Daqing Wan
This paper proves the general rank one case of Dwork's conjecture over the affine space. It generalizes and improves the method of ANT-0141 "Dwork's conjecture on unit root zeta functions" (Ann. Math., 150(1999), 867-929). In addition, explicit information about the zeros and poles (along the Gouvêa-Mazur conjecture direction) for the unit root zeta function is obtained. The paper is to appear in JAMS.
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This is the final version of ANT-0142 ("An embedding approach to Dwork's conjecture"). It reduces the higher rank case of the conjecture over a general base variety to the rank one case over the affine space. The general rank one case is completed in ANT-0235 "Rank one case of Dwork's conjecture". Both papers will appear in JAMS.
This is an expository paper which gives a quick introduction to Dwork's conjecture about p-adic meromorphic continuation of his unit root zeta function arising from algebraic geometry. Special emphasis is given to the case of elliptic family and its relation to p-adic modular forms such as Gouvêa-Mazur conjecture and some sort of p-adic Ramanujuan-Peterson conjecture. There is also a section describing some of my personal interactions with Dwork.
In this article, we introduce a systematic new method to investigate the conjectural p-adic meromorphic continuation of Professor Bernard Dwork's unit root zeta function attached to an ordinary family of algebraic varieties defined over a finite field of characteristic p.
New cases of Dwork's conjecture on asymptotic behaviors of solutions of $p$-adic differential equations without solvability
本文研究Dwork关于无解性条件下p-adic微分方程解的对数增长阶猜想，通过结合微分模下降技术与作者先前秩2情形结果，证明了该猜想的新情形。
Zeta functions over finite fields- An exposition of Dwork's methods
本文系统阐述Dwork在有限域Zeta函数研究中使用的p进解析与余调方法，并将其应用于Dwork超曲面族的一个具体例子。
For a $p$-adic differential equation solvable in an open disc (in a $p$-adic sense), around 1970, Dwork proves that the solutions satisfy a certain growth condition on the boundary. Dwork also conjectures that a similar phenomenon should be observed without assuming the solvability. In this paper, we verify Dwork's conjecture in the rank two case, which is the first non-trivial result on the conjecture. The proof is an application of Kedlaya's decomposition theorem of $p$-adic differential equations defined over annulus.
We extend our previous work on hypergeometric point count formulas by proving that we can express the number of points on families of Dwork hypersurfaces $$X_λ^d: \hspace{.1in} x_1^d+x_2^d+\ldots+x_d^d=dλx_1x_2\cdots x_d$$ over finite fields of order $q\equiv 1\pmod d$ in terms of Greene's finite field hypergeometric functions. We prove that when $d$ is odd, the number of points can be expressed as a sum of hypergeometric functions plus $(q^{d-1}-1)/(q-1)$ and conjecture that this is also true when $d$ is even. The proof rests on a result that equates certain Gauss sum expressions with finite …
On Siegel's problem and Dwork's conjecture for $G$-functions
本文否定回答了G-函数的西格尔问题，构造出无穷多个非超几何代数拉回的几何秩二局部系统，为德沃克猜想提供新反例并回答了Krammer的问题。
We demonstrate that Wan's alternate description of Dwork's unit root L-function in the rank one case may be modified to give a proof of meromorphy that is classical, eliminating the need to study sequences of uniform meromorphic functions.
In this paper, we study the logarithmic growth (log-growth) filtration, a mysterious invariant found by B. Dwork, for $(φ,\nabla)$-modules over the bounded Robba ring. The main result is a proof of a conjecture proposed by B. Chiarellotto and N. Tsuzuki on a comparison between the log-growth filtration and Frobenius slope filtration. One of the ingredients of the proof is a new criterion for pure of bounded quotient, which is a notion introduced by Chiarellotto and Tsuzuki to formulate their conjecture. We also give several applications to log-growth Newton polygons, including a conjecture of …
In this paper, we focus on a family of generalized Kloosterman sums over the torus. With a few changes to Haessig and Sperber's construction, we derive some relative $p$-adic cohomologies corresponding to the $L$-functions. We present explicit forms of bases of top dimensional cohomology spaces, so to obtain a concrete method to compute lower bounds of Newton polygons of the $L$-functions. Using the theory of GKZ system, we derive the Dwork's deformation equation for our family. Furthermore, with the help of Dwork's dual theory and deformation theory, the strong Frobenius structure of this equ…
In this article we review the observation, due originally to Dwork, that the zeta-function of an arithmetic variety, defined originally over the field with p elements, is a superdeterminant. We review this observation in the context of a one parameter family of quintic threefolds, and study the zeta-function as a function of the parameter ϕ. Owing to cancellations, the superdeterminant of an infinite matrix reduces to the (ordinary) determinant of a finite matrix, U(ϕ), corresponding to the action of the Frobenius map on certain cohomology groups. The parameter-dependence of U(ϕ) is given by a…
This is an expository paper which gives a simple arithmetic introduction to the conjectures of Weil and Dwork concerning zeta functions of algebraic varieties over finite fields. A number of further open questions are raised.
Dwork-type $q$-congruences through the $q$-Lucas theorem
本文运用 $q$-Lucas 定理和已知 $q$-超同余，证明了若干 Dwork 型 $q$-同余，证实了三个先前提出的猜想，并验证了 Swisher (H.3) 猜想的较弱形式。
Let $\sum\_{n=0}^\infty a\_n z^n\in \overline{\mathbb Q}[[z]]$ be a $G$-function, and, for any $n\ge0$, let $δ\_n\ge 1$ denote the least integer such that $δ\_n a\_0, δ\_n a\_1, ..., δ\_n a\_n$ are all algebraic integers. By definition of a $G$-function, there exists some constant $c\ge 1$ such that $δ\_n\le c^{n+1}$ for all $n\ge 0$. In practice, it is observed that $δ\_n$ always divides $D\_{bn}^{s} C^{n+1}$ where $D\_n=lcm\{1,2, ..., n\}$, $b, C$ are positive integers and $s\ge 0$ is an integer. We prove that this observation holds for any $G$-function provided the following conjecture is a…
As a consequence of Wan's theorem about Dwork's conjecture, the unit root $L$-functions of the $n$-dimensional Kloosterman family are $p$-adic meromorphic. By studying the symmetric power $L$-functions associated to the Kloosterman family, we prove that for each $0\le j\le n$, the unit root $L$-function coming from slope $j$ has a unique unit root which equals 1.
The Bombieri-Dwork conjecture predicts that the differential equations satisfied by $G$-functions come from geometry. In this paper, we will look at special $G$-functions whose differential equations have a special singularity with maximally unipotent monodromy. We will formulate a Gamma conjecture about such $G$-functions, which has close connections with the mirror symmetry of Calabi-Yau threefolds and the Gamma conjecture in algebraic geometry. We will provide examples to support this conjecture, which involves numerical computations using Mathematica programs.
In this paper, we will study the connections between the mirror symmetry of K3 surfaces and the geometry of the Legendre family of elliptic curves. We will prove that the mirror map of the Dwork family is equal to the period map of the Legendre family. This result provides an interesting explanation to the modularities of counting functions for K3 surfaces from the mirror symmetry point of view. We will also discuss the relations between the arithmetic geometry of smooth fibers of the Fermat pencil (Dwork family) and that of the smooth fibers of the Legendre family, e.g. Shioda-Inose structure…
In this note we study the relationship between the power series expansion of the Dwork exponential and the Mahler expansion of the $p$-adic Gamma function. We exploit this relationship to prove that certain quantities that appeared in our previous computations of the Frobenius map can be expressed in terms of the derivatives of the $p$-adic Gamma function at 0. This is used to prove a conjecture about the non-trivial off-diagonal entry in the Frobenius matrix of the mirror quintic threefold.
Dwork's conjecture, now proven by Wan, states that unit root L-functions "coming from geometry" are p-adic meromorphic. In this paper we study the p-adic variation of a family of unit root L-functions coming from a suitable family of toric exponential sums. In this setting, we find that the unit root L-functions each have a unique p-adic unit root. We then study the variation of this unit root over the family of unit root L-functions. Surprisingly, we find that this unit root behaves similarly to the classical case of families of exponential sums. That is, the unit root is essentially a ratio …
Based on work of Atkin and Swinnerton-Dyer on partition rank difference functions, and more recent work of Lovejoy and Osburn, Mao has proved several inequalities between partition ranks modulo $10$, and additional results modulo $6$ and $10$ for the $M_2$ rank of partitions without repeated odd parts. Mao conjectured some additional inequalities. We prove some of Mao's rank inequality conjectures for both the rank and the $M_2$ rank modulo $10$ using elementary methods.
In this paper we announce a conjecture concerning enumeration of 2n x k n-times persymmetric matrices over F_2 by rank.
Let E be an elliptic curve defined over a number field K. Michael Larsen conjectured that for any finitely generated subgroup G of Gal(\bar K/K), the Mordell-Weil rank of E is unbounded in number fields fixed by G. We prove that the conjecture holds over K=Q for both the analytic rank and the p-infinity Selmer rank of E for every odd prime p. For arbitrary E/K, we show that Larsen's conjecture follows from the standard conjectures for ranks of elliptic curves, provided K has a real place or E has non-integral j-invariant.
Based on an equation for the rank of an elliptic surface over $\mathbb{Q}$ which appears in the work of Nagao, Rosen, and Silverman, we conjecture that 100% of elliptic surfaces have rank $0$ when ordered by the size of the coefficients of their Weierstrass equations, and present a probabilistic heuristic to justify this conjecture. We then discuss how it would follow from either understanding of certain $L$-functions, or from understanding of the local behaviour of the surfaces. Finally, we make a conjecture about ranks of elliptic surfaces over finite fields, and highlight some experimental …
We present an heuristic argument for the prediction of expected Mordell-Weil rank of elliptic curves over number fields, using Birch and Swinnerton-Dyer's original conjecture and Sato-Tate conjectures. We do calculations in some cases and raise questions about their relations, if any, with the predictions of various average rank models that have been considered.
In 1944 Dyson defined the rank of a partition as the largest part minus the number of parts, and conjectured that the residue of the rank mod 5 divides the partitions of 5n+4 into five equal classes. This gave a combinatorial explanation of Ramanujan's famous partition congruence mod 5. He made an analogous conjecture for the rank mod 7 and the partitions of 7n+5. In 1954 Atkin and Swinnerton-Dyer proved Dyson's rank conjectures by constructing several Lambert-series identities basically using the theory of elliptic functions. In 2016 the author gave another proof using the theory of weak harm…
In this article, we present the first half of our project on the Iwasawa theory of higher rank Galois deformations over deformations rings of arbitrary dimension. We develop a theory of Coleman maps for a very general class of coefficient rings, devise a dimension reduction procedure for locally restricted Euler systems and finally, put these into use in order to prove a divisibility in a 3-variable main conjecture for nearly ordinary families of Rankin-Selberg convolutions.
We conjecture that special elements associated with rank-one motives are obtained $p$-adically from Rubin-Stark elements by means of a precise `higher-rank Soulé twist' construction. We show this conjecture incorporates a variety of known results and existing predictions and also gives rise to a concrete strategy for proving the equivariant Tamagawa Number Conjecture for rank-one motives. We then use this approach to obtain new evidence in support of the equivariant Tamagawa Number Conjecture in the setting of CM abelian varieties.
In this paper we prove that the p-adic L-function that interpolates the Rankin-Selberg product of a general modular form and a CM form of higher weight divides the characteristic ideal of the corresponding Selmer group. This is one divisibility of the Iwasawa main conjecture for the p-adic L-function. We prove this conjecture using the congruences between Klingen Eisensteinseries and cusp forms on the group GU(3; 1), following the strategy of a recent work of C. Skinner and E. Urban. This theorem can be used to deduce a converse of Gross-Zagier-Kolyvagin theorem and the precise BSD formula in …
We describe an algorithm to prove the Birch and Swinnerton-Dyer conjectural formula for any given elliptic curve defined over the rational numbers of analytic rank zero or one. With computer assistance we have proved the formula for 16714 of the 16725 such curves of conductor less than 5000.
The Adams conjecture and intersections of local Arthur packets
本文研究局部Arthur包的交集与Adams猜想之间的相互作用，探讨表示在低秩下何时满足该猜想。
We show that Lang's conjecture on error terms in Diophantine approximation implies Honda's conjecture on ranks of elliptic curves over number fields. We also show that even a very weak version of Lang's error term conjecture would be enough to deduce boundedness of ranks for quadratic twists of elliptic curves over number fields. This can be seen as evidence for boundedness of ranks not relying on probabilistic heuristics on elliptic curves.
Recently, Andrews, Chan, Kim and Osburn introduced a $q$-series $h(q)$ for the study of the first positive rank and crank moments for overpartitions. They conjectured that for all integers $m \geq 3$, \begin{equation*}\label{hqcon} \frac{1}{(q)_{\infty}} (h(q) - m h(q^{m})) \end{equation*} has positive power series coefficients for all powers of $q$. Byungchan Kim, Eunmi Kim and Jeehyeon Seo provided a combinatorial interpretation and proved it is asymptotically true by circle method. In this note, we show this conjecture is true if $m$ is any positive power of $2$, and we show that in order t…
It is conjectured that for fixed $A$, $r \ge 1$, and $d \ge 1$, there is a uniform bound on the size of the torsion submodule of a Drinfeld $A$-module of rank $r$ over a degree $d$ extension $L$ of the fraction field $K$ of $A$. We verify the conjecture for $r=1$, and more generally for Drinfeld modules having potential good reduction at some prime above a specified prime of $K$. Moreover, we show that within an $\Lbar$-isomorphism class, there are only finitely many Drinfeld modules up to isomorphism over $L$ which have nonzero torsion. For the case $A=\Fq[T]$, $r=1$, and $L=\Fq(T)$, we give …
Ramanujan's congruence $p(5k+4) \equiv 0 \pmod 5$ led Dyson \cite{dyson} to conjecture the existence of a measure "rank" such that $p(5k+4)$ partitions of $5k+4$ could be divided into sub-classes with equal cardinality to give a direct proof of Ramanujan's congruence. The notion of rank was extended to rank differences by Atkin and Swinnerton-Dyer \cite{atkin}, who proved Dyson's conjecture. More recently, Mao proved several equalities and inequalities, leaving some as conjectures, for rank differences for partitions modulo 10 \cite{mao10} and for $M_2$ rank differences for partitions with no …
Let L be a number field and let E be any subgroup of the units O_L^* of L. If rank(E) = 1, Lehmer's conjecture predicts that the height of any non-torsion element of E is bounded below by an absolute positive constant. If rank(E) = rank(O_L^*), Zimmert proved a lower bound on the regulator of E which grows exponentially with [L:Q]. Fernando Rodriguez Villegas made a conjecture in 2002 that "interpolates" between these two extremes of rank. Here we prove a high-rank case of this conjecture. Namely, it holds if L contains a subfield K for which [L:K] >> [K:Q] and E contains the kernel of the nor…
Watkins' conjecture asserts that for a rational elliptic curve $E$ the degree of the modular parametrization is divisible by $2^r$, where $r$ is the rank of $E$. In this paper we prove that if the modular degree is odd then $E$ has rank $0$. Moreover, we prove that the conjecture holds for all rank two rational elliptic curves of prime conductor and positive discriminant.
We give a new definition of a $p$-adic $L$-function for a mixed signature character of a real quadratic field and for a nontrivial ray class character of an imaginary quadratic field. We then state a $p$-adic Stark conjecture for this $p$-adic $L$-function. We prove our conjecture in the case when $p$ is split in the imaginary quadratic field by relating our construction to Katz's $p$-adic $L$-function. We also provide numerical evidence for our conjecture in three examples.
Larsen's conjecture for elliptic curves over $\mathbb{Q}$ with analytic rank at most $1$
本文证明了Larsen猜想：对解析秩至多1的$\mathbb{Q}$上椭圆曲线，其在任意拓扑有限生成伽罗瓦子群的不动子域上的秩必为无限。
Ranks of Matrices of Logarithms of Algebraic Numbers II: The Matrix Coefficient Conjecture
本文针对代数数对数方阵的行列式非零性这一数论问题，提出矩阵系数猜想，称行列式为零的该类矩阵经左右有理基变换后可得到含零系数的矩阵。
In this paper we prove that the $p$-adic $L$-function that interpolates the Rankin-Selberg product of a general weight two modular form which is unramified and non-ordinary at $p$, and an ordinary CM form of higher weight contains the characteristic ideal of the corresponding Selmer group. This is one divisibility of the Iwasawa-Greenberg main conjecture for the $p$-adic $L$-function. This generalizes an earlier work of the author to the non-ordinary case. The result of this paper plays a crucial role in the proof of Iwasawa main conjecture and refined Birch-Swinnerton-Dyer formula for supersi…
On a conjecture of Mazur predicting the growth of Mordell--Weil ranks in $\mathbb{Z}_p$-extensions
本文研究Mazur猜想，通过显式计算条件验证椭圆曲线在$\mathbb{Z}_p$扩张中Mordell-Weil秩的增长。
Leopoldt's Conjecture is a statement about the relationship between the global and local units of a number field. Approximately the conjecture states that the Z_p-rank of the diagonal embedding of the global units into the product of all local units equals the Z-rank of the global units. The variation we consider asks: Can we say anything about the Z_p-rank of the diagonal embedding of the global units into the product of some local units? We use the p-adic Schanuel Conjecture to answer the question in the affirmative and moreover we give a value for the Z_p-rank (of the diagonal embedding of …
Let $F$ be a totally real number field, $p$ a rational prime, and $χ$ a finite order totally odd abelian character of Gal$(\bar{F}/F)$ such that $χ(\mathfrak{p})=1$ for some $\mathfrak{p}|p$. Motivated by a conjecture of Stark, Gross conjectured a relation between the derivative of the $p$-adic $L$-function associated to $χ$ at its exceptional zero and the $\mathfrak{p}$-adic logarithm of a $p$-unit in the $χ$ component of $F_χ^\times$. In a recent work, Dasgupta, Darmon, and Pollack have proven this conjecture assuming two conditions: that Leopoldt's conjecture holds for $F$ and $p$, and that…
Motivated by the work of Candelas, de la Ossa and Rodriguez-Villegas [6], we study the relations between Hasse-Witt matrices and period integrals of Calabi-Yau hypersurfaces in both toric varieties and partial flag varieties. We prove a conjecture by Vlasenko [23] on higher Hasse-Witt matrices for toric hypersurfaces following Katz's method of local expansion [14, 15]. The higher Hasse-Witt matrices also have close relation with period integrals. The proof gives a way to pass from Katz's congruence relations in terms of expansion coefficients [15] to Dwork's congruence relations [8] about peri…
We prove the $p$-curvature conjecture for rank two vector bundles with connection on generic curves, by combining deformation techniques for families of varieties and topological arguments.
The Manin-Peyre conjecture for smooth spherical Fano varieties of semisimple rank one
本文证明了半单秩一光滑球面法诺簇上的Manin-Peyre猜想，涵盖所有T型光滑球面法诺三维簇及部分高维情形。
Using techniques developed in \cite{KLR}, we verify Sarnak's conjecture for two classes of rank-one subshifts with unbounded cutting parameters. The first class of rank-one subshifts we consider are called {\em almost complete congruency classes} ({\em accc}), the definition of which is motivated by the main result of \cite{GS}, which implies that, when a rank-one subshift carries a unique non-atomic invariant probability measure, it is accc if it is measure-theoretically isomorphic to an odometer. The second class we consider consists of Katok's map and its generalizations.
Murmurations and Sato-Tate Conjectures for High Rank Zetas of Elliptic Curves
本研究将有理数域上椭圆曲线的Sato-Tate猜想与murmuration现象从阿贝尔Hasse-Weil Zeta函数推广至其p约化的非阿贝尔高秩Zeta函数，证明二者在该场景下同样成立。
Recently, much attention has been given to various inequalities among partition functions. For example, Nicolas, {and later DeSavlvo--Pak,} proved that $p(n)$ is eventually log-concave, and Ji--Zang showed that the cranks are eventually unimodal. This has led to a flurry of recent activity generalizing such results in different directions. At the same time, Stanton recently made deep conjectures on the positivity of certain polynomials associated to ranks and cranks of partitions, with the ultimate goal of pointing the way to ``deeper'' structure refining ranks and cranks. These have been show…
Residues of Rankin-Selberg Zeta integrals and the split non-tempered Gan-Gross-Prasad conjectures
本文通过 Zeta 积分残差正则化方法，解决了非温控 Gan-Gross-Prasad 猜想及其细化版本在局部域上的问题。
The celebrated result of Eskin, Margulis and Mozes (1998) and Dani and Margulis (1993) on quantitative Oppenheim conjecture says that for irrational quadratic forms $q$ of rank at least 5, the number of integral vectors $\mathbf v$ such that $q(\mathbf v)$ is in a given bounded interval is asymptotically equal to the volume of the set of real vectors $\mathbf v$ such that $q(\mathbf v)$ is in the same interval. In dimension $3$ or $4$, there are exceptional quadratic forms which fail to satisfy the quantitative Oppenheim conjecture. Even in those cases, one can say that almost all quadratic fo…
Following D. Ramakrishnan, we explain how L. Lafforgue's modularity theorem and an analytic theorem of H. Jacquet and J. Shalika can be applied to prove the following result related to the Tate Conjecture: for a smooth, projective, geometrically-connected variety defined over a global function field, the algebraic rank is less than or equal to the analytic rank. Also discussed is the analogous (open) question for number fields and an easy extension of Lafforgue's theorem to remove the "finite-order character" assumption. All results are likely "known to the experts", but don't appear to be wri…
This article deals with the coherence of the model given by the Cohen-Lenstra heuristic philosophy for class groups and also for their generalizations to Tate-Shafarevich groups. More precisely, our first goal is to extend a previous result due to E. Fouvry and J. Klüners which proves that a conjecture provided by the Cohen-Lenstra philosophy implies another such conjecture. As a consequence of our work, we can deduce, for example, a conjecture for the probability laws of $p^j$-ranks of Selmer groups of elliptic curves. This is compatible with some theoretical works and other classical conject…
We produce explicit elliptic curves over \Bbb F_p(t) whose Mordell-Weil groups have arbitrarily large rank. Our method is to prove the conjecture of Birch and Swinnerton-Dyer for these curves (or rather the Tate conjecture for related elliptic surfaces) and then use zeta functions to determine the rank. In contrast to earlier examples of Shafarevitch and Tate, our curves are not isotrivial. Asymptotically these curves have maximal rank for their conductor. Motivated by this fact, we make a conjecture about the growth of ranks of elliptic curves over number fields.
Moments of the partition rank and crank statistics have been studied for their connections to combinatorial objects such as Durfee symbols, as well as for their connections to harmonic Maass forms. This paper proves a conjecture due to Bringmann and Mahlburg that refined a conjecture of Garvan. Garvan's conjecture states that the moments of the crank function are always larger than the moments of the rank function, even though the moments have the same main asymptotic term. The proof uses the Hardy-Ramanujan method to provide precise asymptotic estimates for rank and crank moments and their di…
Murmurations and Sato-Tate Conjectures for High Rank Zetas of Elliptic Curves II: Beyond Riemann Hypothesis
本文提出不依赖黎曼假设的新方法，利用更强渐近界建立椭圆曲线高阶Zeta函数的Sato-Tate定律并表述Murmuration现象。
Quantitative rank distribution conjecture over $\mathbb{F}_q(t)$
本文基于有限域上椭圆曲线的精确计数和无 torsion 性质，结合Goldfeld和Katz-Sarnak的猜想，提出关于椭圆曲线秩分布的量化陈述和更精细的猜想。
In this paper, we will study the arithmetic geometry of rank-2 attractors, which are Calabi-Yau threefolds whose Hodge structures admit interesting splits. We will develop methods to analyze the algebraic de Rham cohomologies of rank-2 attractors, and we will illustrate how our methods work by focusing on an example in a recent paper by Candelas, de la Ossa, Elmi and van Straten. We will look at the interesting connections between rank-2 attractors in string theory and Deligne's conjecture on the special values of $L$-functions. We will also formulate several open questions concerning the pote…
The aim of this paper is to get a complete list of positive definite symmetric matrices with integer entries $\a&b\b&d\$ such that all complex solutions to the system of equations $1-x_1=x_1^ax_2^b\ 1-x_2=x_1^bx_2^d$ are real. This result is related to Nahm's conjecture in rank 2 case.
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