Local André-Oort conjecture for the universal abelian variety

We prove a p-adic version of the André-Oort conjecture for subvarieties of the universal abelian varieties. Let g and n be integers with n≥3 and p a prime number not dividing n. Let R be a finite extension of , the ring of Witt vectors of the algebraic closure of the field of p elements. The moduli space of g-dimensional principally polarized abelian varieties with full level n-structure as well as the universal abelian variety over may be defined over R. We call a point R-special if is a canonical lift and ξ is a torsion point of its fibre. Employing the model theory of difference fields and work of Moonen on special subvarieties of , we show that an irreducible subvariety of containing a dense set of R-special points must be a special subvariety in the sense of mixed Shimura varieties.     

An unconditional proof of the Andr���COort conjecture for Hilbert modular surfaces

We give an unconditional proof of the André?COort conjecture for Hilbert modular surfaces asserting that an algebraic curve contained in such a surface and containing an infinite set of special points, is special. The proof relies on a combination of Galois-theoretic techniques and results from the theory of o-minimal structures.     

A proof of Sethares'''' conjecture

Let (?)(z) be holomorphic in the unit disk △ and meromorphic on △. Suppose / is a Teichmuller mapping with complex dilatation In 1968, Sethares conjectured that f is extremal if and only if either (i)(?) has a double pole or (ii)(?) has no pole of order exceeding two on (?)△. The "if" part of the conjecture had been solved by himself. We will give the affirmative answer to the "only if" part of the conjecture. In addition, a more general criterion for extremality of quasiconformal mappings is constructed in this paper, which generalizes the "if" part of Sethares' conjecture and improves the result by Reich and Shapiro in 1990.     

The Lang–Trotter conjecture for products of non-CM elliptic curves

The Ramanujan Journal - Inspired by the work of Lang–Trotter on the densities of primes with fixed Frobenius traces for elliptic curves defined over $${mathbb {Q}}$$ and by the subsequent...     

A proof of snevily’s conjecture

We prove Snevily’s conjecture, which states that for any positive integer k and any two k-element subsets {a ₁, …, a _k } and {b ₁, …, b _k } of a finite abelian group of odd order there exists a permutation π ∈ S _k such that all sums a _i + b _π(i) (i ∈ [1, k]) are pairwise distinct.     

A proof of Frankl’s union-closed sets conjecture for dismantlable lattices

For a lattice L, let Princ(L) denote the ordered set of principal congruences of L. In a pioneering paper, G. Grätzer characterized the ordered set Princ(L) of a finite lattice L; here we do the same for a countable lattice. He also showed that every bounded ordered set H is isomorphic to Princ(L) of a bounded lattice L. We prove a related statement: if an ordered set H with a least element is the union of a chain of principal ideals (equivalently, if 0 \({\in}\) H and H has a cofinal chain), then H is isomorphic to Princ(L) of some lattice L.     

Néron-Tate heights on algebraic curves and subgroups of the modular group

Combining Arakelov theory with Belyis theorem we derive that the values of the Néron-Tate height pairing for divisors on algebraic curves defined over number fields are essentially given by linear combinations of scattering constants associated to finite index subgroups of the modular group PSL₂().Mathematics Subject Classification (2000): 14G40, 11G40, 11G50Supported by the TMR-Network Arithmetic Algebraic Geometry     

A proof of Parisi’s conjecture on the random assignment problem

An assignment problem is the optimization problem of finding, in an m by n matrix of nonnegative real numbers, k entries, no two in the same row or column, such that their sum is minimal. Such an optimization problem is called a random assignment problem if the matrix entries are random variables. We give a formula for the expected value of the optimal k-assignment in a matrix where some of the entries are zero, and all other entries are independent exponentially distributed random variables with mean 1. Thereby we prove the formula 1+1/4+1/9+...+1/k ² conjectured by G. Parisi for the case k=m=n, and the generalized conjecture of D. Coppersmith and G. B. Sorkin for arbitrary k, m and n. AcknowledgementWe thank Mireille Bousquet-Mélou and Gilles Schaeffer for introducing the problem to us. We also thank an anonymous referee for suggesting some improvements of the exposition.     

A short proof of a conjecture of Aougab–Huang

Geometriae Dedicata - In response to Sanki–Vadnere, we present a short proof of the following theorem: a pair of simple curves on a hyperbolic surface whose complementary regions are disks...     

A proof of Terras’ conjecture on the radius of convergence of the Ihara zeta function

We prove the conjecture by Audrey Terras (2011) on the inequalities among the spectral radius ${\rho }_X$ of a finite graph $X$, the radius of convergence $R$ of its Ihara zeta function $Z_X\left(u\right)$, and the average degree $\overline{d_X}$ of $X$. Relating to Terras’ conjecture, we propose a new conjecture between $R$ and certain Rayleigh quotients.     

On the Bloch–Kato conjecture for elliptic modular forms of square-free level

Let $\kappa \ge 6$ be an even integer, $M$ an odd square-free integer, and $f \in S_{2\kappa -2}(\Gamma _0(M))$ a newform. We prove that under some reasonable assumptions that half of the $\lambda $ -part of the Bloch–Kato conjecture for the near central critical value $L(\kappa ,f)$ is true. We do this by bounding the $\ell $ -valuation of the order of the appropriate Bloch–Kato Selmer group below by the $\ell $ -valuation of algebraic part of $L(\kappa ,f)$ . We prove this by constructing a congruence between the Saito–Kurokawa lift of $f$ and a cuspidal Siegel modular form.     

Krahn’s proof of the Rayleigh conjecture revisited

The paper is a discussion of Krahn’s proof of the Rayleigh conjecture that amongst all membranes of the same area and the same physical properties, the circular one has the lowest ground frequency. We show how his approach coincides with the modern techniques of geometric measure theory using the co-area formula. We furthermore discuss some issues and generalisations of his proof.     

On the Lang–Trotter conjecture for two elliptic curves

The Ramanujan Journal - We extend the author’s earlier computation and give coefficient formulas for the (quasimodular) Poincaré square series of weight 3 / 2 and...     

A note on egoryčev's proof of the van der waerden conjecture

Two variants are given of Egory?ev's proof of the van der Waerden permanent conjecture.     

A new proof of Faber?s intersection number conjecture

We give a new proof of Faber?s intersection number conjecture concerning the top intersections in the tautological ring of the moduli space of curves Mg. The proof is based on a very straightforward geometric and combinatorial computation with double ramification cycles.