##### Department of Mathematics,

University of California San Diego

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### Math 209 - Number Theory

## Everett Howe

#### CCR

## Forbidden Frobenius: Using the Brauer relations to show a curve doesn't exist

##### Abstract:

To every genus-2 curve C over a finite field $F_q$, one can associate the characteristic polynomials of the Frobenius endomorphism acting on the Jacobian of C. This polynomial --- also known as the *Weil polynomial* of C --- is of the form $$x^4 + a*x^3 + b*x^2 + a*q*x + q^2$$ where a and b are integers. We will use the Brauer relations, applied to a certain biquadratic number field, to show that no curve over $F_q$ gives rise to the Weil polynomial with $a = 0$ and $b = 2 - 2*q$. The same method can be used to show that the Weil polynomials with $a = 0$ and with other values of b (subject to certain elementary restrictions) *do* occur; this was carried out by Daniel Maisner. These results, combined with earlier work, allow us to easily determine the Weil polynomials that arise from genus-2 curves with ordinary Jacobians.

Host: Audrey Terras

### May 5, 2005

### 3:00 PM

### AP&M 7321

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