# On the Euler characteristics of signed Selmer groups

## TL;DR

This paper computes the Euler characteristics of signed Selmer groups for elliptic curves over cyclotomic extensions, accommodating mixed reduction types and signs, advancing understanding in Iwasawa theory.

## Contribution

It introduces a method to compute Euler characteristics of signed Selmer groups for elliptic curves with mixed reduction and sign conditions, broadening previous results.

## Key findings

- Euler characteristics computed for signed Selmer groups
- Handles elliptic curves with mixed reduction types
- Allows mixed signs in Selmer group definitions

## Abstract

Let $p$ be an odd prime number, and $E$ an elliptic curve defined over a number field with good reduction at every prime of $F$ above $p$. In this short note, we compute the Euler characteristics of the signed Selmer groups of $E$ over the cyclotomic $\Zp$-extension. The novelty of our result is that we allow the elliptic curve to have mixed reduction types for primes above $p$ and that we allow mixed signs in the definition of the signed Selmer groups.