arXiv subjects
Publications and source records attributed to Brooke Feigon.
We introduce the subsum polynomial of a partition $\lambda=(\lambda_1, \lambda_2, \ldots, \lambda_k)$ defined by $\mathrm{sp}(\lambda, x)=\prod_{i=1}^k(1+x^{\lambda_i})$. We study the sum of reciprocals of $\mathrm{sp}(\lambda, x)$ over all partitions of $n$. We prove arithmetic properties of related polynomials and offer connections to other combinatorial objects.
The work of Andrews and Merca on the truncated Euler's pentagonal number theorem led to a resurgence in research on truncated theta series identities. In particular, Yee proved a truncated version of the Jacobi Triple Product (JTP) identity. Recently, Merca conjectured a stronger form of the truncated JTP identity. In this article we prove the first three cases of the conjecture and several related truncated identities. We prove combinatorially an identity related to the JTP identity which in particular cases reduces to identities conjectured by Merca and proved analytically by Krattenthaler, Merca and Radu. Moreover, we introduce a new combinatorial interpretation for the number of distinct 5-regular partitions of n.
In their seminal paper, Lubotzky, Phillips and Sarnak (LPS) defined the notion of regular Ramanujan graphs and gave an explicit construction of infinite families of $(p+1)$-regular Ramanujan Cayley graphs, for infinitely many primes $p$. In this paper we extend the work of LPS and its successors to bigraphs (biregular bipartite graphs), in several aspects: we investigate the combinatorial properties of various generalizations of the notion of Ramanujan graphs, define a notion of Cayley bigraphs, and give explicit constructions of infinite families of $(p^3+1,p+1)$-regular Ramanujan Cayley bigraphs, for infinitely many $p$. Both the LPS graphs and our ones are arithmetic, arising as quotients of Bruhat-Tits trees by congruence subgroups of arithmetic lattices in a $p$-adic group, $PGL_2(\mathbb{Q}_p)$ for LPS and $PU_3(\mathbb{Q}_p)$ for us. In both cases the Ramanujan property relates to the Ramanujan Conjecture (RC) on the respective groups. But while for $PGL_2$ the RC holds unconditionally, this is not so in the case of $PU_3$. We find explicit cases where the RC does and does not hold, and use this to construct arithmetic non-Ramanujan Cayley bigraphs as well, and prove that nevertheless they satisfy the Sarnak-Xue density hypothesis. On the combinatorial side, we present a pseudorandomness characterization of Ramanujan bigraphs, and a more general notion of biexpanders. We show that our Ramanujan bigraphs exhibit the cutoff phenomenon with bounded window size for non-backtracking random walks, either as a consequence of the Ramanujan property, or of the Sarnak-Xue density hypothesis. Finally, we present some other applications of our work: golden gates for $PU_3$, Ramanujan and non-Ramanujan complexes of type $\widetilde{A}_2$, optimal strong approximation for $p$-arithmetic subgroups of $PSU_3$ and vanishing of first Betti numbers of Picard modular surfaces.
In this paper we study the security of a proposal for Post-Quantum Cryptography from both a number theoretic and cryptographic perspective. Charles-Goren-Lauter in 2006 [CGL06] proposed two hash functions based on the hardness of finding paths in Ramanujan graphs. One is based on Lubotzky-Phillips-Sarnak (LPS) graphs and the other one is based on Supersingular Isogeny Graphs. A 2008 paper by Petit-Lauter-Quisquater breaks the hash function based on LPS graphs. On the Supersingular Isogeny Graphs proposal, recent work has continued to build cryptographic applications on the hardness of finding isogenies between supersingular elliptic curves. A 2011 paper by De Feo-Jao-Plût proposed a cryptographic system based on Supersingular Isogeny Diffie-Hellman as well as a set of five hard problems. In this paper we show that the security of the SIDH proposal relies on the hardness of the SIG path-finding problem introduced in [CGL06]. In addition, similarities between the number theoretic ingredients in the LPS and Pizer constructions suggest that the hardness of the path-finding problem in the two graphs may be linked. By viewing both graphs from a number theoretic perspective, we identify the similarities and differences between the Pizer and LPS graphs.
We study fluctuations in the number of points of $\ell$-cyclic covers of the projective line over the finite field $\mathbb{F}_q$ when $q \equiv 1 \mod \ell$ is fixed and the genus tends to infinity. The distribution is given as a sum of $q+1$ i.i.d. random variables. This was settled for hyperelliptic curves by Kurlberg and Rudnick, while statistics were obtained for certain components of the moduli space of $\ell$-cyclic covers by Bucur, David, Feigon and Lalín. In this paper, we obtain statistics for the distribution of the number of points as the covers vary over the full moduli space of $\ell$-cyclic covers of genus $g$. This is achieved by relating $\ell$-covers to cyclic function field extensions, and counting such extensions with prescribed ramification and splitting conditions at a finite number of primes.
We construct explicitly an infinite family of Ramanujan graphs which are bipartite and biregular. Our construction starts with the Bruhat-Tits building of an inner form of $SU_3(\mathbb Q_p)$. To make the graphs finite, we take successive quotients by infinitely many discrete co-compact subgroups of decreasing size.
Let $\pi$ be a cuspidal automorphic representation of PGL($2n$) over a number field $F$, and $\eta$ the quadratic idele class character attached to a quadratic extension $E/F$. Guo and Jacquet conjectured a relation between the nonvanishing of $L(1/2,\pi)L(1/2, \pi \otimes \eta)$ for $\pi$ of symplectic type and the nonvanishing of certain GL($n,E$) periods. When $n=1$, this specializes to a well-known result of Waldspurger. We prove this conjecture, and related global results, under some local hypotheses using a simple relative trace formula. We then apply these global results to obtain local results on distinguished supercuspidal representations, which partially establish a conjecture of Prasad and Takloo-Bighash.
We study the distribution of the number of points and of the zeroes of the zeta function in different $p$-rank strata of Artin-Schreier covers over $\F_q$ when $q$ is fixed and the genus goes to infinity. The $p$-rank strata considered include the ordinary family, the whole family, and the family of curves with $p$-rank equal to $p-1.$ While the zeta zeroes always approach the standard Gaussian distribution, the number of points over $\F_q$ has a distribution that varies with the specific family.
We study the distribution of the zeroes of the zeta functions of the family of Artin-Schreier covers of the projective line over $\mathbb{F}_q$ when $q$ is fixed and the genus goes to infinity. We consider both the global and the mesoscopic regimes, proving that when the genus goes to infinity, the number of zeroes with angles in a prescribed non-trivial subinterval of $[-π,π)$ has a standard Gaussian distribution (when properly normalized).
We obtain exact formulas for central values of triple product L-functions averaged over newforms of weight 2 and prime level. We apply these formulas to non-vanishing problems. This paper uses a period formula for the triple product L-function proved by Gross and Kudla.
In this note, we study the fluctuations in the number of points of smooth projective plane curves over finite fields $\mathbb{F}_q$ as $q$ is fixed and the genus varies. More precisely, we show that these fluctuations are predicted by a natural probabilistic model, in which the points of the projective plane impose independent conditions on the curve. The main tool we use is a geometric sieving process introduced by Poonen.
We study the variation of the trace of the Frobenius endomorphism associated to a cyclic trigonal curve of genus g over a field of q elements as the curve varies in an irreducible component of the moduli space. We show that for q fixed and g increasing, the limiting distribution of the trace of the Frobenius equals the sum of q+1 independent random variables taking the value 0 with probability 2/(q+2) and 1, e^{(2pi i)/3}, e^{(4pi i)/3} each with probability q/(3(q+2)). This extends the work of Kurlberg and Rudnick who considered the same limit for hyperelliptic curves. We also show that when both g and q go to infinity, the normalized trace has a standard complex Gaussian distribution and how to generalize these results to p-fold covers of the projective line.
We use the relative trace formula to obtain exact formulas for central values of certain twisted quadratic base change L-functions averaged over Hilbert modular forms of a fixed weight and level. We apply these formulas to the subconvexity problem for these L-functions. We also establish an equidistribution result for the Hecke eigenvalues weighted by these L-values.