After formulating Conjecture A for p-adic L-functions defined over ordinary Hilbert modular Hida deformations on a totally real field F of degree d, we construct two p-adic L-functions of d+1-variable depending on the parity of weight as a partial result on Conjecture A. We will also state Conjecture B which is a corollary of Conjecture A but is important by itself. Main issues of the construct...

The parity type of a Latin square is defined in terms of the numbers of even and odd rows and columns. It is related to an Alon-Tarsi-like conjecture that applies to Latin squares of odd order. Parity types are used to derive upper bounds for the size of autotopy groups. A new algorithm for finding the autotopy group of a Latin square, based on the cycle decomposition of its rows, is presented,...

In this lecture, we will talk about circuit lower bound for constant-depth circuit with MODp gates. Using switching lemma, we can prove exponential size lower bound for constant-depth circuit computing parity and majority. What if parity (=MOD2 gates) are allowed? It was conjectured that majority still needs exponential size to compute in constant-depth circuit. Razborov (1987) solves this conj...

Global root numbers have played an important role in the study of rational points on abelian varieties since the discovery of the conjecture of Birch and Swinnerton-Dyer. The aim of this paper is to throw some new light on this intriguing and still largely conjectural relationship. The simplest avatar of this phenomenon is the parity conjecture which asserts that for an abelian variety A over a...

We study the condensation of localized tachyon in non-supersymmetric orbifold. We first show that the G-parity of chiral primaries are preserved under the condensation of localized tachyon(CLT). Using this, we finalize the proof of the conjecture that the lowest-tachyon-mass-squared increases under CLT at the level of type II string with full consideration of GSO projection. We also show the eq...

Let p(n) denote the ordinary partition function. In 1966, Subbarao [18] conjectured that in every arithmetic progression r (mod t) there are infinitely many integers N (resp. M) ≡ r (mod t) for which p(N) is even (resp. odd). We prove Subbarao’s conjecture for all moduli t of the form m · 2 where m ∈ {1, 5, 7, 17}. To obtain this theorem we make use of recent results of Ono and Taguchi [14] on ...

For each of the torsion groups Z/2Z ⊕ Z/10Z, Z/2Z ⊕ Z/12Z,Z/15Z we find the quadratic field with the smallest absolute value of its discriminant such that there exists an elliptic curve with that torsion and positive rank. For the torsion groups Z/11Z, Z/14Z we solve the analogous problem after assuming the Parity conjecture.

I conjecture on the phase structure expected for lattice gauge theory with two avors of Wilson fermions, concentrating on large values of the hopping parameter. Numerous phases are expected, including the conventional connnement and deconnnement phases, as well as an Aoki phase with spontaneous breaking of avor and parity and a large hopping phase corresponding to negative quark masses.

A formula for Glynn’s hyperdeterminant detp (p prime) of a square matrix shows that the number of ways to decompose any integral doubly stochastic matrix with row and column sums p− 1 into p− 1 permutation matrices with even product, minus the number of ways with odd product, is 1 (mod p). It follows that the number of even Latin squares of order p− 1 is not equal to the number of odd Latin squ...

In order to prove the Strong Perfect Graph Conjecture, the existence of a ”simple” property P holding for any minimal non-quasi-parity Berge graph G would really reduce the difficulty of the problem. We prove here that this property cannot be of type ”G is F-free”, where F is any fixed family of Berge graphs.