Chandan Singh Dalawat
Using Kummer theory for a finite extension K of \Qp(ζ)(where p is a prime number and ζa primitive p-th root of~1), we compute the ramification filtration and the discriminant of an arbitrary elementary abelian p-extension of K. We also develop the analogous Artin-Schreier theory for finite extensions of \Fp((π)) and derive similar results for their elementary abelian p-extensions.
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We show how the ramification filtration on the maximal elementary abelian p-extension (p prime) on a local number field of residual characteristic p can be derived using only Kummer theory and a certain orthogonality relation for the Kummer pairing, even in the absence of a primitive p-th root of 1; the case of other local fields with finite residue fields was treated earlier. In all cases, we compute the contribution of cyclic extensions to Serre's degree-$p$ mass formula.
We adapt an old local-to-global technique of Ore to compute, under certain mild assumptions, an integral basis of a number field without a previous factorization of the discriminant of the defining polynomial. In a first phase, the method yields as a by-product successive splittings of the discriminant. When this phase concludes, it requires a squarefree factorization of some base factors of the discriminant to terminate.
We classify all the number fields with signature (4,2), (6,1), (1,4) and (3,3) having discriminant lower than a specific upper bound. This completes the search for minimum discriminants for fields of degree 8 and continues it in the degree 9 case. We recall the theoretical tools and the algorithmic steps upon which our procedure is based, then we focus on the novelties due to a new implementation of this process on the computer algebra system PARI/GP; finally, we make some remarks about the final results, among which the existence of a number field with signature $(3,3)$ and small discriminant…
We obtain several results on the computation of different and discriminant ideals of finite extensions of local fields. As an application, we deduce routines to compute the $\p$-adic valuation of the discriminant $\dsc(f)$, and the resultant $\res(f,g)$, for polynomials $f(x),g(x)\in A[x]$, where $A$ is a Dedekind domain and $\p$ is a non-zero prime ideal of $A$ with finite residue field. These routines do not require the computation of neither $\dsc(f)$ nor $\res(f,g)$; hence, they are useful in cases where this latter computation is inefficient because the polynomials have a large degree or …
Let s(n) be the number of representations of n as the sum of three squares. We prove a remarkable new identity for s(p^2n)- ps(n) with p being an odd prime. This identity makes nontrivial use of ternary quadratic forms with discriminants p^2 and 16p^2. These forms are related by Watson's transformations. To prove this identity we employ the Siegel--Weil and the Smith--Minkowski product formulas.
Some thoughts on the congruence D=0,1(mod 4) for the absolute discriminant D of a number field
In 1997, Y. Ohno empirically stumbled on an astoundingly simple identity relating the number of cubic rings $h(D)$ of a given discriminant $D$, over the integers, to the number of cubic rings $h'(D)$ of discriminant $-27D$ in which every element has trace divisible by 3: $h'(D) = h(D)$, if $D < 0$, and $h'(D) = 3h(D)$ if $D > 0$, where in each case, rings are weighted by the reciprocal of their number of automorphisms. This allows the functional equations governing the analytic continuation of the Shintani zeta functions (the Dirichlet series built from the values of $h$ and $h'$) to be put in…
Using the action of the Galois group of a normal extension of number fields, we generalize and symmetrize various fundamental statements in algebra and algebraic number theory concerning splitting types of prime ideals, factorization types of polynomials modulo primes, and cycle types of the Galois groups of polynomials. One remarkable example is the removal of all artificial constraints from the Kummer-Dedekind Theorem that relates splitting and factorization patterns. Finally, we present an elementary proof that the discriminant of the splitting field of a monic irreducible polynomial with i…
We determine, asymptotically, the number of quintic fields having bounded discriminant. Specifically, we prove that the asymptotic number of quintic fields having absolute discriminant at most X is a constant times X. In contrast with the quartic case, we also show that a density of 100% of quintic fields, when ordered by absolute discriminant, have Galois closure with full Galois group $S_5$. The analogues of these results are also proven for orders in quintic fields. Finally, we give an interpretation of the various constants appearing in these theorems in terms of local masses of quintic ri…
Geometry-of-numbers over number fields and the density of ADE families of curves having squarefree discriminant
本文研究ADE型Dynkin图所产生曲线族中判别式无平方因子的密度，借助Thorne–Laga框架并将Bhargava的数的几何方法推广到数域，证明其全局密度等于各局部密度之积。
In this paper we describe how to use the algorithmic methods provided by Hunter and Pohst in order to give a complete classification of number fields of degree 8 and signature (2,3) with absolute discriminant less than a certain bound. The choice of this bound comes from the local corrections given by prime ideals to the lower estimates for discriminants obtained with the Odlyzko-Poitou-Serre method.
Consider tuples of separable algebras over a common local or global number field, related to each other by specified resolvent constructions. Under the assumption that all ramification is tame, simple group-theoretic calculations give best possible divisibility relations among the discriminants. We show that for many resolvent constructions, these divisibility relations continue to hold even in the presence of wild ramification.
In this paper we consider a tower of number fields $\cdots \supseteq K(1) \supseteq K(0) \supseteq K$ arising naturally from a continuous $p$-adic representation of $\mathrm{Gal}(\bar{\mathbb{Q}}/K)$, referred to as a $p$-adic Lie tower over $K$. A recent conjecture of Daqing Wan hypothesizes, for certain $p$-adic Lie towers of curves over $\mathbb{F}_p$, a stable (polynomial) growth formula for the genus. Here we prove the analogous result in characteristic zero, namely: the $p$-adic valuation of the discriminant of the extension $K(i)/K$ is given by a polynomial in $i,p^i$ for $i$ sufficient…
Prime isogenous discriminant ideal twins
本文研究了素数等价判别理想双生的分类，证明在数域中，当p∈{2,3,5,7,13}时，存在有限个这样的双生。
This paper was motivated by a recent paper by Krumm and Pollack investigating modulo-$p$ behaviour of quadratic twists with rational points of a given hyperelliptic curve, conditional on the abc-conjecture. We extend those results to twisted Galois covers with arbitrary Galois groups. The main point of this generalization is to interpret those results as statements about the sets of specializations of a given Galois cover under restrictions on the discriminant. In particular, we make a connection with existing heuristics about the distribution of discriminants of Galois extensions such as the …
On the asymptotics of elementary-abelian extensions of local and global function fields
本文研究特征 $p$ 下局部与整体函数域的野分歧初等阿贝尔扩张，确定了判别式的分布，并对局部及有理函数域给出了固定判别式除子下扩张数量的精确公式，揭示局部-整体原则。
We give an exact formula for the number of $G$-extensions of local function fields $\mathbb{F}_q((t))$ for finite abelian groups $G$ up to a conductor bound. As an application we give a lower bound for the corresponding counting problem by discriminant.
The Euclidean minimum $M(K)$ of a number field $K$ is an important numerical invariant that indicates whether $K$ is norm-Euclidean. When $K$ is a non-CM field of unit rank 2 or higher, Cerri showed $M(K)$, as the supremum in the Euclidean spectrum $Spec(K)$, is isolated and attained and can be computed in finite time. We extend Cerri's works by applying recent dynamical results of Lindenstrauss and Wang. In particular, the following facts are proved: (1) For any number field $K$ of unit rank 3 or higher, $M(K)$ is isolated and attained and Cerri's algorithm computes $M(K)$ in finite time. (2)…
We investigate how various invariants of elliptic curves, such as the discriminant, Kodaira type, Tamagawa number and real and complex periods, change under an isogeny of prime degree p. For elliptic curves over l-adic fields, the classification is almost complete (the exception is wild potentially supersingular reduction when l=p), and is summarised in a table.
We show that for a large class of finite groups G, the number of Galois extensions E/Q of group G and discriminant $|d_E|\leq y$ grows like a power of $y$ (for some specified exponent). The groups G are the regular Galois groups over Q and the extensions E/Q that we count are obtained by specialization from a given regular Galois extension F/Q(T). The extensions E/Q can further be prescribed any unramified local behavior at each suitably large prime $p\leq \log (y)/δ$ for some $δ\geq 1$. This result is a step toward the Malle conjecture on the number of Galois extensions of given group and bou…
Let $f \in { \mathbb R} ( t) [x]$ be given by $ f(t, x) = x^n + t \cdot g(x) $ and $β_1 < \dots < β_m$ the distinct real roots of the discriminant $Δ_{(f, x)} (t)$ of $f(t, x)$ with respect to $x$. Let $γ$ be the number of real roots of $g(x)=\sum_{k=0}^s t_{s-k} x^{s-k}$. For any $ξ> | β_m |$, if $n-s$ is odd then the number of real roots of $f(ξ, x)$ is $γ+1$, and if $n-s$ is even then the number of real roots of $f(ξ, x)$ is $γ$, $γ+2$ if $t_s>0$ or $t_s < 0$ respectively. A special case of the above result is constructing a family of totally complex polynomials which are reducible over $\m…
Erdős-Kac theorems for discriminants of number fields
本文证明了当 G 为阿贝尔群时，随机 G-扩张中分歧素数个数的中心极限定理，推广了 Lemke Oliver 和 Thorne 在 G=S_d (2≤d≤5) 情形的工作，并首次给出不同素数处分歧事件不独立的例子。
Counting Frobenius extensions over local function fields
研究局部函数域在特征p下扩展的渐进行为，特别考虑伽罗瓦群为AGL_1(p)子群的情况，并解决更一般情况下的计数问题，包括非阿贝尔群如S_3、二面体群和许多弗罗贝尼乌斯群。
We give a new approach for the local class field theory of Serre and Hazewinkel. We also discuss two-dimensional local class field theory in this framework.
This note is an elaboration of the ideas and intuitions of Grothendieck and Weil concerning the "arithmetic topology". Given 3-dimensional manifold M fibering over the circle we introduce an real quadratic number field K with discriminant d, where d>0 is an integer number uniquely determined by M. The idea is to relate geometry of M to the arithmetic of field K. On this way, we show that Vol M is a limit density of ideals of given norm in the field K (Dirichlet density). The second statement says that the number of cusp points of manifold M is equal to the class number of the field K. It is re…
We give a description of the set of exceptional pairs for a number field $K$, that is the set of pairs $(\ell, j(E))$, where $\ell$ is a prime and $j(E)$ is the $j$-invariant of an elliptic curve $E$ over $K$ which admits an $\ell$-isogeny locally almost everywhere but not globally. We obtain an upper bound for $\ell$ in such pairs in terms of the degree and the discriminant of $K$. Moreover, we prove finiteness results about the number of exceptional pairs.
The Hasse norm principle for abelian extensions
该研究聚焦数域上判别式有界且不满足哈塞范数原理的阿贝尔扩张分布，分类了对应正比例失效的有限阿贝尔群，还得到范数一环面弱逼近失效的类似分类，借助调和分析工具完成带无穷多局部条件的阿贝尔扩张计数。
Lifts of unramified twists and local-global principles
本文证明了特征为p的有理全局函数域的两步nilpotent p-扩展在按最大上分支断裂（'最后跳跃'）计数时满足定量的局部-整体原则。该结果扩展了之前仅适用于p≠2的情况，并提供了一个关于特征为2的D4扩展最后跳跃分布的应用。同时，作者还展示了按判别式计数时局部-整体原则的反例。
In this article we study motives corresponding to the moduli stacks of G-shtukas and their local models. In particular we deal with the question of describing their motivic fundamental invariants. As an application, we provide a criterion for mixed Tateness of the local model, and discuss the semi-simplicity of Frobenius on their cohomology. We then use the theory of local models to reformulate a purity result for these moduli stacks in the motivic context.
If K_0 is the fraction field of the Witt vectors over an algebraically closed field k of characteristic p, we calculate upper bounds on the conductor of higher ramification for (the Galois closure of) extensions K_0(zeta_{p^r}, sqrt[p^r]{a})/K_0, where a is in K_0(zeta_{p^r}). Here zeta_{p^r} is a primitive p^r-th root of unity. In certain cases, including when a is in K_0 and p=2, we calculate the conductor exactly. These calculations can be used to determine the discriminants of various extensions of Q obtained by adjoining roots of unity and radicals.
In this paper we study a relation between conductor, depth, and the level of principal congruence subgroups for irreducible admissible representations of ${\rm GL}_n(F)$ for a non-archimedean local field $F$ of characteristic zero. Some global and local applications are also discussed.
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