Number Field Case Research Articles

Ideal solutions in the Prouhet–Tarry–Escott problem

For given positive integers m m and n n with m > n m>n , the Prouhet–Tarry–Escott problem asks if there exist two disjoint multisets of integers of size n n having identical k k th moments for 1 ≤ k ≤ m 1\leq k\leq m ; in the ideal case one requires m = n − 1 m=n-1 , which is maximal. We describe some searches for ideal solutions to the Prouhet–Tarry–Escott problem, especially solutions possessing a particular symmetry, both over Z \mathbb {Z} and over the ring of integers of several imaginary quadratic number fields. Over Z \mathbb {Z} , we significantly extend searches for symmetric ideal solutions at sizes 9 9 , 10 10 , 11 11 , and 12 12 , and we conduct extensive searches for the first time at larger sizes up to 16 16 . For the quadratic number field case, we find new ideal solutions of sizes 10 10 and 12 12 in the Gaussian integers, of size 9 9 in Z [ i 2 ] \mathbb {Z}[i\sqrt {2}] , and of sizes 9 9 and 12 12 in the Eisenstein integers.

On the abc$abc$ conjecture in algebraic number fields

AbstractIn this paper, we prove a weak form of the conjecture generalised to algebraic number fields. Given integers satisfying , Stewart and Yu were able to give an exponential bound in terms of the radical over the integers (Stewart and Yu [Math. Ann. 291 (1991), 225–230], Stewart and Yu [Duke Math. J. 108 (2001), no. 1, 169–181]), whereas Győry was able to give an exponential bound in the algebraic number field case for the projective height in terms of the radical for algebraic numbers (Győry [Acta Arith. 133 (2008), 281–295]). We generalise Stewart and Yu's method to give an improvement on Győry's bound for algebraic integers over the Hilbert Class Field of the initial number field K. Given algebraic integers in a number field K satisfying , we give an upper bound for the logarithm of the projective height in terms of norms of prime ideals dividing , where L is the Hilbert Class Field of K. In many cases, this allows us to give a bound in terms of the modified radical as given by Masser (Proc. Amer. Math. Soc. 130 (2002), no. 11, 3141–3150). Furthermore, by employing a recent bound of Győry (Publ. Math. Debrecen 94 (2019), 507–526) on the solutions of S‐unit equations, our estimates imply the upper bound where is an effectively computable constant. Further, given conditions on the largest prime ideal dividing , we obtain a sub‐exponential bound for in terms of the radical. Independently, as a direct application of his bounds on the solutions of S‐unit equations(Győry ([Publ. Math. Debrecen 94 (2019), 507–526]), Győry (Publ. Math. Debrecen 100 (2022), 499–511) also attains results mentioned above, including the above inequality, but over the base field K, as discussed in Section 6. As a consequence of our results, we will give an application to the effective Skolem–Mahler–Lech problem and give an improvement to a result by Lagarias and Soundararajan (J. Théor. Nombres Bordeaux 23 (2011), no. 1, 209–234) on the XYZ conjecture.

Tate Duality in Positive Dimension over Function Fields

We extend the classical duality results of Poitou and Tate for finite discrete Galois modules over local and global fields (local duality, nine-term exact sequence, etc.) to all affine commutative group schemes of finite type, building on the recent work of Česnavičius (“Poitou-Tate without restrictions on the order,” 2015) extending these results to all finite commutative group schemes. We concentrate mainly on the more difficult function field setting, giving some remarks about the number field case along the way.

Third homology of SL2 over number fields: The norm-Euclidean quadratic imaginary case

In the article The third homology ofSL2(Q) ([7]), Hutchinson determined the structure of H3(SL2(Q),Z[12]) by expressing it in terms of K3ind(Q)≅Z/24 and the scissor congruence group of the residue field Fp with p a prime number. In this paper, we develop further the properties of the refined scissors congruence group in order to extend this result to the case of imaginary quadratic number fields whose ring of integers is a Euclidean domain with respect to the norm.

On the meromorphic continuation of Eisenstein series

Eisenstein series are ubiquitous in the theory of automorphic forms. The traditional proofs of the meromorphic continuation of Eisenstein series, due to Selberg and Langlands, start with cuspidal Eisenstein series as a special case, and deduce the general case from spectral theory. We present a “soft” proof which relies only on rudimentary Fredholm theory (needed only in the number field case). It is valid for Eisenstein series induced from an arbitrary automorphic form. The proof relies on the principle of meromorphic continuation. It is close in spirit to Selberg’s later proofs.

The Minimal Ramification Problem for Rational Function Fields over Finite Fields

Abstract We study the minimal number of ramified primes in Galois extensions of rational function fields over finite fields with prescribed finite Galois group. In particular, we obtain a general conjecture in analogy with the well studied case of number fields, which we establish for abelian, symmetric, and alternating groups in many cases.

Inequities in the Shanks–Renyi prime number race over function fields

Fix a prime p > 2 $p >2$ and a finite field F q $\mathbb {F}_{q}$ with q elements, where q is a power of p. Let m be a monic polynomial in the polynomial ring F q [ T ] $\mathbb {F}_{q}[T]$ such that deg ( m ) $\deg (m)$ is large. Fix an integer r ⩾ 2 $r\geqslant 2$ , and let a 1 , ⋯ , a r $a_1,\dots ,a_r$ be distinct residue classes modulo m that are relatively prime to m. In this paper, we derive an asymptotic formula for the natural density δ m ; a 1 , ⋯ , a r $\delta _{m;a_1,\dots ,a_r}$ of the set of all positive integers X such that ∑ N = 1 X π q ( a 1 , m , N ) > ∑ N = 1 X π q ( a 2 , m , N ) > ⋯ > ∑ N = 1 X π q ( a r , m , N ) $\sum _{N=1}^{X} \pi _{q}(a_1,m,N) > \sum _{N=1}^{X} \pi _{q}(a_2,m,N)>\cdots > \sum _{N=1}^{X} \pi _{q}(a_r,m,N)$ , where π q ( a i , m , N ) $\pi _{q}(a_i,m,N)$ denotes the number of irreducible monic polynomials in F q [ T ] $\mathbb {F}_{q}[T]$ of degree N that are congruent to a i mod m $ a_i \bmod m$ , under the assumption of LI (Linear Independence Hypothesis). Many consequences follow from our results. First, we deduce the exact rate at which δ m ; a 1 , a 2 $\delta _{m;a_1,a_2}$ converges to 1 2 $\frac{1}{2}$ as deg ( m ) $\deg (m)$ grows, where a1 is a quadratic non-residue and a2 is a quadratic residue modulo m, generalizing the work of Fiorilli and Martin. Furthermore, similarly to the number field setting, we show that two-way races behave differently than races involving three or more competitors, once deg ( m ) $\deg (m)$ is large. In particular, biases do appear in races involving three or more quadratic residues (or quadratic non-residues) modulo m. This work is a function field analog of the work of Lamzouri, who established similar results in the number field case. However, we exhibit some examples of races in function fields where LI is false, and where the associated densities vanish, or behave differently than in the number field setting.

Lifting and automorphy of reducible mod p Galois representations over global fields

We prove the modularity of most reducible, odd representations \({\bar{\rho }}: \Gamma _{{\mathbb {Q}}} \rightarrow \mathrm {GL}_2(k)\) with k a finite field of characteristic an odd prime p. This is an analogue of Serre’s celebrated modularity conjecture (which concerned irreducible, odd representations \({\bar{\rho }}: \Gamma _{{\mathbb {Q}}} \rightarrow \mathrm {GL}_2(k)\)) for reducible, odd representations. Our proof lifts \({\bar{\rho }}\) to an irreducible geometric p-adic representation \(\rho \) which is known to arise from a newform by results of Skinner–Wiles and Pan. We likewise prove automorphy of many reducible representations \({\bar{\rho }}:\Gamma _{F} \rightarrow \mathrm {GL}_n(k)\) when F is a global function field of characteristic different from p, by establishing a p-adic lifting theorem and invoking the work of L. Lafforgue. Crucially, in both cases we show that the actual representation \({\bar{\rho }}\), rather than just its semisimplification, arises from reduction of the geometric representation attached to a cuspidal automorphic representation. Our main theorem establishes a geometric lifting result for mod p representations \({\bar{\rho }}:\Gamma _{F} \rightarrow G(k)\) of Galois groups of global fields F, valued in reductive groups G(k), and assumed to be odd when F is a number field. Thus we find that lifting theorems, combined with automorphy lifting results pioneered by Wiles in the number field case and the results in the global Langlands correspondence proved by Drinfeld and L. Lafforgue in the function field case, give the only known method to access modularity of mod p Galois representations both in reducible and irreducible cases.

Isolation of the cuspidal spectrum: The function field case

Isolating cuspidal automorphic representations from the whole automorphic spectrum is a basic problem in the trace formula approach. For example, matrix coefficients of supercuspidal representations can be used as test functions for this. However, they kill a large class of interesting cuspidal automorphic representations. For the case of number fields, multipliers of the Schwartz algebra are used in the recent work (see Beuzart-Plessis et al. (2019)) to isolate all the cuspidal spectrum. In particular, they are suitable for the comparison of orbital integrals. These multipliers are then applied to the proof of the Gan-Gross-Prasad conjecture for unitary groups (see Beuzart-Plessis et al. (2019, 2020)). In this article, we prove the similar result on isolating the cuspidal spectrum in Beuzart-Plessis et al. (2019) for the function field case.

Nonvanishing for cubic L-functions

Abstract We prove that there is a positive proportion of L-functions associated to cubic characters over $\mathbb F_q[T]$ that do not vanish at the critical point $s=1/2$ . This is achieved by computing the first mollified moment using techniques previously developed by the authors in their work on the first moment of cubic L-functions, and by obtaining a sharp upper bound for the second mollified moment, building on work of Lester and Radziwiłł, which in turn develops further ideas from the work of Soundararajan, Harper and Radziwiłł. We work in the non-Kummer setting when $q\equiv 2 \,(\mathrm {mod}\,3)$ , but our results could be translated into the Kummer setting when $q\equiv 1\,(\mathrm {mod}\,3)$ as well as into the number-field case (assuming the generalised Riemann hypothesis). Our positive proportion of nonvanishing is explicit, but extremely small, due to the fact that the implied constant in the upper bound for the mollified second moment is very large.

ON THE GROWTH OF LINEAR RECURRENCES IN FUNCTION FIELDS

AbstractLet $ (G_n)_{n=0}^{\infty } $ be a nondegenerate linear recurrence sequence whose power sum representation is given by $ G_n = a_1(n) \alpha _1^n + \cdots + a_t(n) \alpha _t^n $ . We prove a function field analogue of the well-known result in the number field case that, under some nonrestrictive conditions, $ |{G_n}| \geq ( \max _{j=1,\ldots ,t} |{\alpha _j}| )^{n(1-\varepsilon )} $ for $ n $ large enough.

Ihara-Type Results for Siegel Modular Forms

Let p be a prime not dividing the integer n. By an Ihara result, we mean existence of a cokernel torsion-free injection from a full lattice in the space of p-old modular forms into a full lattice in the space of all modular forms of level pn. In this paper, we will prove an Ihara result in the number field case, for Siegel modular forms. The case of elliptic modular forms is discussed in Ihara (Discrete subgroups of Lie groups and applications to moduli, Oxford University Press, Bombay, 1975). We will use a geometric formulation for the notion of p-old Siegel modular forms (Rastegar in BIMS 43(7):1–23, 2017). Then, we apply an argument by Pappas, and prove the Ihara result using density of Hecke orbits (Chai in Invent Math 121(3):439–479, 1995). This result is meant to pave the way for modularity results in higher genera.

Rational Conjugacy Classes of Maximal Tori in Groups of Type Dn

We give a concrete characterization of the rational conjugacy classes of maximal tori in groups of type [Formula: see text], with specific emphasis on the case of number fields and [Formula: see text]-adic fields. This includes the forms associated to quadratic spaces, all of their inner and outer forms as well as the Spin groups, their simply connected covers. In particular, in this work, we handle all (simply connected) outer forms of [Formula: see text].

On elementary estimates of arithmetic sums for polynomial rings over finite fields

In this paper, a simple and elementary method is given for deriving estimates of sums of arithmetic functions in Fq[t]. The method is the function field analogue of a result first proved by Stefan A. Burr in 1973 in the number field case. A novelty of this paper is that we are able to extend Burr's result, in the function field context, and obtain secondary main terms for the appropriate sums involving the divisor functions dr(f) with an error term that improves the one given by Burr.

Cohen-Lenstra heuristics and local conditions

We prove function field theorems supporting the Cohen–Lenstra heuristics for real quadratic fields, and natural strengthenings of these analogs from the affine class group to the Picard group of the associated curve. Our function field theorems also support a conjecture of Bhargava on how local conditions on the quadratic field do not affect the distribution of class groups. Our results lead us to make further conjectures refining the Cohen–Lenstra heuristics, including on the distribution of certain elements in class groups. We prove instances of these conjectures in the number field case. Our function field theorems use a homological stability result of Ellenberg, Venkatesh, and Westerland.

Squarefree doubly primitive divisors in dynamical sequences

AbstractLetKbe a number field or a function field of characteristic 0, letφ∈K(z) with deg(φ) ⩾ 2, and letα∈ ℙ1(K). LetSbe a finite set of places ofKcontaining all the archimedean ones and the primes whereφhas bad reduction. After excluding all the natural counterexamples, we define a subsetA(φ,α) of ℤ⩾0× ℤ>0and show that for all but finitely many (m,n) ∈A(φ,α) there is a prime 𝔭 ∉Ssuch that ord𝔭(φm+n(α)−φm(α)) = 1 andαhas portrait (m,n) under the action ofφmodulo 𝔭. This latter condition implies ord𝔭(φu+v(α)−φu(α)) ⩽ 0 for (u,v) ∈ ℤ⩾0× ℤ>0satisfyingu<morv<n. Our proof assumes a conjecture of Vojta for ℙ1× ℙ1in the number field case and is unconditional in the function field case thanks to a deep theorem of Yamanoi. This paper extends earlier work of Ingram–Silverman, Faber–Granville and the authors.

Reflection principles for class groups

We present several new examples of reflection principles which apply to both class groups of number fields and picard groups of curves over P1/Fp. This proves a conjecture of Lemmermeyer [3] about equality of 2-rank in subfields of A4, up to a constant not depending on the discriminant in the number field case, and exactly in the function field case. More generally we prove similar relations for subfields of a Galois extension with group G for the cases when G is S3, S4, A4, D2l and Z/lZ⋊Z/rZ. The method of proof uses sheaf cohomology on 1-dimensional schemes, which reduces to Galois module computations.

On a conjecture of Wan about limiting Newton polygons

We show that for a monic polynomial f(x) over a number field K containing a global permutation polynomial of degree >1 as its composition factor, the Newton Polygon of fmodp does not converge for p passing through all finite places of K. In the rational number field case, our result is the “only if” part of a conjecture of Wan about limiting Newton polygons.

Preperiodic points for rational functions defined over a global field in terms of good reduction

Let $phi$ be an endomorphism of the projective line defined over a global field $K.$ We prove a bound for the cardinality of the set of $K$–rational preperiodic points for φ in terms of the number of places of bad reduction. The result is completely new in the function field case and is an improvement of the number field case.

Prime number races for elliptic curves over function fields

We study the prime number race for elliptic curves over the function field of a proper, smooth and geometrically connected curve over a finite field. This constitutes a function field analogue of prior work by Mazur, Sarnak and the second author. In this geometric setting we can prove unconditional results whose counterparts in the number field case are conditional on a Riemann Hypothesis and a linear independence hypothesis on the zeros of the implied L-functions. Notably we show that in certain natural families of elliptic curves, the bias generically dissipates as the conductor grows. This is achieved by proving a central limit theorem and combining it with generic linear independence results that will appear in a separate paper. Also we study in detail a particular family of elliptic curves that have been considered by Ulmer. In contrast to the generic case we show that the race exhibits very diverse outcomes, some of which are believed to be impossible in the number field setting. Such behaviors are possible in the function field case because the zeros of Hasse-Weil L-functions for those elliptic curves can be proven to be highly dependent among themselves, which is a very non generic situation.