On the Ranks of the 2-Selmer Groups of Twists of a Given Elliptic Curve
We extend work of Swinnerton-Dyer on the density of the number of twists of a given elliptic curve that have 2-Selmer group of a particular rank.
math.NT↗
arXiv subjects
Daniel M. Kane
Publications and source records attributed to Daniel M. Kane.
Modeling the distribution of ranks, Selmer groups, and Shafarevich-Tate groups of elliptic curves
Using maximal isotropic submodules in a quadratic module over Z_p, we prove the existence of a natural discrete probability distribution on the set of isomorphism classes of short exact sequences of co-finite type Z_p-modules, and then conjecture that as E varies over elliptic curves over a fixed global field k, the distribution of 0 --> E(k) tensor Q_p/Z_p --> Sel_{p^infty} E --> Sha[p^infty] --> 0 is that one. We show that this single conjecture would explain many of the known theorems and conjectures on ranks, Selmer groups, and Shafarevich-Tate groups of elliptic curves. We also prove the existence of a discrete probability distribution of the set of isomorphism classes of finite abelian p-groups equipped with a nondegenerate alternating pairing, defined in terms of the cokernel of a random alternating matrix over Z_p, and we prove that the two probability distributions are compatible with each other and with Delaunay's predicted distribution for Sha. Finally, we prove new theorems on the fppf cohomology of elliptic curves in order to give further evidence for our conjecture.
math.NT↗
A Low-Depth Monotone Function that is not an Approximate Junta
We provide an example of a monotone Boolean function on the hypercube given by a low depth decision tree that is not well approximated by any k-junta for small k.
cs.DM↗
Closed expressions for averages of set partition statistics
In studying the enumerative theory of super characters' of the group of upper triangular matrices over a finite field we found that the moments (mean, variance and higher moments) of novel statistics on set partitions have simple closed expressions as linear combinations of shifted bell numbers. It is shown here that families of other statistics have similar moments. The coefficients in the linear combinations are polynomials in $n$. This allows exact enumeration of the moments for small $n$ to determine exact formulae for all $n$.
math.CO↗
An Asymptotic for the Number of Solutions to Linear Equations in Prime Numbers from Specified Chebotarev Classes
We extend known results on the number of solutions to a linear equation in at least three prime numbers when the primes involved are required to lie in specified Chebotarev classes. We prove asymptotic results similar to previous ones only now taking into account corrections coming form the Chebotarev Density Theorem and Global Class Field Theory. We then apply these results to find elliptic curves whose discriminants split completely of a given number field.
math.NT↗
A Pseudorandom Generator for Polynomial Threshold Functions of Gaussian with Subpolynomial Seed Length
We develop a pseudorandom generator that fools degree-$d$ polynomial threshold functions in $n$ variables with respect to the Gaussian distribution and has seed length $O_{c,d}(\log(n) ε^{-c})$.
cs.CC↗
The Correct Exponent for the Gotsman-Linial Conjecture
We prove a new bound on the average sensitivity of polynomial threshold functions. In particular we show that a polynomial threshold function of degree $d$ in at most $n$ variables has average sensitivity at most $\sqrt{n}(\log(n))^{O(d\log(d))}2^{O(d^2\log(d)}$. For fixed $d$ the exponent in terms of $n$ in this bound is known to be optimal. This bound makes significant progress towards the Gotsman-Linial Conjecture which would put the correct bound at $Θ(d\sqrt{n})$.
math.CO↗
A Structure Theorem for Poorly Anticoncentrated Gaussian Chaoses and Applications to the Study of Polynomial Threshold Functions
We prove a structural result for degree-$d$ polynomials. In particular, we show that any degree-$d$ polynomial, $p$ can be approximated by another polynomial, $p_0$, which can be decomposed as some function of polynomials $q_1,...,q_m$ with $q_i$ normalized and $m=O_d(1)$, so that if $X$ is a Gaussian random variable, the probability distribution on $(q_1(X),...,q_m(X))$ does not have too much mass in any small box. Using this result, we prove improved versions of a number of results about polynomial threshold functions, including producing better pseudorandom generators, obtaining a better invariance principle, and proving improved bounds on noise sensitivity.
math.PR↗
Small Designs for Path Connected Spaces and Path Connected Homogeneous Spaces
We prove the existence of designs of small size in a number of contexts. In particular our techniques can be applied to prove the existence of $n$-designs on $S^{d}$ of size $O_d(n^{d}\log(n)^{d-1})$.
math.CO↗
A proof of Andrews' conjecture on Partitions with no short sequences
Holroyd, Liggett, and Romik introduced the following probability model. Let $C_1, C_2,...$ be independent events with probabilities $¶_s(C_n)= 1-e^{-ns}$ under a probability measure $¶_s$ with $0<s<1$. Let $A_k$ be the event that there is no sequence of $k$ consecutive $C_i$ that do not occur. We given an asymptotic for $¶_s(A_k)$ with a relative error term that goes to 0 as $s\to 0$. This establishes a conjecture of Andrews.
math.NT↗
$k$-Independent Gaussians Fool Polynomial Threshold Functions
We show that any $O_d(ε^{-4d 7^d})$-independent family of Gaussians $ε$-fools any degree-$d$ polynomial threshold function.
cs.CC↗
A Small PRG for Polynomial Threshold Functions of Gaussians
We develop a pseudo-random generator to fool degree-$d$ polynomial threshold functions with respect to the Gaussian distribution. For $c>0$ any constant, we construct a pseudo-random generator that fools such functions to within $ε$ and has seed length $\log(n) 2^{O(d)} ε^{-4-c}$.
cs.CC↗
Minimal S-universality criteria may vary in size
In this note, we give simple examples of sets S of quadratic forms that have minimal S-universality criteria of multiple cardinalities. This answers a question of Kim, Kim, and Oh in the negative.
math.NT↗
Canonical Projective Embeddings of the Deligne-Lusztig Curves Associated to $2A2$, $2B2$ and $2G2$
The Deligne-Lusztig varieties associated to the Coxeter classes of the algebraic groups 2A2, 2B2 and 2G2 are affine algebraic curves. We produce explicit projective models of the closures of these curves. Furthermore for $d$ the Coxeter number of these groups, we find polynomials for each of these models that cut out the $\F_q$-points, the $\F_{q^d}$-points and the $\F_{q^{d+1}}$-points, and demonstrate a relation satisfied by these polynomials.
math.NT↗
A Derandomized Sparse Johnson-Lindenstrauss Transform
Recent work of [Dasgupta-Kumar-Sarlos, STOC 2010] gave a sparse Johnson-Lindenstrauss transform and left as a main open question whether their construction could be efficiently derandomized. We answer their question affirmatively by giving an alternative proof of their result requiring only bounded independence hash functions. Furthermore, the sparsity bound obtained in our proof is improved. The main ingredient in our proof is a spectral moment bound for quadratic forms that was recently used in [Diakonikolas-Kane-Nelson, FOCS 2010].
cs.DS↗
Fast Moment Estimation in Data Streams in Optimal Space
We give a space-optimal algorithm with update time O(log^2(1/eps)loglog(1/eps)) for (1+eps)-approximating the pth frequency moment, 0 < p < 2, of a length-n vector updated in a data stream. This provides a nearly exponential improvement in the update time complexity over the previous space-optimal algorithm of [Kane-Nelson-Woodruff, SODA 2010], which had update time Omega(1/eps^2).
cs.DS↗
Quantum interpolation of polynomials
We consider quantum interpolation of polynomials. We imagine a quantum computer with black-box access to input/output pairs (x_i, f(x_i)), where f is a degree-d polynomial, and we wish to compute f(0). We give asymptotically tight quantum lower bounds for this problem, even in the case where 0 is among the possible values of x_i.
quant-ph↗
Bounded Independence Fools Degree-2 Threshold Functions
Let x be a random vector coming from any k-wise independent distribution over {-1,1}^n. For an n-variate degree-2 polynomial p, we prove that E[sgn(p(x))] is determined up to an additive epsilon for k = poly(1/epsilon). This answers an open question of Diakonikolas et al. (FOCS 2009). Using standard constructions of k-wise independent distributions, we obtain a broad class of explicit generators that epsilon-fool the class of degree-2 threshold functions with seed length log(n)*poly(1/epsilon). Our approach is quite robust: it easily extends to yield that the intersection of any constant number of degree-2 threshold functions is epsilon-fooled by poly(1/epsilon)-wise independence. Our results also hold if the entries of x are k-wise independent standard normals, implying for example that bounded independence derandomizes the Goemans-Williamson hyperplane rounding scheme. To achieve our results, we introduce a technique we dub multivariate FT-mollification, a generalization of the univariate form introduced by Kane et al. (SODA 2010) in the context of streaming algorithms. Along the way we prove a generalized hypercontractive inequality for quadratic forms which takes the operator norm of the associated matrix into account. These techniques may be of independent interest.
cs.CC↗