Marilyn Myers
Kummer's Conjecture predicts the rate of growth of the relative class numbers of cyclotomic fields of prime conductor. We extend Kummer's Conjecture to cyclotomic fields of conductor n, where n is any natural number. We show that the Elliott-Halberstam Conjecture implies that this Generalised Kummer's Conjecture is true for almost all n but is false for infinitely many n.
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The existence of rational points on Kummer varieties associated to 2-coverings of abelian varieties over number fields can sometimes be proved through the variation of the Selmer group in the family of quadratic twists of the underlying abelian variety, using an idea of Swinnerton-Dyer. Following Mazur and Rubin, we consider the case when the Galois action on the 2-torsion has a large image. Under mild additional hypotheses we prove the Hasse principle for the associated Kummer varieties assuming the finiteness of relevant Shafarevich-Tate groups.
We use a generalised Kummer construction to realise all but one known weight four newforms with complex multiplication and rational Fourier coefficients in smooth Calabi-Yau threefolds defined over the rational numbers. The Calabi-Yau manifolds are smooth models of quotients of the Weil restrictions of elliptic curves with CM of class number three.
The classification of generalised Kummer surfaces in positive characteristic
该研究在Katsura与Rybakov工作基础上，完成了正特征下作用于阿贝尔曲面且商消解为K3曲面的群分类，证明超奇异阿贝尔曲面相关商无法为广义Kummer曲面，并构造了具体例子。
Transcendental Brauer groups of cubic generalised Kummer surfaces
本文研究由三次曲线构造的广义 Kummer K3 曲面，通过联系其与积曲面的超越 Brauer 群，显式计算有理对角三次曲线情形，并据此探讨处处局部可解曲线的 Galois 三次点存在性。
We prove several results on the number of rational points on open subsets of Kummer varieties of arbitrary dimension. Some of our results are unconditional, and others depend on the Parity Conjecture (a corollary of the Conjecture of Birch and Swinnerton-Dyer). As examples, we show that (conditional on the Parity Conjecture) all odd-dimension Kummer varieties over Q which are quotients of absolutely simple abelian varieties have dense rational points, and we construct an infinite family of K3 surfaces with dense rational points.
Moment of Kummer sums weighted by $L$-functions
该研究运用特征和估计及解析方法分析 $L$-函数加权 Kummer 和的高阶矩，补充并类比了相关历史猜想。
The paper contains at the end a proof of the conjecture of Gross - Kuz'min, for CM extensions of Q. The main topic of the paper is the investigation of the growth of order and ranks at finite levels of some Lambda modules (p-parts of ideal class groups).
In this paper we formulate generalised Isogeny, Mumford-Tate and Shafarevich conjectures and prove that they are equivalent.
The Kummer ratio of the relative class number for prime cyclotomic fields
本文在无Siegel零点、存在Siegel零点及仅对奇特征Dirichlet L-级数假设Riemann Hypothesis三种情形下，显著改进了素数分圆域相对类数Kummer比值的已知界，并通过数值计算验证了理论结果。
Mazur's Conjecture and An Unexpected Rational Curve on Kummer Surfaces and their Superelliptic Generalisations
本文证明了Mazur猜想的一个特殊情形，通过研究Kummer曲面上的奇异有理曲线并推广至更一般的超椭圆曲面，扩展了前人结果并给出了无限制条件下的证明。
Kummer's conjecture states that the relative class number of the $p$-th cyclotomic field follows a strict asymptotic law. Granville has shown it unlikely to be true -- it cannot be true if we assume the truth of two other widely believed conjectures. We establish a new bound for the error term in Kummer's conjecture, and more precisely we prove that $\log(h_p^-)=\frac{p+3}{4}\log(p) + \frac{p}{2}\log(2π)+\log(1-β)+O(\log_2 p)$, where $β$ is a possible Siegel zero of an $L(s,χ)$, $χ$ odd.
The p-adic Kummer--Leopoldt constant kappa\_K of a number field K is (assuming the Leopoldt conjecture) the least integer c such that for all n \textgreater{}\textgreater{} 0, any global unit of K, which is locally a p^(n+c)th power at the p-places, is necessarily the p^nth power of a global unit of K. This constant has been computed by Assim \& Nguyen Quang Do using Iwasawa's techniques,after intricate studies and calculations by many authors. We give an elementary p-adic proof and an improvement of these results, then a class field theory interpretation of kappa\_K. We give some applications…
Kummer (1851) and, many years later, Ihara (2005) both posed conjectures on invariants related to the cyclotomic field $\mathbb Q(ζ_q)$ with $q$ a prime. Kummer's conjecture concerns the asymptotic behaviour of the first factor of the class number of $\mathbb Q(ζ_q)$ and Ihara's the positivity of the Euler-Kronecker constant of $\mathbb Q(ζ_q)$ (the ratio of the constant and the residue of the Laurent series of the Dedekind zeta function $ζ_{\mathbb Q(ζ_q)}(s)$ at $s=1$). If certain standard conjectures in analytic number theory hold true, then one can show that both conjectures are true for a…
We consider a Laurent series defined by infinite products $g_u(t) = \prod_{n=0}^\infty (1 + ut^{-2^n})$, where $u\in \mathbb{F}$ is a parameter and $\mathbb{F}$ is a field. We show that for all $u\in\mathbb{Q}\setminus\{-1,0,1\}$ the series $g_u(t)$ does not satisfy the $t$-adic Littlewood conjecture. On the other hand, if $\mathbb{F}$ is finite then $g_u(t)\in \mathbb{F}((t^{-1}))$ is either a rational function or it satisfies the $t$-adic Littlewood conjecture.
In this paper, we introduce and study a variant of Kummer's notion of (ir)regularity of primes which we call G-irregularity. It is based on Genocchi numbers $G_n$, rather than Bernoulli number $B_n.$ We say that an odd prime $p$ is G-irregular if it divides at least one of the integers $G_2,G_4,\ldots, G_{p-3}$, and G-regular otherwise. We show that, as in Kummer's case, G-irregularity is related to the divisibility of some class number. Furthermore, we obtain some results on the distribution of G-irregular primes. In particular, we show that each primitive residue class contains infinitely ma…
Jacobsthal's function was recently generalised for the case of paired progressions. It was proven that a specific bound of this function is sufficient for the truth of Goldbach's conjecture and of the prime pairs conjecture as well. We extended and adapted algorithms described for the computation of the common Jacobsthal function, and computed respective function values of the paired Jacobsthal function for primorial numbers for primes up to 73. All these values fulfil the conjectured specific bound. In addition to this note, we provide a detailed review of the algorithmic approaches and the c…
We compute all irregular primes less than 163,577,356. For all of these primes we verify that the Kummer-Vandiver conjecture holds and that the lambda-invariant is equal to the index of irregularity.
We prove the generalised Tate conjecture for H^3 of products of elliptic curves over finite fields, by slightly modifying an argument of M. Spiess concerning the Tate conjecture. We prove it fully if the elliptic curves run among at most 3 isogeny classes. We also show how things become more intricate from H^4 onwards, for more that 3 isogeny classes.
Rational curves and the Hilbert Property on Jacobian Kummer varieties
本文在特征零的有限生成域上，证明了与亏格2曲线的Jacobian相关联的所有Kummer曲面具有Hilbert性质，并推广证明与奇数次亏格≥2超椭圆曲线相关联的所有Jacobian Kummer簇也具有该性质。
Height functions on Hecke orbits and the generalised André-Pink-Zannier conjecture
本文在Shimura簇中定义广义Hecke轨道及其高度函数，在Mumford-Tate猜想版本下给出Galois轨道大小下界，并借Pila-Zannier策略证明广义André-Pink-Zannier猜想。
Let $X$ be a smooth projective variety over an algebraically closed field of arbitrary characteristic, and $f$ a dynamical correspondence of $X$. In 2016, the second author conjectured that the dynamical degrees of $f$ defined by the pullback actions on étale cohomology groups and on numerical cycle class groups are equivalent, which we call the dynamical degree comparison (DDC) conjecture. It contains the generalized Weil's Riemann hypothesis (for polarized endomorphisms) as a special case. To proceed, we introduce the so-called Conjecture $G_r$, which is a quantitative strengthening of the s…
In this paper we introduce generalised Markov numbers and extend the classical Markov theory for the discrete Markov spectrum to the case of generalised Markov numbers. In particular we show recursive properties for these numbers and find corresponding values in the Markov spectrum. Further we construct a counterexample to the generalised Markov uniqueness conjecture. The proposed generalisation is based on geometry of numbers. It substantively uses lattice trigonometry and geometric theory of continued numbers.
In this article we study an abelian analogue of Schanuel's conjecture. This conjecture falls in the realm of the generalised period conjecture of Y. Andr{é}. As shown by C. Bertolin, the generalised period conjecture includes Schanuel's conjecture as a special case. Extending methods of Bertolin, it can be shown that the abelian analogue of Schanuel's conjecture we consider, also follows from Andr{é}'s conjecture. C. Cheng et al. showed that the classical Schanuel's conjecture implies the algebraic independence of the values of the iterated exponential function and the values of the iterated l…
The cyclic insertion conjecture of Borwein, Bradley, Broadhurst and Lisoněk states that by inserting all cyclic permutations of some initial blocks of 2's into the multiple zeta value $ ζ(1,3,\ldots,1,3) $ and summing, one obtains an explicit rational multiple of a power of $ π$. Hoffman gives a conjectural identity of a similar flavour concerning $ 2 ζ(3,3,\{2\}^m) - ζ(3,\{2\}^m,(1,2)) $. In this paper we introduce the 'generalised cyclic insertion conjecture', which we describe using a new combinatorial structure on iterated integrals -- the so-called alternating block decomposition. We see …
Nonvanishing of generalised Kato classes and Iwasawa main conjectures
本文针对正偶数解析秩椭圆曲线的广义Kato类非零性猜想，给出κ_p(E)≠0蕴含Selmer群二维的新证明，并证明逆蕴含成立当且仅当局部化映射非零，将作者在CM情形的方法推广到非CM情形。
The conjecture of Leopoldt states that the $p$ - adic regulator of a number field does not vanish. It was proved for the abelian case in 1967 by Brumer, using Baker theory. We prove this conjecture for CM number fields $\K$. The proof uses Iwasawa's methods -- especially Takagi Theory -- for deriving his skew symmetric pairing, together with Kummer- and Class Field Theory.
Relative class numbers and Euler-Kronecker constants of maximal real cyclotomic subfields
研究最大实环形 cyclotomic 子域的欧拉-克罗内克常数分布及其与库默猜想的关系，改进了库默比率的平均界并证明了精确界。
Generalised André-Pink-Zannier Conjecture for Shimura varieties of abelian type
本文证明了所有阿贝尔型 Shimura 簇的广义 André-Pink-Zannier 猜想，提出了适用于所有 Shimura 簇的通用命题，并证明了关于 p-adic 整数上齐次空间良好约化与 Mumford 稳定性关系的 p-adic Kempf-Ness 定理。
Let p be a prime number. The p-adic case of the Mixed Littlewood Conjecture states that liminf_{q \to \infty} q . |q|_p . ||q x|| = 0 for all real numbers x. We show that with the additional factor of log q.loglog q the statement is false. Indeed, our main result implies that the set of x for which liminf_{q\to\infty} q . log q . loglog q. |q|_p . ||qx|| > 0 is of full dimension. The result is obtained as an application of a general framework for Cantor sets developed in this paper.
Modular transform of free fermion generalised Gibbs ensembles and generalised power partitions
该研究将仅含单个KdV荷的自由费米子广义吉布斯系综模变换猜想推广到任意有限KdV荷情形，证明了有限荷下的猜想成立并讨论无限荷情形，这类系综是幂分拆生成函数的推广。
Following the work of Harris and Kudla we prove a more general form of a conjecture of Jacquet relating the non-vanishing of a certain period integral to non-vanishing of the central critical value of a certain $L$-function. As a consequence we deduce certain local results about the existence of $GL_2(k)$-invariant linear forms on irreducible, admissible representations of $GL_2({\Bbb K})$ for ${\Bbb K}$ a commutative semi-simple cubic algebra over a non-archimedean local field $k$ in terms of certain local epsilon factors which were proved only in certain cases by the first author in his earl…
Iwasawa Theory for K3 Surfaces over Finite Fields
本文为有限域上K3曲面建立岩泽理论，证明Mazur控制定理类比、Brauer群的岩泽类数公式及主猜想。
We compute all irregular primes less than 2^31 = 2 147 483 648. We verify the Kummer-Vandiver conjecture for each of these primes, and we check that the p-part of the class group of Q(zeta_p) has the simplest possible structure consistent with the index of irregularity of p. Our method for computing the irregular indices saves a constant factor in time relative to previous methods, by adapting Rader's algorithm for evaluating discrete Fourier transforms.
We characterise the modular behaviour of (generalised) Apéry number modulo $9$, thereby in particular establishing two conjectures in "A method for determining the mod-$3^k$ behaviour of recursive sequences" [arXiv:1308.2856].
Hodge-Tate prismatic crystals and Sen theory
本文通过Kummer塔重建Sen理论，证明Min-Wang猜想：Hodge-Tate棱镜晶体由Sen算子决定，从而等价于近Hodge-Tate表示范畴。
Generalised Whittaker models as instances of relative Langlands duality II: Plancherel density and global periods
本文研究了一类广义Whittaker模型在相对Langlands对偶框架下的局部Plancherel密度数值猜想及其在全局周期猜想中的应用。
We conjecture that bounded generalised polynomial functions cannot be generated by finite automata, except for the trivial case when they are periodic away from a finite set. Using methods from ergodic theory, we are able to partially resolve this conjecture, proving that any hypothetical counterexample is periodic away from a very sparse and structured set. In particular, we show that for a polynomial $p(n)$ with at least one irrational coefficient (except for the constant one) and integer $m$, the sequence $\lfloor p(n) \rfloor \bmod{m}$ is never automatic. We also obtain a conditional resul…
We conjecture that bounded generalised polynomial functions cannot be generated by finite automata, except for the trivial case when they are ultimately periodic. Using methods from ergodic theory, we are able to partially resolve this conjecture, proving that any hypothetical counterexample is periodic away from a very sparse and structured set. In particular, we show that for a polynomial $p(n)$ with at least one irrational coefficient (except for the constant one) and integer $m\geq 2$, the sequence $\lfloor p(n) \rfloor \bmod{m}$ is never automatic. We also prove that the conjecture is equ…
In this paper we prove, assuming the Generalised Riemann Hypothesis, a conjecture of Yves Andre that that asserts that a curve in a Shimura variety containing an infinite set of special points is of Hodge type.
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