seminars:arit
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| * **March 3** \\ **//Speaker//**: Connor Stewart (CUNY) \\ **//Title//**: Conductor–Discriminant Inequality for Tamely Ramified Cyclic Covers \\ **//Abstract//**: We consider $\mathbb{Z}/n$-covers $X\to\mathbb{P}^1$ defined over discretely valued fields $K$ with excellent valuation ring $\mathcal{O}_K$ and perfect residue field of characteristic not dividing $n$. Two standard measures of bad reduction for such a curve $X$ are the Artin conductor of its minimal regular model over $\mathcal{O}_K$ and the valuation of the discriminant of a Weierstrass equation for $X$. We prove an inequality relating these two measures. Specifically, if $X$ is given by an affine equation $y^n = f(x)$ with $f(x) \in \mathcal{O}_K[x]$, and if $\mathcal{X}$ is its minimal regular model over $\mathcal{O}_K$, then the negative of the Artin conductor of $\mathcal{X}$ is bounded above by $(n-1)v_K(\textrm{disc (rad}\ f))$. This extends previous work of Ogg, Saito, Liu, Srinivisan, and Obus-Srinivasan on elliptic and hyperelliptic curves. (Joint work with Andrew Obus and Padmavathi Srinivasan.) \\ | * **March 3** \\ **//Speaker//**: Connor Stewart (CUNY) \\ **//Title//**: Conductor–Discriminant Inequality for Tamely Ramified Cyclic Covers \\ **//Abstract//**: We consider $\mathbb{Z}/n$-covers $X\to\mathbb{P}^1$ defined over discretely valued fields $K$ with excellent valuation ring $\mathcal{O}_K$ and perfect residue field of characteristic not dividing $n$. Two standard measures of bad reduction for such a curve $X$ are the Artin conductor of its minimal regular model over $\mathcal{O}_K$ and the valuation of the discriminant of a Weierstrass equation for $X$. We prove an inequality relating these two measures. Specifically, if $X$ is given by an affine equation $y^n = f(x)$ with $f(x) \in \mathcal{O}_K[x]$, and if $\mathcal{X}$ is its minimal regular model over $\mathcal{O}_K$, then the negative of the Artin conductor of $\mathcal{X}$ is bounded above by $(n-1)v_K(\textrm{disc (rad}\ f))$. This extends previous work of Ogg, Saito, Liu, Srinivisan, and Obus-Srinivasan on elliptic and hyperelliptic curves. (Joint work with Andrew Obus and Padmavathi Srinivasan.) \\ | ||
| - | * **March 10** \\ **//Speaker//**: Eric Yin (Binghamton) \\ **//Title//**: TBA \\ **//Abstract//**: TBA \\ | + | * **March 10** \\ **//Speaker//**: Eric Yin (Binghamton) \\ **//Title//**: Generating abelian extensions with elliptic curves with complex multiplication. \\ **//Abstract//**: An elliptic curve with complex multiplication is one with extra endomorphisms, ones that are not simply given by multiplication-by-m maps. In this talk we discuss how this extra structure allows us to find an analogy to Kronecker-Weber, generating abelian extensions of imaginary quadratic fields through torsion points on elliptic curves with CM. In addition, we discuss the role the ideal class group plays in both measuring ramification and classifying elliptic curves with a given endomorphism ring. This lets us describe both the Hilbert class field, the maximal unramified abelian extension, and the maximal abelian extension of any imaginary quadratic field.\\ |
| * **March 17** \\ **//Speaker//**: Anitha Srinivasan (Comillas University, Madrid), [[https://binghamton.zoom.us/j/92745369515?pwd=gg9R8gOQrFpFOwe4T3c6nUbUcNrLPq.12|by Zoom]] \\ **//Title//**: The generalized Markoff equation \\ **//Abstract//**: The talk will look at various aspects of the generalized Markoff equation $a^2+b^2+c^2=3abc+m$ ($m\ge 0$), giving an overview of all the exciting work in the area. A few examples of topics that will be mentioned are: the classification of solution triples $(a, b, c)$ that come from $k$-Fibonacci sequences, open conjectures (which $m's$ have no solutions?), counting algorithms for the number of solutions (trees) and the Markoff equation mod $p$. \\ | * **March 17** \\ **//Speaker//**: Anitha Srinivasan (Comillas University, Madrid), [[https://binghamton.zoom.us/j/92745369515?pwd=gg9R8gOQrFpFOwe4T3c6nUbUcNrLPq.12|by Zoom]] \\ **//Title//**: The generalized Markoff equation \\ **//Abstract//**: The talk will look at various aspects of the generalized Markoff equation $a^2+b^2+c^2=3abc+m$ ($m\ge 0$), giving an overview of all the exciting work in the area. A few examples of topics that will be mentioned are: the classification of solution triples $(a, b, c)$ that come from $k$-Fibonacci sequences, open conjectures (which $m's$ have no solutions?), counting algorithms for the number of solutions (trees) and the Markoff equation mod $p$. \\ | ||
seminars/arit.1771764638.txt · Last modified: 2026/02/22 07:50 by borisov