1. Slice the ginger and jalapeños. In a saucepan over medium heat, bring the sugar and 1 cup water to a boil, stirring to dissolve the sugar, about 5 minutes total. Remove from heat and add ginger and jalapeños; let cool at room temperature, about 1 hour.
2. Juice 2 limes; cut the third into wedges. In a cocktail shaker or bowl, combine the juice, lime wedges and mint leaves. Remove the ginger from the sugar syrup. Using a garlic press, squeeze the ginger juice over the lime and mint. Using a wooden spoon, lightly crush and mix the ingredients to release the flavors. Divide the ginger-lime-mint mixture among 4 tall glasses.
3. Strain the remaining syrup through a wire-mesh sieve and add 1 tbsp ginger-jalapeño syrup to each glass. Divide rum and sparkling water among glasses, fill with ice cubes and serve.
Per serving: 352 calories, 0g fat, 0g saturated fat, 0mg cholesterol, 19mg sodium, 56g carbohydrate, 2g fiber, 51g sugar, 1g protein