed a priori, when averaging over elliptic curves defined over the rationals. We discuss the relation of this result with the Lang-Trotter conjecture, and some other questions on the curve modulo p. [ABSTRACT FROM AUTHOR]

An elementary but useful fact is that the numerator of the difference of two consecutive Farey fractions is equal to one. For triples of consecutive fractions, the numerators of the differences are well understood and have applications to several interesting problems. In this paper, we investigate numerators of differences of fractions which are farther apart. We establish algebraic identities between such differences which then allow us to calculate their average values by using properties of a measure preserving transformation of the Farey triangle. [ABSTRACT FROM AUTHOR]

In this paper, we study the Diophantine equation x2 + C = 2yn in positive integers x,y with gcd(x,y) = 1, where n ≥ 3 and C is a positive integer. If C ≡ 1 (mod 4), we give a very sharp bound for prime values of the exponent n; our main tool here is the result on existence of primitive divisors in Lehmer sequences due to Bilu, Hanrot and Voutier. We illustrate our approach by solving completely the equations x2 + 17a1 = 2yn, x2 + 5a113a2 = 2yn and x2 + 3a111a2 = 2yn. [ABSTRACT FROM AUTHOR]

Let k be an arbitrary even integer, and Ek(s;z) denote the non-holomorphic Eisenstein series (of weight k attached to SL2(ℤ)), defined by (1.1) below. In the present paper we first establish a complete asymptotic expansion of Ek(s;z) in the descending order of y as y → + ∞ (Theorem 2.1), upon transferring from the previously derived asymptotic expansion of E0(s;z) (due to the first author [16]) to that of Ek(s;z) through successive use of Maass' weight change operators. Theorem 2.1 yields various results on Ek(s;z), including its functional properties (Corollaries 2.1–2.3), its relevant specific values (Corollaries 2.4–2.7), and its asymptotic aspects as z → 0 (Corollary 2.8). We then apply the non-Euclidean Laplacian ΔH,k (of weight k attached to the upper-half plane) to the resulting expansion, in order to justify the eigenfunction equation for Ek(s;z) in (1.6), where the justification can be made uniformly in the whole s-plane (Theorem 2.2). [ABSTRACT FROM AUTHOR]

Nonlinear congruential pseudorandom number generators can have unexpectedly short periods. Shamir and Tsaban introduced the class of counter-dependent generators which admit much longer periods. In this paper, using a technique developed by Niederreiter and Shparlinski, we present discrepancy bounds for sequences of s-tuples of successive pseudorandom numbers generated by counter-dependent generators modulo a composite M. [ABSTRACT FROM AUTHOR]

We show that a p-adic limit of a Siegel–Eisenstein series of prime level q becomes a Siegel modular form of level pq. This paper contains a simple formula for Fourier coefficients of a Siegel–Eisenstein series of degree two and prime levels. [ABSTRACT FROM AUTHOR]

This paper proves a conjecture proposed by Chen and Ru in [1] on the finiteness of the number of integer solutions to decomposable form inequalities. Let k be a number field and let F(X1,...,Xm) be a non-degenerate decomposable form with coefficients in k. We show that for every finite set of places S of k containing the archimedean places of k, for each real number λ < 1 and each constant c > 0, the inequality $0 < \Pi_{\upsilon \in S}\|F(x_1, \ldots, x_m)\|_{\upsilon}\leq cH_S^{\lambda}(x_1,\ldots,x_m)$ has only finitely many $\mathcal{O}_{S}^{*}$-non-proportional solutions, where HS(x1,...,xm) = Πυ∈Smax1≤i≤m ||xi||υ is the S-height. [ABSTRACT FROM AUTHOR]

The rank of an overpartition pair is a generalization of Dyson's rank of a partition. The purpose of this paper is to investigate the role that this statistic plays in the congruence properties of $\overline{pp}(n)$, the number of overpartition pairs of n. Some generating functions and identities involving this rank are also presented. [ABSTRACT FROM AUTHOR]

Let bℓ(n) denote the number of ℓ-regular partitions of n, where ℓ is prime and 3 ≤ ℓ ≤ 23. In this paper we prove results on the distribution of bℓ(n) modulo m for any odd integer m > 1 with 3 ∤ m if ℓ ≠ 3. [ABSTRACT FROM AUTHOR]

It is well known that bounds on moments of a specific L-function can lead to zero-density result for that L-function. In this paper, we generalize this argument to all L-functions in the Selberg class by assuming a certain second power moment. As an application, it is shown that in the case of symmetric-square L-function, this result improves the existing one. [ABSTRACT FROM AUTHOR]

In this paper we prove an analogue of Mertens' theorem for primes of each of the forms a2+27b2 and 4a2+2ab+7b2 and then use this result to determine an asymptotic formula for the number of positive integers n ≤ x which are discriminants of cyclic cubic fields with each such field having field index 2. [ABSTRACT FROM AUTHOR]

The classical Euler sum \[ S_{p, q} = \sum_{k=1}^{\infty} \frac{1}{k^{q}} \sum_{j=1}^{k} \frac{1}{j^{p}} \] cannot be evaluated when the weight p + q is even unless p = 1 or p = q or (p, q) = (2, 4) or (p, q) = (4, 2) [7]. However it is a different story if instead we consider the alternating sums \[ G_{p, q}^{- -} = \sum_{k=0}^{\infty} \frac{(-1)^{k}}{(2k+1)^{q}} \sum_{j=1}^{k} \frac{(-1)^{j+1}}{j^{p}} \] and \[ G_{p, q}^{+ -} = \sum_{k=0}^{\infty} \frac{(-1)^{k}}{(2k+1)^{q}} \sum_{j=1}^{k} \frac{1}{j^{p}}. \] They can be evaluated for even weight p + q. In this paper, we shall evaluate a family of generalized Euler sums containing $G_{p,q}^{- -}$ when the weight p + q is even via integral transforms of Bernoulli identities. [ABSTRACT FROM AUTHOR]

We give an alternative and relatively simple proof of the functional equation (Funct. Approx. Comment. Math. 55(2) (2016) 227–241) of Mordell–Tornheim multiple zeta function. [ABSTRACT FROM AUTHOR]

Let denote the classical Dedekind sum, where is a positive integer and , . For a given positive integer , we describe a set of at most numbers for which may be , provided that is sufficiently large. For the numbers not in this set, . [ABSTRACT FROM AUTHOR]

We extend the Series [The modular surface and continued fractions, J. London Math. Soc. (2) 31(1) (1985) 69–80] connection between the modular surface ℳ = P S L (2 , ℤ) \ ℍ , cutting sequences, and regular continued fractions to the slow converging Lehner and Farey continued fractions with digits (1 , + 1) and (2 , − 1) in the notation used for the Lehner continued fractions. We also introduce an alternative insertion and singularization algorithm for Farey expansions and other non-semiregular continued fractions, and an alternative dual expansion to the Farey expansions so that d x d y (1 + x y) 2 is invariant under the natural extension map. [ABSTRACT FROM AUTHOR]

A theorem of Gauss extending Wilson's theorem states the congruence (n - 1)n! ≡ -1 (mod n) whenever n has a primitive root, and ≡ 1 (mod n) otherwise, where Nn! denotes the product of all integers up to N that are relatively prime to n. In the spirit of this theorem, we study the multiplicative orders of $(\frac{n-1}{M})_n! (mod n) for odd prime powers pα. We prove a general result about the connection between the order for pα and for pα+1 and study exceptions to the general rule. Particular emphasis is given to the cases M = 3, M = 4 and M = 6, while the case M = 2 is already known. [ABSTRACT FROM AUTHOR]

We give an unconditional proof of a result on sign changes of Fourier coefficients of cusp forms of half-integral weight that before was proved only under the hypothesis of Chowla's conjecture. [ABSTRACT FROM AUTHOR]

In this paper, we study the Diophantine equation x2 + C = 2yn in positive integers x,y with gcd(x,y) = 1, where n ≥ 3 and C is a positive integer. If C ≡ 1 (mod 4), we give a very sharp bound for prime values of the exponent n; our main tool here is the result on existence of primitive divisors in Lehmer sequences due to Bilu, Hanrot and Voutier. We illustrate our approach by solving completely the equations x2 + 17a1 = 2yn, x2 + 5a113a2 = 2yn and x2 + 3a111a2 = 2yn. [ABSTRACT FROM AUTHOR]

Consider the moduli space of smooth curves of genus g and p-rank f defined over an algebraically closed field k of characteristic p. It is an open problem to classify which group schemes occur as the p-torsion of the Jacobians of these curves for f < g - 1. We prove that the generic point of every component of this moduli space has a-number 1 when f = g - 2 and f = g - 3. Likewise, we show that a generic hyperelliptic curve with p-rank g - 2 has a-number 1 when p ≥ 3. We also show that the locus of curves with p-rank g - 2 and a-number 2 is non-empty with codimension 3 in $\mathcal{M}_g$ when p ≥ 5. We include some other results when f = g - 3. The proofs are by induction on g while fixing g - f. They use computations about certain components of the boundary of $\mathcal{M}_g$. [ABSTRACT FROM AUTHOR]