Remark on Tono's theorem about cuspidal curves

Rational Cuspidal Curves

It is the product of my playing with beautiful geometric objects called rational cuspidal curves over the past two years. I would like to thank everyone who has contributed to this thesis. I owe so much to everyone who has ever taught me mathematics. Thank you for inspiring me and for providing me with the skills necessary to complete this thesis. To my friends and fellow students at Abel, than...

Abelian Vortices on Nodal and Cuspidal Curves

We compute the Euler characteristics of the moduli spaces of abelian vortices on curves with nodal and cuspidal singularities. This generalizes our previous work where only nodes were taken into account. The result we obtain is again consistent with the expected reconciliation between the vortex picture of D2-D0 branes and the proposal by Gopakumar and Vafa.

A Remark on a Remark by Macaulay or Enhancing Lazard Structural Theorem

A Remark on Zoloterav’s Theorem

Let n ≥ 3 be an odd integer. For any integer a prime to n, define the permutation γ a,n of {1,. .. , (n − 1)/2} by γ a,n (x) = n − {ax} n if {ax} n ≥ (n + 1)/2, {ax} n if {ax} n ≤ (n − 1)/2, where {x} n denotes the least nonnegative residue of x modulo n. In this note, we show that the sign of γ a,n coincides with the Jacobi symbol a n if n ≡ 1 (mod 4), and 1 if n ≡ 3 (mod 4).

Remark on the rank of elliptic curves

A covariant functor on the elliptic curves with complex multiplication is constructed. The functor takes values in the noncommutative tori with real multiplication. A conjecture on the rank of an elliptic curve is formulated.